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1.0.0 | problem:theoremdb-p8732 | problem | theoremdb-p8732 | Existence of Large Simultaneous Double Product Configurations in Abelian Groups | {
"main": "math",
"sub": "combinatorics",
"slug": "combinatorics",
"name": "Combinatorics",
"original_discipline": "mathematics"
} | [] | [
{
"code": "05",
"basis": "community-map",
"evidence": "Inferred from community field: combinatorics"
}
] | For every real number $\varepsilon > 0$, do there exist arbitrarily large positive integers $n$, a finite abelian group $H$, and families of finite subsets $A_1, \dots, A_n \subseteq H$ and $B_1, \dots, B_n \subseteq H$ such that all of the following hold?
\begin{itemize}
\item $n^{2-\varepsilon} \leq |H| \leq n^{2+\va... | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P8732",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P8732",
"citation": null,
"doi": null,
"arxiv_id": null,
"license": "Selected packets label their content CC0-1.0; original references retain their own terms",
"notes": "Selected problem... | theorem green_36 :
answer(sorry) ↔
∀ ε > (0 : ℝ), ∃ᶠ n in atTop,
∃ (H : Type) (_ : AddCommGroup H) (_ : Finite H) (A B : Fin n → Finset H),
(n : ℝ) ^ (2 - ε) ≤ Nat.card H ∧ Nat.card H ≤ (n : ℝ) ^ (2 + ε) ∧
(∀ i, (n : ℝ) ^ (2 - ε) ≤ (A i).card * (B i).card) ∧
Green36Proper... | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem green_36 :\n answer(sorry) ↔\n ∀ ε > (0 : ℝ), ∃ᶠ n in atTop,\n ∃ (H : Type) (_ : AddCommGroup H) (_ : Finite H) (A B : Fin n → Finset H),\n (n : ℝ) ^ (2 - ε) ≤ Nat.card H ∧ Nat.card H ≤ (n : ℝ) ^ (2 + ε) ∧\n (∀ i, (n : ℝ) ^ (2 - ε) ≤ ... | [] | TheoremDB P8732 · Candidate awaiting source review | null | Source-reported assessment, retrieved 8 September 2026. Frontier Science has not independently reviewed this status.
Open. No signed Lean verification is attached to this problem or its selected resolution.
This record is in the source backlog; its novelty and unresolved status have not passed source review. | null | published | incremental | hard | [
"source-candidate",
"theoremdb"
] | null | null | null | null | [
{
"kind": "website",
"citation": "TheoremDB P8732",
"url": "https://theoremdb.org/statements/P8732",
"note": null,
"sort_order": 0
},
{
"kind": "formalization",
"citation": "Source formal statement (not locally verified)",
"url": "https://theoremdb.org/statements/P8732",
"not... | {
"source_record": "P8732",
"imported_at": "2026-09-08",
"status_as_of": "2026-09-08",
"evidence_label": "Source candidate · not reviewed",
"note": "Full public directory statement. Source lane: backlog; source resolution: open. Source verification claims are attributed only; no local Lean verification has be... | {"created_at":"2026-08-17T19:11:40+00:00","display_standard":{"editorial_review":"submitter_supplied","formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":8732,"public_status":{"evidence":{"hasFormalization":false,"hasInformalProof":false,"hasRecordedWork":false,"solutionRecord":null},"lean":{... | [
"formalization-contains-placeholder",
"formalization-not-compiled-during-export",
"current-status-unassessed",
"source-full-text-not-present"
] | [] | {
"created_at": "2026-09-08T03:32:15.935Z",
"updated_at": "2026-09-08T03:32:15.935Z",
"published_at": "2026-09-08T03:32:58.774Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"8e002057-58c3-4c83-bbd2-26b548f60d73","format_issues":null,"number":5252} |
1.0.0 | problem:theoremdb-p8734 | problem | theoremdb-p8734 | Non-existence of monochromatic quantum graphs with 10 vertices and 3 colours | {
"main": "math",
"sub": "combinatorics",
"slug": "combinatorics",
"name": "Combinatorics",
"original_discipline": "mathematics"
} | [] | [
{
"code": "05",
"basis": "community-map",
"evidence": "Inferred from community field: combinatorics"
}
] | Let $N = 10$ and $D = 3$. A weight function on the edge-coloured complete graph $K_{10}$ with colours from $\{1, 2, 3\}$ is a function $W$ that assigns a complex number to every edge together with a colour assignment. For each vertex colouring $\iota : \{1, \dots, 10\} \to \{1, 2, 3\}$, define the perfect-matching sum ... | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P8734",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P8734",
"citation": null,
"doi": null,
"arxiv_id": null,
"license": "Selected packets label their content CC0-1.0; original references retain their own terms",
"notes": "Selected problem... | theorem eqSystem10_no_solution_d3 :
answer(sorry) ↔
¬ ∃ W : WeightsN 10 3 ℂ, EqSystemN 10 3 W | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem eqSystem10_no_solution_d3 :\n answer(sorry) ↔\n ¬ ∃ W : WeightsN 10 3 ℂ, EqSystemN 10 3 W",
"target": null,
"source_url": "https://theoremdb.org/statements/P8734",
"format": "declaration-or-fragment",
"sha256": "3f0a863ae1957d435ff5fbc43e45e65... | [] | TheoremDB P8734 · Candidate awaiting source review | null | Source-reported assessment, retrieved 8 September 2026. Frontier Science has not independently reviewed this status.
Open. No signed Lean verification is attached to this problem or its selected resolution.
This record is in the source backlog; its novelty and unresolved status have not passed source review. | null | published | incremental | hard | [
"source-candidate",
"theoremdb"
] | null | null | null | null | [
{
"kind": "website",
"citation": "TheoremDB P8734",
"url": "https://theoremdb.org/statements/P8734",
"note": null,
"sort_order": 0
},
{
"kind": "formalization",
"citation": "Source formal statement (not locally verified)",
"url": "https://theoremdb.org/statements/P8734",
"not... | {
"source_record": "P8734",
"imported_at": "2026-09-08",
"status_as_of": "2026-09-08",
"evidence_label": "Source candidate · not reviewed",
"note": "Full public directory statement. Source lane: backlog; source resolution: open. Source verification claims are attributed only; no local Lean verification has be... | {"created_at":"2026-08-17T19:11:48+00:00","display_standard":{"editorial_review":"submitter_supplied","formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":8734,"public_status":{"evidence":{"hasFormalization":false,"hasInformalProof":false,"hasRecordedWork":false,"solutionRecord":null},"lean":{... | [
"formalization-contains-placeholder",
"formalization-not-compiled-during-export",
"current-status-unassessed",
"source-full-text-not-present"
] | [] | {
"created_at": "2026-09-08T03:32:15.935Z",
"updated_at": "2026-09-08T03:32:15.935Z",
"published_at": "2026-09-08T03:33:04.295Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"bb3f76d6-0fe4-4cdf-806c-59a92733628c","format_issues":null,"number":6281} |
1.0.0 | problem:theoremdb-p8736 | problem | theoremdb-p8736 | Covering Cyclic Groups by Random Translates | {
"main": "math",
"sub": "combinatorics",
"slug": "combinatorics",
"name": "Combinatorics",
"original_discipline": "mathematics"
} | [] | [
{
"code": "05",
"basis": "community-map",
"evidence": "Inferred from community field: combinatorics"
}
] | For a prime $p$, let $k = \lfloor \sqrt{p} \rfloor$ and $c = 100k$. For a subset $A \subseteq \mathbb{Z}/p\mathbb{Z}$ with $|A| = k$, say that $A$ is coverable with $c$ translates if there exists a set $T \subseteq \mathbb{Z}/p\mathbb{Z}$ with $|T| \leq c$ such that $A + T = \mathbb{Z}/p\mathbb{Z}$, where $A + T = \{a ... | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P8736",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P8736",
"citation": null,
"doi": null,
"arxiv_id": null,
"license": "Selected packets label their content CC0-1.0; original references retain their own terms",
"notes": "Selected problem... | theorem green_39 : answer(sorry) ↔
Tendsto
(fun p : {q : ℕ // q.Prime} ↦
let k := Nat.sqrt p
let c := 100 * k
(proportionCoverable p k c : ℝ))
atTop (𝓝 1) | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem green_39 : answer(sorry) ↔\n Tendsto\n (fun p : {q : ℕ // q.Prime} ↦\n let k := Nat.sqrt p\n let c := 100 * k\n (proportionCoverable p k c : ℝ))\n atTop (𝓝 1)",
"target": null,
"source_url": "https://theoremdb.org/statements... | [] | TheoremDB P8736 · Candidate awaiting source review | null | Source-reported assessment, retrieved 8 September 2026. Frontier Science has not independently reviewed this status.
Open. No signed Lean verification is attached to this problem or its selected resolution.
This record is in the source backlog; its novelty and unresolved status have not passed source review. | null | published | incremental | hard | [
"source-candidate",
"theoremdb"
] | null | null | null | null | [
{
"kind": "website",
"citation": "TheoremDB P8736",
"url": "https://theoremdb.org/statements/P8736",
"note": null,
"sort_order": 0
},
{
"kind": "formalization",
"citation": "Source formal statement (not locally verified)",
"url": "https://theoremdb.org/statements/P8736",
"not... | {
"source_record": "P8736",
"imported_at": "2026-09-08",
"status_as_of": "2026-09-08",
"evidence_label": "Source candidate · not reviewed",
"note": "Full public directory statement. Source lane: backlog; source resolution: open. Source verification claims are attributed only; no local Lean verification has be... | {"created_at":"2026-08-17T19:12:01+00:00","display_standard":{"editorial_review":"submitter_supplied","formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":8736,"public_status":{"evidence":{"hasFormalization":false,"hasInformalProof":false,"hasRecordedWork":false,"solutionRecord":null},"lean":{... | [
"formalization-contains-placeholder",
"formalization-not-compiled-during-export",
"current-status-unassessed",
"source-full-text-not-present"
] | [] | {
"created_at": "2026-09-08T03:32:15.935Z",
"updated_at": "2026-09-08T03:32:15.935Z",
"published_at": "2026-09-08T03:32:52.832Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"71cf5d24-d028-421c-a493-580e29603f5d","format_issues":null,"number":3945} |
1.0.0 | problem:theoremdb-p8738 | problem | theoremdb-p8738 | Infinitely many preimages under iterated totient-plus-one map | {
"main": "math",
"sub": "number theory",
"slug": "number-theory",
"name": "Number theory",
"original_discipline": "mathematics"
} | [] | [
{
"code": "11",
"basis": "community-map",
"evidence": "Inferred from community field: number-theory"
}
] | For a positive integer $n$, define the map $f(n) = \phi(n) + 1$, where $\phi$ denotes Euler's totient function. For $i \geq 0$, let $f^{[i]}(n)$ denote the $i$-th iterate of $f$ applied to $n$, with $f^{[0]}(n) = n$. Determine whether there exists a prime number $p$ such that the set $\{ n \in \mathbb{N} : \exists i \g... | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P8738",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P8738",
"citation": null,
"doi": null,
"arxiv_id": null,
"license": "Selected packets label their content CC0-1.0; original references retain their own terms",
"notes": "Selected problem... | theorem erdos_409.parts.ii :
answer(sorry) ↔ ∃ (p : ℕ) (hp : p.Prime), { n | ∃ i, (φ · + 1)^[i] n = p }.Infinite | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem erdos_409.parts.ii :\n answer(sorry) ↔ ∃ (p : ℕ) (hp : p.Prime), { n | ∃ i, (φ · + 1)^[i] n = p }.Infinite",
"target": null,
"source_url": "https://theoremdb.org/statements/P8738",
"format": "declaration-or-fragment",
"sha256": "57fa6e94f1e02ae0180... | [] | TheoremDB P8738 · Candidate awaiting source review | null | Source-reported assessment, retrieved 8 September 2026. Frontier Science has not independently reviewed this status.
Open. No signed Lean verification is attached to this problem or its selected resolution.
This record is in the source backlog; its novelty and unresolved status have not passed source review. | null | published | incremental | hard | [
"source-candidate",
"theoremdb"
] | null | null | null | null | [
{
"kind": "website",
"citation": "TheoremDB P8738",
"url": "https://theoremdb.org/statements/P8738",
"note": null,
"sort_order": 0
},
{
"kind": "formalization",
"citation": "Source formal statement (not locally verified)",
"url": "https://theoremdb.org/statements/P8738",
"not... | {
"source_record": "P8738",
"imported_at": "2026-09-08",
"status_as_of": "2026-09-08",
"evidence_label": "Source candidate · not reviewed",
"note": "Full public directory statement. Source lane: backlog; source resolution: open. Source verification claims are attributed only; no local Lean verification has be... | {"created_at":"2026-08-17T19:12:04+00:00","display_standard":{"editorial_review":"submitter_supplied","formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":8738,"public_status":{"evidence":{"hasFormalization":false,"hasInformalProof":false,"hasRecordedWork":false,"solutionRecord":null},"lean":{... | [
"formalization-contains-placeholder",
"formalization-not-compiled-during-export",
"current-status-unassessed",
"source-full-text-not-present"
] | [] | {
"created_at": "2026-09-08T03:32:15.935Z",
"updated_at": "2026-09-08T03:32:15.935Z",
"published_at": "2026-09-08T03:33:00.966Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"977bd9f2-3153-4926-9cb3-79fff47a8bd2","format_issues":null,"number":5629} |
1.0.0 | problem:theoremdb-p8740 | problem | theoremdb-p8740 | Existence of Sidon sets of optimal size in cyclic groups of prime order | {
"main": "math",
"sub": "combinatorics",
"slug": "combinatorics",
"name": "Combinatorics",
"original_discipline": "mathematics"
} | [] | [
{
"code": "05",
"basis": "community-map",
"evidence": "Inferred from community field: combinatorics"
}
] | Let $p$ denote a prime number. A subset $S$ of an abelian group is called a Sidon set if all pairwise sums $a+b$ with $a,b \in S$ (allowing $a=b$) are distinct, except for the trivial symmetry $a+b = b+a$. For each prime $p$, consider the cyclic group $\mathbb{Z}/p\mathbb{Z}$ of order $p$. Does there exist a family of ... | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P8740",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P8740",
"citation": null,
"doi": null,
"arxiv_id": null,
"license": "Selected packets label their content CC0-1.0; original references retain their own terms",
"notes": "Selected problem... | theorem green_31.variants.zmod_p : answer(sorry) ↔
∃ S : (n : ℕ) → Finset (ZMod n),
∃ o : ℕ → ℝ,
(o =o[atTop] fun _ : ℕ ↦ (1 : ℝ)) ∧
(∀ p, p.Prime → IsSidon (S p : Set (ZMod p))) ∧
(∀ p, p.Prime → ((S p).card : ℝ) = (1 + o p) * Real.sqrt (p : ℝ)) | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem green_31.variants.zmod_p : answer(sorry) ↔\n ∃ S : (n : ℕ) → Finset (ZMod n),\n ∃ o : ℕ → ℝ,\n (o =o[atTop] fun _ : ℕ ↦ (1 : ℝ)) ∧\n (∀ p, p.Prime → IsSidon (S p : Set (ZMod p))) ∧\n (∀ p, p.Prime → ((S p).card : ℝ) = (1 + o p) * Real.sqrt (p : ℝ... | [] | TheoremDB P8740 · Candidate awaiting source review | null | Source-reported assessment, retrieved 8 September 2026. Frontier Science has not independently reviewed this status.
Open. No signed Lean verification is attached to this problem or its selected resolution.
This record is in the source backlog; its novelty and unresolved status have not passed source review. | null | published | incremental | hard | [
"source-candidate",
"theoremdb"
] | null | null | null | null | [
{
"kind": "website",
"citation": "TheoremDB P8740",
"url": "https://theoremdb.org/statements/P8740",
"note": null,
"sort_order": 0
},
{
"kind": "formalization",
"citation": "Source formal statement (not locally verified)",
"url": "https://theoremdb.org/statements/P8740",
"not... | {
"source_record": "P8740",
"imported_at": "2026-09-08",
"status_as_of": "2026-09-08",
"evidence_label": "Source candidate · not reviewed",
"note": "Full public directory statement. Source lane: backlog; source resolution: open. Source verification claims are attributed only; no local Lean verification has be... | {"created_at":"2026-08-17T19:12:21+00:00","display_standard":{"editorial_review":"submitter_supplied","formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":8740,"public_status":{"evidence":{"hasFormalization":false,"hasInformalProof":false,"hasRecordedWork":false,"solutionRecord":null},"lean":{... | [
"formalization-contains-placeholder",
"formalization-not-compiled-during-export",
"current-status-unassessed",
"source-full-text-not-present"
] | [] | {
"created_at": "2026-09-08T03:32:15.935Z",
"updated_at": "2026-09-08T03:32:15.935Z",
"published_at": "2026-09-08T03:32:58.816Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"d3dc0e5c-54d7-4030-8f43-8d836749556e","format_issues":null,"number":5261} |
1.0.0 | problem:theoremdb-p8742 | problem | theoremdb-p8742 | Zhi-Wei Sun's Four-Square Conjecture with Powers of 2, 3, and 5 | {
"main": "math",
"sub": "number theory",
"slug": "number-theory",
"name": "Number theory",
"original_discipline": "mathematics"
} | [] | [
{
"code": "11",
"basis": "community-map",
"evidence": "Inferred from community field: number-theory"
}
] | Let $A(n)$ denote the property that a positive integer $n$ can be expressed as $(2^a \cdot 3^b)^2 + (2^c \cdot 5^d)^2 + x^2 + y^2$ for some nonnegative integers $a, b, c, d, x, y$. Determine whether $A(n)$ holds for every integer $n > 1$. | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P8742",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P8742",
"citation": null,
"doi": null,
"arxiv_id": null,
"license": "Selected packets label their content CC0-1.0; original references retain their own terms",
"notes": "Selected problem... | theorem conjecture (n : ℕ) (hn : 1 < n) : A n | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem conjecture (n : ℕ) (hn : 1 < n) : A n",
"target": null,
"source_url": "https://theoremdb.org/statements/P8742",
"format": "declaration-or-fragment",
"sha256": "9c3552ea3b574a89c5f6b8cd5a92ea437c055998af76a271622edaaf8ee3d6e7",
"license": "Selected... | [] | TheoremDB P8742 · Candidate awaiting source review | null | Source-reported assessment, retrieved 8 September 2026. Frontier Science has not independently reviewed this status.
Open. No signed Lean verification is attached to this problem or its selected resolution.
This record is in the source backlog; its novelty and unresolved status have not passed source review. | null | published | incremental | hard | [
"source-candidate",
"theoremdb"
] | null | null | null | null | [
{
"kind": "website",
"citation": "TheoremDB P8742",
"url": "https://theoremdb.org/statements/P8742",
"note": null,
"sort_order": 0
},
{
"kind": "formalization",
"citation": "Source formal statement (not locally verified)",
"url": "https://theoremdb.org/statements/P8742",
"not... | {
"source_record": "P8742",
"imported_at": "2026-09-08",
"status_as_of": "2026-09-08",
"evidence_label": "Source candidate · not reviewed",
"note": "Full public directory statement. Source lane: backlog; source resolution: open. Source verification claims are attributed only; no local Lean verification has be... | {"created_at":"2026-08-17T19:12:28+00:00","display_standard":{"formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":8742,"public_status":{"evidence":{"hasFormalization":false,"hasInformalProof":false,"hasRecordedWork":false,"solutionRecord":null},"lean":{"kernelAccepted":false,"label":"","reaso... | [
"formalization-not-compiled-during-export",
"current-status-unassessed",
"source-full-text-not-present"
] | [] | {
"created_at": "2026-09-08T03:32:15.935Z",
"updated_at": "2026-09-08T03:32:15.935Z",
"published_at": "2026-09-08T03:33:07.658Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"65287157-f25b-421c-8715-571aed85b58f","format_issues":null,"number":7007} |
1.0.0 | problem:theoremdb-p8744 | problem | theoremdb-p8744 | Erdős–Szekeres strengthening of a binomial coefficient divisibility problem | {
"main": "math",
"sub": "number theory",
"slug": "number-theory",
"name": "Number theory",
"original_discipline": "mathematics"
} | [] | [
{
"code": "11",
"basis": "community-map",
"evidence": "Inferred from community field: number-theory"
}
] | For a natural number $n$ and integers $i, j$ with $1 \le i < j \le n/2$, let $\binom{n}{i}$ and $\binom{n}{j}$ denote the binomial coefficients. The Sylvester–Schur theorem asserts that for each $i$ in this range, there exists a prime $p > i$ dividing $\binom{n}{i}$. Erdős Problem 699 asks whether for each such triple ... | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P8744",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P8744",
"citation": null,
"doi": null,
"arxiv_id": null,
"license": "Selected packets label their content CC0-1.0; original references retain their own terms",
"notes": "Selected problem... | theorem erdos_szekeres_strengthening : answer(sorry) ↔
∃ E : Finset (ℕ × ℕ × ℕ), ∀ n i j : ℕ,
1 ≤ i →
i < j →
j ≤ n / 2 →
(n, i, j) ∉ E →
∃ p : ℕ, p.Prime ∧ i < p ∧ p ∣ Nat.gcd (Nat.choose n i) (Nat.choose n j) | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem erdos_szekeres_strengthening : answer(sorry) ↔\n ∃ E : Finset (ℕ × ℕ × ℕ), ∀ n i j : ℕ,\n 1 ≤ i →\n i < j →\n j ≤ n / 2 →\n (n, i, j) ∉ E →\n ∃ p : ℕ, p.Prime ∧ i < p ∧ p ∣ Nat.gcd (Nat.choose n i) (Nat.choose n j)",
"target": null,
... | [] | TheoremDB P8744 · Candidate awaiting source review | null | Source-reported assessment, retrieved 8 September 2026. Frontier Science has not independently reviewed this status.
Open. No signed Lean verification is attached to this problem or its selected resolution.
This record is in the source backlog; its novelty and unresolved status have not passed source review. | null | published | incremental | hard | [
"source-candidate",
"theoremdb"
] | null | null | null | null | [
{
"kind": "website",
"citation": "TheoremDB P8744",
"url": "https://theoremdb.org/statements/P8744",
"note": null,
"sort_order": 0
},
{
"kind": "formalization",
"citation": "Source formal statement (not locally verified)",
"url": "https://theoremdb.org/statements/P8744",
"not... | {
"source_record": "P8744",
"imported_at": "2026-09-08",
"status_as_of": "2026-09-08",
"evidence_label": "Source candidate · not reviewed",
"note": "Full public directory statement. Source lane: backlog; source resolution: open. Source verification claims are attributed only; no local Lean verification has be... | {"created_at":"2026-08-17T19:12:37+00:00","display_standard":{"editorial_review":"submitter_supplied","formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":8744,"public_status":{"evidence":{"hasFormalization":false,"hasInformalProof":false,"hasRecordedWork":false,"solutionRecord":null},"lean":{... | [
"formalization-contains-placeholder",
"formalization-not-compiled-during-export",
"current-status-unassessed",
"source-full-text-not-present"
] | [] | {
"created_at": "2026-09-08T03:32:15.935Z",
"updated_at": "2026-09-08T03:32:15.935Z",
"published_at": "2026-09-08T03:32:56.711Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"8103731d-078b-4376-8a3b-830dae8989f4","format_issues":null,"number":4798} |
1.0.0 | problem:theoremdb-p8746 | problem | theoremdb-p8746 | Asymptotic ratio of values in the range of Euler's totient function | {
"main": "math",
"sub": "number theory",
"slug": "number-theory",
"name": "Number theory",
"original_discipline": "mathematics"
} | [] | [
{
"code": "11",
"basis": "community-map",
"evidence": "Inferred from community field: number-theory"
}
] | For a positive real number $x$, let $V(x)$ denote the number of positive integers $n \leq x$ such that the equation $\phi(m) = n$ has a solution, where $\phi$ is Euler's totient function. Does $\displaystyle\frac{V(2x)}{V(x)} \to 2$ as $x \to \infty$? | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P8746",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P8746",
"citation": null,
"doi": null,
"arxiv_id": null,
"license": "Selected packets label their content CC0-1.0; original references retain their own terms",
"notes": "Selected problem... | theorem erdos_416.parts.i :
Filter.Tendsto (fun x => (V (2 * x) / V (x))) Filter.atTop (𝓝 2) | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem erdos_416.parts.i :\n Filter.Tendsto (fun x => (V (2 * x) / V (x))) Filter.atTop (𝓝 2)",
"target": null,
"source_url": "https://theoremdb.org/statements/P8746",
"format": "declaration-or-fragment",
"sha256": "27d00293fa7e7ebc59e056d66a653d16ce24bf... | [] | TheoremDB P8746 · Candidate awaiting source review | null | Source-reported assessment, retrieved 8 September 2026. Frontier Science has not independently reviewed this status.
Open. No signed Lean verification is attached to this problem or its selected resolution.
This record is in the source backlog; its novelty and unresolved status have not passed source review. | null | published | incremental | hard | [
"source-candidate",
"theoremdb"
] | null | null | null | null | [
{
"kind": "website",
"citation": "TheoremDB P8746",
"url": "https://theoremdb.org/statements/P8746",
"note": null,
"sort_order": 0
},
{
"kind": "formalization",
"citation": "Source formal statement (not locally verified)",
"url": "https://theoremdb.org/statements/P8746",
"not... | {
"source_record": "P8746",
"imported_at": "2026-09-08",
"status_as_of": "2026-09-08",
"evidence_label": "Source candidate · not reviewed",
"note": "Full public directory statement. Source lane: backlog; source resolution: open. Source verification claims are attributed only; no local Lean verification has be... | {"created_at":"2026-08-17T19:12:52+00:00","display_standard":{"editorial_review":"submitter_supplied","formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":8746,"public_status":{"evidence":{"hasFormalization":false,"hasInformalProof":false,"hasRecordedWork":false,"solutionRecord":null},"lean":{... | [
"formalization-not-compiled-during-export",
"current-status-unassessed",
"source-full-text-not-present"
] | [] | {
"created_at": "2026-09-08T03:32:15.935Z",
"updated_at": "2026-09-08T03:32:15.935Z",
"published_at": "2026-09-08T03:32:50.387Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"c886d022-35df-41cf-88fa-821815104734","format_issues":null,"number":3409} |
1.0.0 | problem:theoremdb-p8748 | problem | theoremdb-p8748 | A conjecture on large least prime factors of shifted integers | {
"main": "math",
"sub": "number theory",
"slug": "number-theory",
"name": "Number theory",
"original_discipline": "mathematics"
} | [] | [
{
"code": "11",
"basis": "community-map",
"evidence": "Inferred from community field: number-theory"
}
] | For a positive integer $m$, let $p(m)$ denote the least prime factor of $m$, that is, the smallest prime number $p$ such that $p$ divides $m$. Is it true that for all sufficiently large positive integers $n$, there exists a positive integer $k$ such that $n + k$ is composite and $p(n + k) > k^2$? | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P8748",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P8748",
"citation": null,
"doi": null,
"arxiv_id": null,
"license": "Selected packets label their content CC0-1.0; original references retain their own terms",
"notes": "Selected problem... | theorem erdos_681 : answer(sorry) ↔
∀ᶠ n in .atTop, ∃ k > 0, (n + k).Composite ∧ ∀ p, IsLPF p (n + k) → p > k ^ 2 | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem erdos_681 : answer(sorry) ↔\n ∀ᶠ n in .atTop, ∃ k > 0, (n + k).Composite ∧ ∀ p, IsLPF p (n + k) → p > k ^ 2",
"target": null,
"source_url": "https://theoremdb.org/statements/P8748",
"format": "declaration-or-fragment",
"sha256": "db4e743946348e68a7... | [] | TheoremDB P8748 · Candidate awaiting source review | null | Source-reported assessment, retrieved 8 September 2026. Frontier Science has not independently reviewed this status.
Open. No signed Lean verification is attached to this problem or its selected resolution.
This record is in the source backlog; its novelty and unresolved status have not passed source review. | null | published | incremental | hard | [
"source-candidate",
"theoremdb"
] | null | null | null | null | [
{
"kind": "website",
"citation": "TheoremDB P8748",
"url": "https://theoremdb.org/statements/P8748",
"note": null,
"sort_order": 0
},
{
"kind": "formalization",
"citation": "Source formal statement (not locally verified)",
"url": "https://theoremdb.org/statements/P8748",
"not... | {
"source_record": "P8748",
"imported_at": "2026-09-08",
"status_as_of": "2026-09-08",
"evidence_label": "Source candidate · not reviewed",
"note": "Full public directory statement. Source lane: backlog; source resolution: open. Source verification claims are attributed only; no local Lean verification has be... | {"created_at":"2026-08-17T19:13:03+00:00","display_standard":{"editorial_review":"submitter_supplied","formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":8748,"public_status":{"evidence":{"hasFormalization":false,"hasInformalProof":false,"hasRecordedWork":false,"solutionRecord":null},"lean":{... | [
"formalization-contains-placeholder",
"formalization-not-compiled-during-export",
"current-status-unassessed",
"source-full-text-not-present"
] | [] | {
"created_at": "2026-09-08T03:32:15.935Z",
"updated_at": "2026-09-08T03:32:15.935Z",
"published_at": "2026-09-08T03:32:49.041Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"2025ccd0-9748-4a0d-8fba-38670035b0ab","format_issues":null,"number":3119} |
1.0.0 | problem:theoremdb-p8750 | problem | theoremdb-p8750 | Countable ordinals with the ordinal Ramsey property for cliques versus triangles | {
"main": "math",
"sub": "logic and foundations",
"slug": "logic-and-foundations",
"name": "Logic and foundations",
"original_discipline": "mathematics"
} | [] | [
{
"code": "03",
"basis": "community-map",
"evidence": "Inferred from community field: logic-and-foundations"
}
] | For a countable ordinal $\beta$, let $\alpha = \omega^\beta$. Determine precisely which such $\beta$ have the property that in every red/blue coloring of the edges of the complete graph $K_\alpha$, there exists either a red $K_\alpha$ (a complete subgraph of order type $\alpha$ with all edges red) or a blue $K_3$ (a mo... | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P8750",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P8750",
"citation": null,
"doi": null,
"arxiv_id": null,
"license": "Selected packets label their content CC0-1.0; original references retain their own terms",
"notes": "Selected problem... | theorem erdos_592 (β : Ordinal.{u}) : β.card ≤ ℵ₀ →
OrdinalCardinalRamsey (ω ^ β) (ω ^ β) 3 ↔ (answer(sorry) : Ordinal.{u} → Prop) β | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem erdos_592 (β : Ordinal.{u}) : β.card ≤ ℵ₀ →\n OrdinalCardinalRamsey (ω ^ β) (ω ^ β) 3 ↔ (answer(sorry) : Ordinal.{u} → Prop) β",
"target": null,
"source_url": "https://theoremdb.org/statements/P8750",
"format": "declaration-or-fragment",
"sha256": ... | [] | TheoremDB P8750 · Candidate awaiting source review | null | Source-reported assessment, retrieved 8 September 2026. Frontier Science has not independently reviewed this status.
Open. No signed Lean verification is attached to this problem or its selected resolution.
This record is in the source backlog; its novelty and unresolved status have not passed source review. | null | published | incremental | hard | [
"source-candidate",
"theoremdb"
] | null | null | null | null | [
{
"kind": "website",
"citation": "TheoremDB P8750",
"url": "https://theoremdb.org/statements/P8750",
"note": null,
"sort_order": 0
},
{
"kind": "formalization",
"citation": "Source formal statement (not locally verified)",
"url": "https://theoremdb.org/statements/P8750",
"not... | {
"source_record": "P8750",
"imported_at": "2026-09-08",
"status_as_of": "2026-09-08",
"evidence_label": "Source candidate · not reviewed",
"note": "Full public directory statement. Source lane: backlog; source resolution: open. Source verification claims are attributed only; no local Lean verification has be... | {"created_at":"2026-08-17T19:13:17+00:00","display_standard":{"editorial_review":"submitter_supplied","formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":8750,"public_status":{"evidence":{"hasFormalization":false,"hasInformalProof":false,"hasRecordedWork":false,"solutionRecord":null},"lean":{... | [
"formalization-contains-placeholder",
"formalization-not-compiled-during-export",
"current-status-unassessed",
"source-full-text-not-present"
] | [] | {
"created_at": "2026-09-08T03:32:15.935Z",
"updated_at": "2026-09-08T03:32:15.935Z",
"published_at": "2026-09-08T03:32:52.767Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"6dd8fef9-0cb4-4252-80b5-72266d67ef3e","format_issues":null,"number":3931} |
1.0.0 | problem:theoremdb-p8752 | problem | theoremdb-p8752 | Erdős–Hajnal exceptionality of the pair $(K_4, K_3)$ | {
"main": "math",
"sub": "combinatorics",
"slug": "combinatorics",
"name": "Combinatorics",
"original_discipline": "mathematics"
} | [] | [
{
"code": "05",
"basis": "community-map",
"evidence": "Inferred from community field: combinatorics"
}
] | A pair of graphs $(G_1, G_2)$ is called Erdős–Hajnal exceptional if it satisfies the following two conditions: (1) for every positive integer $n$, there exists a graph $H$ containing no copy of $G_1$ such that every $n$-colouring of the edges of $H$ contains a monochromatic copy of $G_2$; and (2) for every graph $H$ co... | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P8752",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P8752",
"citation": null,
"doi": null,
"arxiv_id": null,
"license": "Selected packets label their content CC0-1.0; original references retain their own terms",
"notes": "Selected problem... | theorem erdos_596.variants.K4_K3_exceptional_iff : answer(sorry) ↔
IsErdosHajnalExceptional (completeGraph (Fin 4)) (completeGraph (Fin 3)) | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem erdos_596.variants.K4_K3_exceptional_iff : answer(sorry) ↔\n IsErdosHajnalExceptional (completeGraph (Fin 4)) (completeGraph (Fin 3))",
"target": null,
"source_url": "https://theoremdb.org/statements/P8752",
"format": "declaration-or-fragment",
"sh... | [] | TheoremDB P8752 · Candidate awaiting source review | null | Source-reported assessment, retrieved 8 September 2026. Frontier Science has not independently reviewed this status.
Open. No signed Lean verification is attached to this problem or its selected resolution.
This record is in the source backlog; its novelty and unresolved status have not passed source review. | null | published | incremental | hard | [
"source-candidate",
"theoremdb"
] | null | null | null | null | [
{
"kind": "website",
"citation": "TheoremDB P8752",
"url": "https://theoremdb.org/statements/P8752",
"note": null,
"sort_order": 0
},
{
"kind": "formalization",
"citation": "Source formal statement (not locally verified)",
"url": "https://theoremdb.org/statements/P8752",
"not... | {
"source_record": "P8752",
"imported_at": "2026-09-08",
"status_as_of": "2026-09-08",
"evidence_label": "Source candidate · not reviewed",
"note": "Full public directory statement. Source lane: backlog; source resolution: open. Source verification claims are attributed only; no local Lean verification has be... | {"created_at":"2026-08-17T19:13:23+00:00","display_standard":{"editorial_review":"submitter_supplied","formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":8752,"public_status":{"evidence":{"hasFormalization":false,"hasInformalProof":false,"hasRecordedWork":false,"solutionRecord":null},"lean":{... | [
"formalization-contains-placeholder",
"formalization-not-compiled-during-export",
"current-status-unassessed",
"source-full-text-not-present"
] | [] | {
"created_at": "2026-09-08T03:32:15.935Z",
"updated_at": "2026-09-08T03:32:15.935Z",
"published_at": "2026-09-08T03:32:56.669Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"54e8ff56-2478-4d9b-8421-01c8486d98d3","format_issues":null,"number":4789} |
1.0.0 | problem:theoremdb-p8754 | problem | theoremdb-p8754 | Finiteness of rational elliptic curves with rank exceeding 21 | {
"main": "math",
"sub": "number theory",
"slug": "number-theory",
"name": "Number theory",
"original_discipline": "mathematics"
} | [] | [
{
"code": "11",
"basis": "community-map",
"evidence": "Inferred from community field: number-theory"
}
] | A rational elliptic curve is an elliptic curve over the rational numbers $\mathbb{Q}$, represented up to isomorphism by a Weierstrass equation $y^2 = x^3 + Ax + B$ with $A, B \in \mathbb{Z}$ satisfying the reduced condition: for every prime $p$, it is not the case that both $p^4 \mid A$ and $p^6 \mid B$, and with nonze... | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P8754",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P8754",
"citation": null,
"doi": null,
"arxiv_id": null,
"license": "Selected packets label their content CC0-1.0; original references retain their own terms",
"notes": "Selected problem... | theorem finite_twentyone_lt_finrank : {E : RatEllipticCurve | 21 < E.rank}.Finite | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem finite_twentyone_lt_finrank : {E : RatEllipticCurve | 21 < E.rank}.Finite",
"target": null,
"source_url": "https://theoremdb.org/statements/P8754",
"format": "declaration-or-fragment",
"sha256": "75f45366f960072dee274892216c88c8f737895668dcf454ebedcda... | [] | TheoremDB P8754 · Candidate awaiting source review | null | Source-reported assessment, retrieved 8 September 2026. Frontier Science has not independently reviewed this status.
Open. No signed Lean verification is attached to this problem or its selected resolution.
This record is in the source backlog; its novelty and unresolved status have not passed source review. | null | published | incremental | hard | [
"source-candidate",
"theoremdb"
] | null | null | null | null | [
{
"kind": "website",
"citation": "TheoremDB P8754",
"url": "https://theoremdb.org/statements/P8754",
"note": null,
"sort_order": 0
},
{
"kind": "formalization",
"citation": "Source formal statement (not locally verified)",
"url": "https://theoremdb.org/statements/P8754",
"not... | {
"source_record": "P8754",
"imported_at": "2026-09-08",
"status_as_of": "2026-09-08",
"evidence_label": "Source candidate · not reviewed",
"note": "Full public directory statement. Source lane: backlog; source resolution: open. Source verification claims are attributed only; no local Lean verification has be... | {"created_at":"2026-08-17T19:13:33+00:00","display_standard":{"editorial_review":"submitter_supplied","formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":8754,"public_status":{"evidence":{"hasFormalization":false,"hasInformalProof":false,"hasRecordedWork":false,"solutionRecord":null},"lean":{... | [
"formalization-not-compiled-during-export",
"current-status-unassessed",
"source-full-text-not-present"
] | [] | {
"created_at": "2026-09-08T03:32:15.935Z",
"updated_at": "2026-09-08T03:32:15.935Z",
"published_at": "2026-09-08T03:32:59.076Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"67911399-f07e-49cb-94b6-6fbb94b6ea98","format_issues":null,"number":5317} |
1.0.0 | problem:theoremdb-p8756 | problem | theoremdb-p8756 | Non-existence of positive derivatives for the distribution function of Euler's totient ratio | {
"main": "math",
"sub": "number theory",
"slug": "number-theory",
"name": "Number theory",
"original_discipline": "mathematics"
} | [] | [
{
"code": "11",
"basis": "community-map",
"evidence": "Inferred from community field: number-theory"
}
] | Let $\varphi$ denote Euler's totient function. A function $f : \mathbb{R} \to \mathbb{R}$ is called the asymptotic distribution function of $\varphi(n)/n$ if for every $c \in [0, 1]$, the set $\{n \in \mathbb{N} : \varphi(n) < cn\}$ has a natural density, and this density equals $f(c)$. Determine whether the following ... | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P8756",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P8756",
"citation": null,
"doi": null,
"arxiv_id": null,
"license": "Selected packets label their content CC0-1.0; original references retain their own terms",
"notes": "Selected problem... | theorem erdos_50 : answer(sorry) ↔ ∀ᵉ (f : ℝ → ℝ) (hf : IsDistributionOfPhiRatio f),
¬∃ x ∈ Icc (0 : ℝ) 1, ∃ y > 0, HasDerivWithinAt f y (Icc 0 1) x | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem erdos_50 : answer(sorry) ↔ ∀ᵉ (f : ℝ → ℝ) (hf : IsDistributionOfPhiRatio f),\n ¬∃ x ∈ Icc (0 : ℝ) 1, ∃ y > 0, HasDerivWithinAt f y (Icc 0 1) x",
"target": null,
"source_url": "https://theoremdb.org/statements/P8756",
"format": "declaration-or-fragment"... | [] | TheoremDB P8756 · Candidate awaiting source review | null | Source-reported assessment, retrieved 8 September 2026. Frontier Science has not independently reviewed this status.
Open. No signed Lean verification is attached to this problem or its selected resolution.
This record is in the source backlog; its novelty and unresolved status have not passed source review. | null | published | incremental | hard | [
"source-candidate",
"theoremdb"
] | null | null | null | null | [
{
"kind": "website",
"citation": "TheoremDB P8756",
"url": "https://theoremdb.org/statements/P8756",
"note": null,
"sort_order": 0
},
{
"kind": "formalization",
"citation": "Source formal statement (not locally verified)",
"url": "https://theoremdb.org/statements/P8756",
"not... | {
"source_record": "P8756",
"imported_at": "2026-09-08",
"status_as_of": "2026-09-08",
"evidence_label": "Source candidate · not reviewed",
"note": "Full public directory statement. Source lane: backlog; source resolution: open. Source verification claims are attributed only; no local Lean verification has be... | {"created_at":"2026-08-17T19:13:42+00:00","display_standard":{"editorial_review":"submitter_supplied","formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":8756,"public_status":{"evidence":{"hasFormalization":false,"hasInformalProof":false,"hasRecordedWork":false,"solutionRecord":null},"lean":{... | [
"formalization-contains-placeholder",
"formalization-not-compiled-during-export",
"current-status-unassessed",
"source-full-text-not-present"
] | [] | {
"created_at": "2026-09-08T03:32:15.935Z",
"updated_at": "2026-09-08T03:32:15.935Z",
"published_at": "2026-09-08T03:33:04.405Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"5d5d94a3-1481-4857-b226-a7c5adb05663","format_issues":null,"number":6303} |
1.0.0 | problem:theoremdb-p8758 | problem | theoremdb-p8758 | Existence of a universal exponential decay constant for constrained power sums | {
"main": "math",
"sub": "number theory",
"slug": "number-theory",
"name": "Number theory",
"original_discipline": "mathematics"
} | [] | [
{
"code": "11",
"basis": "community-map",
"evidence": "Inferred from community field: number-theory"
}
] | For each integer $n \geq 2$, consider sequences $z_1, z_2, \ldots, z_n$ of complex numbers satisfying $z_1 = 1$ and $|z_i| \geq 1$ for all $1 \leq i \leq n$. For such a sequence, define the maximum power sum deviation
$$
S_n(z) = \max_{2 \leq k \leq n+1} \left| \sum_{i=1}^{n} z_i^k \right|.
$$
Does there exist a real... | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P8758",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P8758",
"citation": null,
"doi": null,
"arxiv_id": null,
"license": "Selected packets label their content CC0-1.0; original references retain their own terms",
"notes": "Selected problem... | theorem erdos_973 :
answer(sorry) ↔
∃ C : ℝ, C > 1 ∧
∀ n : ℕ, n ≥ 2 → ∃ z : ℕ → ℂ,
z 1 = 1 ∧
(∀ i ∈ Icc 1 n, 1 ≤ ‖z i‖) ∧
(∀ k ∈ Icc 2 (n + 1), ‖∑ i ∈ Icc 1 n, z i ^ k‖ < C ^ (-(n : ℝ))) | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem erdos_973 :\n answer(sorry) ↔\n ∃ C : ℝ, C > 1 ∧\n ∀ n : ℕ, n ≥ 2 → ∃ z : ℕ → ℂ,\n z 1 = 1 ∧\n (∀ i ∈ Icc 1 n, 1 ≤ ‖z i‖) ∧\n (∀ k ∈ Icc 2 (n + 1), ‖∑ i ∈ Icc 1 n, z i ^ k‖ < C ^ (-(n : ℝ)))",
"target": null,
"source_... | [] | TheoremDB P8758 · Candidate awaiting source review | null | Source-reported assessment, retrieved 8 September 2026. Frontier Science has not independently reviewed this status.
Open. No signed Lean verification is attached to this problem or its selected resolution.
This record is in the source backlog; its novelty and unresolved status have not passed source review. | null | published | incremental | hard | [
"source-candidate",
"theoremdb"
] | null | null | null | null | [
{
"kind": "website",
"citation": "TheoremDB P8758",
"url": "https://theoremdb.org/statements/P8758",
"note": null,
"sort_order": 0
},
{
"kind": "formalization",
"citation": "Source formal statement (not locally verified)",
"url": "https://theoremdb.org/statements/P8758",
"not... | {
"source_record": "P8758",
"imported_at": "2026-09-08",
"status_as_of": "2026-09-08",
"evidence_label": "Source candidate · not reviewed",
"note": "Full public directory statement. Source lane: backlog; source resolution: open. Source verification claims are attributed only; no local Lean verification has be... | {"created_at":"2026-08-17T19:13:56+00:00","display_standard":{"editorial_review":"submitter_supplied","formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":8758,"public_status":{"evidence":{"hasFormalization":false,"hasInformalProof":false,"hasRecordedWork":false,"solutionRecord":null},"lean":{... | [
"formalization-contains-placeholder",
"formalization-not-compiled-during-export",
"current-status-unassessed",
"source-full-text-not-present"
] | [] | {
"created_at": "2026-09-08T03:32:15.935Z",
"updated_at": "2026-09-08T03:32:15.935Z",
"published_at": "2026-09-08T03:32:58.574Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"e9b96486-1679-4752-be67-ed7c1dedc596","format_issues":null,"number":5211} |
1.0.0 | problem:theoremdb-p8760 | problem | theoremdb-p8760 | Existence of Sums of Coprime $r$-Powerful Numbers That Are $r$-Powerful | {
"main": "math",
"sub": "number theory",
"slug": "number-theory",
"name": "Number theory",
"original_discipline": "mathematics"
} | [] | [
{
"code": "11",
"basis": "community-map",
"evidence": "Inferred from community field: number-theory"
}
] | For an integer $r \geq 2$, a positive integer $n$ is called $r$-powerful if every prime factor of $n$ appears with exponent at least $r$ in the prime factorization of $n$; that is, if $p$ is prime and $p \mid n$, then $p^r \mid n$. For a finite set $S$ of positive integers, we say that $S$ is coprime if $\gcd(S) = 1$, ... | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P8760",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P8760",
"citation": null,
"doi": null,
"arxiv_id": null,
"license": "Selected packets label their content CC0-1.0; original references retain their own terms",
"notes": "Selected problem... | theorem erdos_939 : answer(sorry) ↔ ∀ r ≥ 4, (Erdos939Sums r).Nonempty | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem erdos_939 : answer(sorry) ↔ ∀ r ≥ 4, (Erdos939Sums r).Nonempty",
"target": null,
"source_url": "https://theoremdb.org/statements/P8760",
"format": "declaration-or-fragment",
"sha256": "3df94bb7c16067578707bf9b29fad377e796d2a3003e25a92b35e09ffd6cd5e7",... | [] | TheoremDB P8760 · Candidate awaiting source review | null | Source-reported assessment, retrieved 8 September 2026. Frontier Science has not independently reviewed this status.
Open. No signed Lean verification is attached to this problem or its selected resolution.
This record is in the source backlog; its novelty and unresolved status have not passed source review. | null | published | incremental | hard | [
"source-candidate",
"theoremdb"
] | null | null | null | null | [
{
"kind": "website",
"citation": "TheoremDB P8760",
"url": "https://theoremdb.org/statements/P8760",
"note": null,
"sort_order": 0
},
{
"kind": "formalization",
"citation": "Source formal statement (not locally verified)",
"url": "https://theoremdb.org/statements/P8760",
"not... | {
"source_record": "P8760",
"imported_at": "2026-09-08",
"status_as_of": "2026-09-08",
"evidence_label": "Source candidate · not reviewed",
"note": "Full public directory statement. Source lane: backlog; source resolution: open. Source verification claims are attributed only; no local Lean verification has be... | {"created_at":"2026-08-17T19:14:09+00:00","display_standard":{"editorial_review":"submitter_supplied","formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":8760,"public_status":{"evidence":{"hasFormalization":false,"hasInformalProof":false,"hasRecordedWork":false,"solutionRecord":null},"lean":{... | [
"formalization-contains-placeholder",
"formalization-not-compiled-during-export",
"current-status-unassessed",
"source-full-text-not-present"
] | [] | {
"created_at": "2026-09-08T03:32:15.935Z",
"updated_at": "2026-09-08T03:32:15.935Z",
"published_at": "2026-09-08T03:32:58.820Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"312427c8-43ed-4e10-8020-c3d5595d119a","format_issues":null,"number":5262} |
1.0.0 | problem:theoremdb-p8762 | problem | theoremdb-p8762 | Primitive Roots of the Form k² + 1 | {
"main": "math",
"sub": "number theory",
"slug": "number-theory",
"name": "Number theory",
"original_discipline": "mathematics"
} | [] | [
{
"code": "11",
"basis": "community-map",
"evidence": "Inferred from community field: number-theory"
}
] | For a prime number $p$, a primitive root modulo $p$ is an integer $g$ with $0 < g < p$ whose multiplicative order modulo $p$ equals $p - 1$; equivalently, $g$ generates the multiplicative group $(\mathbb{Z}/p\mathbb{Z})^\times$. For each prime $p$, must there exist an integer $k$ such that $k^2 + 1$ is a primitive root... | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P8762",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P8762",
"citation": null,
"doi": null,
"arxiv_id": null,
"license": "Selected packets label their content CC0-1.0; original references retain their own terms",
"notes": "Selected problem... | theorem conjecture (p : ℕ) (hp : p.Prime) : A p | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem conjecture (p : ℕ) (hp : p.Prime) : A p",
"target": null,
"source_url": "https://theoremdb.org/statements/P8762",
"format": "declaration-or-fragment",
"sha256": "6b6f3caf3e31e2917e09271fc2e726fc69935983b815432b1fb344f658d305fb",
"license": "Select... | [] | TheoremDB P8762 · Candidate awaiting source review | null | Source-reported assessment, retrieved 8 September 2026. Frontier Science has not independently reviewed this status.
Open. No signed Lean verification is attached to this problem or its selected resolution.
This record is in the source backlog; its novelty and unresolved status have not passed source review. | null | published | incremental | hard | [
"source-candidate",
"theoremdb"
] | null | null | null | null | [
{
"kind": "website",
"citation": "TheoremDB P8762",
"url": "https://theoremdb.org/statements/P8762",
"note": null,
"sort_order": 0
},
{
"kind": "formalization",
"citation": "Source formal statement (not locally verified)",
"url": "https://theoremdb.org/statements/P8762",
"not... | {
"source_record": "P8762",
"imported_at": "2026-09-08",
"status_as_of": "2026-09-08",
"evidence_label": "Source candidate · not reviewed",
"note": "Full public directory statement. Source lane: backlog; source resolution: open. Source verification claims are attributed only; no local Lean verification has be... | {"created_at":"2026-08-17T19:14:25+00:00","display_standard":{"formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":8762,"public_status":{"evidence":{"hasFormalization":false,"hasInformalProof":false,"hasRecordedWork":false,"solutionRecord":null},"lean":{"kernelAccepted":false,"label":"","reaso... | [
"formalization-contains-placeholder",
"formalization-not-compiled-during-export",
"current-status-unassessed",
"source-full-text-not-present"
] | [] | {
"created_at": "2026-09-08T03:32:15.935Z",
"updated_at": "2026-09-08T03:32:15.935Z",
"published_at": "2026-09-08T03:33:05.273Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"7c198393-bbaf-4b66-8abc-5df2c1066793","format_issues":null,"number":6492} |
1.0.0 | problem:theoremdb-p8764 | problem | theoremdb-p8764 | Existence of Transversals in Odd-Order Latin Squares | {
"main": "math",
"sub": "combinatorics",
"slug": "combinatorics",
"name": "Combinatorics",
"original_discipline": "mathematics"
} | [] | [
{
"code": "05",
"basis": "community-map",
"evidence": "Inferred from community field: combinatorics"
}
] | A Latin square of order $n$ is an $n \times n$ array filled with $n$ distinct symbols such that each symbol occurs exactly once in each row and exactly once in each column. A transversal in a Latin square is a set of $n$ cells, one from each row, one from each column, and containing each of the $n$ symbols exactly once... | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P8764",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P8764",
"citation": null,
"doi": null,
"arxiv_id": null,
"license": "Selected packets label their content CC0-1.0; original references retain their own terms",
"notes": "Selected problem... | theorem oddOrderLatinSquareTransversal : answer(sorry) ↔
Odd n → ∀ (L : LatinSquare n), ∃ σ, IsTransversal L σ | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem oddOrderLatinSquareTransversal : answer(sorry) ↔\n Odd n → ∀ (L : LatinSquare n), ∃ σ, IsTransversal L σ",
"target": null,
"source_url": "https://theoremdb.org/statements/P8764",
"format": "declaration-or-fragment",
"sha256": "2cba804e24df3d78b57b0... | [] | TheoremDB P8764 · Candidate awaiting source review | null | Source-reported assessment, retrieved 8 September 2026. Frontier Science has not independently reviewed this status.
Open. No signed Lean verification is attached to this problem or its selected resolution.
This record is in the source backlog; its novelty and unresolved status have not passed source review. | null | published | incremental | hard | [
"source-candidate",
"theoremdb"
] | null | null | null | null | [
{
"kind": "website",
"citation": "TheoremDB P8764",
"url": "https://theoremdb.org/statements/P8764",
"note": null,
"sort_order": 0
},
{
"kind": "formalization",
"citation": "Source formal statement (not locally verified)",
"url": "https://theoremdb.org/statements/P8764",
"not... | {
"source_record": "P8764",
"imported_at": "2026-09-08",
"status_as_of": "2026-09-08",
"evidence_label": "Source candidate · not reviewed",
"note": "Full public directory statement. Source lane: backlog; source resolution: open. Source verification claims are attributed only; no local Lean verification has be... | {"created_at":"2026-08-17T19:14:34+00:00","display_standard":{"editorial_review":"submitter_supplied","formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":8764,"public_status":{"evidence":{"hasFormalization":false,"hasInformalProof":false,"hasRecordedWork":false,"solutionRecord":null},"lean":{... | [
"formalization-contains-placeholder",
"formalization-not-compiled-during-export",
"current-status-unassessed",
"source-full-text-not-present"
] | [] | {
"created_at": "2026-09-08T03:32:15.935Z",
"updated_at": "2026-09-08T03:32:15.935Z",
"published_at": "2026-09-08T03:32:58.866Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"6b4927c5-3523-43af-9dad-a51a85ed8d14","format_issues":null,"number":5272} |
1.0.0 | problem:theoremdb-p8766 | problem | theoremdb-p8766 | Infinitely many coprime 3-powerful triples summing to a 3-powerful number | {
"main": "math",
"sub": "number theory",
"slug": "number-theory",
"name": "Number theory",
"original_discipline": "mathematics"
} | [] | [
{
"code": "11",
"basis": "community-map",
"evidence": "Inferred from community field: number-theory"
}
] | A positive integer $n$ is called $3$-powerful if for every prime $p$ dividing $n$, the exponent of $p$ in the prime factorization of $n$ is at least $3$. Equivalently, $n$ is $3$-powerful if $p^3 \mid n$ for every prime $p \mid n$. A finite set of natural numbers is called coprime if the greatest common divisor of all ... | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P8766",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P8766",
"citation": null,
"doi": null,
"arxiv_id": null,
"license": "Selected packets label their content CC0-1.0; original references retain their own terms",
"notes": "Selected problem... | theorem erdos_939.variants.triples :
answer(sorry) ↔ {(a,b,c) | ({a, b, c} : Finset ℕ).Coprime ∧
(3).Full a ∧ (3).Full b ∧ (3).Full c ∧
a + b = c}.Infinite | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem erdos_939.variants.triples :\n answer(sorry) ↔ {(a,b,c) | ({a, b, c} : Finset ℕ).Coprime ∧\n (3).Full a ∧ (3).Full b ∧ (3).Full c ∧\n a + b = c}.Infinite",
"target": null,
"source_url": "https://theoremdb.org/statements/P8766",
"format": "dec... | [] | TheoremDB P8766 · Candidate awaiting source review | null | Source-reported assessment, retrieved 8 September 2026. Frontier Science has not independently reviewed this status.
Open. No signed Lean verification is attached to this problem or its selected resolution.
This record is in the source backlog; its novelty and unresolved status have not passed source review. | null | published | incremental | hard | [
"source-candidate",
"theoremdb"
] | null | null | null | null | [
{
"kind": "website",
"citation": "TheoremDB P8766",
"url": "https://theoremdb.org/statements/P8766",
"note": null,
"sort_order": 0
},
{
"kind": "formalization",
"citation": "Source formal statement (not locally verified)",
"url": "https://theoremdb.org/statements/P8766",
"not... | {
"source_record": "P8766",
"imported_at": "2026-09-08",
"status_as_of": "2026-09-08",
"evidence_label": "Source candidate · not reviewed",
"note": "Full public directory statement. Source lane: backlog; source resolution: open. Source verification claims are attributed only; no local Lean verification has be... | {"created_at":"2026-08-17T19:14:53+00:00","display_standard":{"editorial_review":"submitter_supplied","formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":8766,"public_status":{"evidence":{"hasFormalization":false,"hasInformalProof":false,"hasRecordedWork":false,"solutionRecord":null},"lean":{... | [
"formalization-contains-placeholder",
"formalization-not-compiled-during-export",
"current-status-unassessed",
"source-full-text-not-present"
] | [] | {
"created_at": "2026-09-08T03:32:15.935Z",
"updated_at": "2026-09-08T03:32:15.935Z",
"published_at": "2026-09-08T03:33:00.859Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"fe0351c9-97d5-407a-923f-9683bff0ef80","format_issues":null,"number":5605} |
1.0.0 | problem:theoremdb-p8768 | problem | theoremdb-p8768 | Multiplicative Triples in Open Subsets of the Unit Interval | {
"main": "math",
"sub": "number theory",
"slug": "number-theory",
"name": "Number theory",
"original_discipline": "mathematics"
} | [] | [
{
"code": "11",
"basis": "community-map",
"evidence": "Inferred from community field: number-theory"
}
] | Suppose that $A \subset [0,1]$ is an open set and has Lebesgue measure greater than $\frac{1}{3}$. Does there always exist a multiplicative triple $(x,y,z)$ with $x \in A$, $y \in A$, $z \in A$, and $x \cdot y = z$? | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P8768",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P8768",
"citation": null,
"doi": null,
"arxiv_id": null,
"license": "Selected packets label their content CC0-1.0; original references retain their own terms",
"notes": "Selected problem... | theorem green_3 :
answer(sorry) ↔ ∀ A : Set ℝ,
IsOpen A → A ⊆ Icc 0 1 → volume A > 1/3 →
∃ x y z, x ∈ A ∧ y ∈ A ∧ z ∈ A ∧ x * y = z | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem green_3 :\n answer(sorry) ↔ ∀ A : Set ℝ,\n IsOpen A → A ⊆ Icc 0 1 → volume A > 1/3 →\n ∃ x y z, x ∈ A ∧ y ∈ A ∧ z ∈ A ∧ x * y = z",
"target": null,
"source_url": "https://theoremdb.org/statements/P8768",
"format": "declaration-or-fragment",... | [] | TheoremDB P8768 · Candidate awaiting source review | null | Source-reported assessment, retrieved 8 September 2026. Frontier Science has not independently reviewed this status.
Open. No signed Lean verification is attached to this problem or its selected resolution.
This record is in the source backlog; its novelty and unresolved status have not passed source review. | null | published | incremental | hard | [
"source-candidate",
"theoremdb"
] | null | null | null | null | [
{
"kind": "website",
"citation": "TheoremDB P8768",
"url": "https://theoremdb.org/statements/P8768",
"note": null,
"sort_order": 0
},
{
"kind": "formalization",
"citation": "Source formal statement (not locally verified)",
"url": "https://theoremdb.org/statements/P8768",
"not... | {
"source_record": "P8768",
"imported_at": "2026-09-08",
"status_as_of": "2026-09-08",
"evidence_label": "Source candidate · not reviewed",
"note": "Full public directory statement. Source lane: backlog; source resolution: open. Source verification claims are attributed only; no local Lean verification has be... | {"created_at":"2026-08-17T19:15:01+00:00","display_standard":{"editorial_review":"submitter_supplied","formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":8768,"public_status":{"evidence":{"hasFormalization":false,"hasInformalProof":false,"hasRecordedWork":false,"solutionRecord":null},"lean":{... | [
"formalization-contains-placeholder",
"formalization-not-compiled-during-export",
"current-status-unassessed",
"source-full-text-not-present"
] | [] | {
"created_at": "2026-09-08T03:32:15.935Z",
"updated_at": "2026-09-08T03:32:15.935Z",
"published_at": "2026-09-08T03:33:04.176Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"af8817b4-53b0-4564-b31a-9ff028799c0f","format_issues":null,"number":6255} |
1.0.0 | problem:theoremdb-p8770 | problem | theoremdb-p8770 | Agrawal's primality conjecture | {
"main": "math",
"sub": "number theory",
"slug": "number-theory",
"name": "Number theory",
"original_discipline": "mathematics"
} | [] | [
{
"code": "11",
"basis": "community-map",
"evidence": "Inferred from community field: number-theory"
}
] | Let $n$ and $r$ be natural numbers with $n > 1$, $r > 0$, and $\gcd(n,r) = 1$. Let $R = (\mathbb{Z}/n\mathbb{Z})[X]$ be the polynomial ring over the integers modulo $n$, and let $I$ be the ideal of $R$ generated by $X^r - 1$. Consider the quotient ring $R/I$ and the canonical projection map from $R$ to $R/I$. Is it tru... | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P8770",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P8770",
"citation": null,
"doi": null,
"arxiv_id": null,
"license": "Selected packets label their content CC0-1.0; original references retain their own terms",
"notes": "Selected problem... | theorem agrawal_conjecture :
answer(sorry) ↔
∀ (n r : ℕ), n > 1 → r > 0 → n.gcd r = 1 →
let R := Polynomial (ZMod n)
let X : R := Polynomial.X
let I : Ideal R := Ideal.span ({X^r - 1} : Set R)
Ideal.Quotient.mk I ((X - 1)^n) = Ideal.Quotient.mk I (X^n - 1) →
(n.Prime ∨ (n^2 : ZMod r)... | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem agrawal_conjecture :\n answer(sorry) ↔\n ∀ (n r : ℕ), n > 1 → r > 0 → n.gcd r = 1 →\n let R := Polynomial (ZMod n)\n let X : R := Polynomial.X\n let I : Ideal R := Ideal.span ({X^r - 1} : Set R)\n Ideal.Quotient.mk I ((X - 1)^n) = Ideal.Quotie... | [] | TheoremDB P8770 · Candidate awaiting source review | null | Source-reported assessment, retrieved 8 September 2026. Frontier Science has not independently reviewed this status.
Open. No signed Lean verification is attached to this problem or its selected resolution.
This record is in the source backlog; its novelty and unresolved status have not passed source review. | null | published | incremental | hard | [
"source-candidate",
"theoremdb"
] | null | null | null | null | [
{
"kind": "website",
"citation": "TheoremDB P8770",
"url": "https://theoremdb.org/statements/P8770",
"note": null,
"sort_order": 0
},
{
"kind": "formalization",
"citation": "Source formal statement (not locally verified)",
"url": "https://theoremdb.org/statements/P8770",
"not... | {
"source_record": "P8770",
"imported_at": "2026-09-08",
"status_as_of": "2026-09-08",
"evidence_label": "Source candidate · not reviewed",
"note": "Full public directory statement. Source lane: backlog; source resolution: open. Source verification claims are attributed only; no local Lean verification has be... | {"created_at":"2026-08-17T19:15:14+00:00","display_standard":{"editorial_review":"submitter_supplied","formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":8770,"public_status":{"evidence":{"hasFormalization":false,"hasInformalProof":false,"hasRecordedWork":false,"solutionRecord":null},"lean":{... | [
"formalization-contains-placeholder",
"formalization-not-compiled-during-export",
"current-status-unassessed",
"source-full-text-not-present"
] | [] | {
"created_at": "2026-09-08T03:32:15.935Z",
"updated_at": "2026-09-08T03:32:15.935Z",
"published_at": "2026-09-08T03:32:49.501Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"0f209688-ef39-4d29-9325-36ac83c6488d","format_issues":null,"number":3221} |
1.0.0 | problem:theoremdb-p8772 | problem | theoremdb-p8772 | Asymptotic growth of the least missing even prime gap | {
"main": "math",
"sub": "number theory",
"slug": "number-theory",
"name": "Number theory",
"original_discipline": "mathematics"
} | [] | [
{
"code": "11",
"basis": "community-map",
"evidence": "Inferred from community field: number-theory"
}
] | Let $p_n$ denote the $n$th prime number, and define $d_n = p_{n+1} - p_n$ to be the $n$th prime gap. For each natural number $x$, define $r(x)$ to be the smallest positive even integer $t$ such that there exists no index $n \leq x$ with $d_n = t$. Does $r(x) / \log x \to \infty$ as $x \to \infty$? | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P8772",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P8772",
"citation": null,
"doi": null,
"arxiv_id": null,
"license": "Selected packets label their content CC0-1.0; original references retain their own terms",
"notes": "Selected problem... | theorem erdos_853.parts.ii :
atTop.Tendsto (fun n ↦ r n / Real.log n) atTop | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem erdos_853.parts.ii :\n atTop.Tendsto (fun n ↦ r n / Real.log n) atTop",
"target": null,
"source_url": "https://theoremdb.org/statements/P8772",
"format": "declaration-or-fragment",
"sha256": "efcf03263a1d3a25b9087053daf913c164a801e95182fb406897e857... | [] | TheoremDB P8772 · Candidate awaiting source review | null | Source-reported assessment, retrieved 8 September 2026. Frontier Science has not independently reviewed this status.
Open. No signed Lean verification is attached to this problem or its selected resolution.
This record is in the source backlog; its novelty and unresolved status have not passed source review. | null | published | incremental | hard | [
"source-candidate",
"theoremdb"
] | null | null | null | null | [
{
"kind": "website",
"citation": "TheoremDB P8772",
"url": "https://theoremdb.org/statements/P8772",
"note": null,
"sort_order": 0
},
{
"kind": "formalization",
"citation": "Source formal statement (not locally verified)",
"url": "https://theoremdb.org/statements/P8772",
"not... | {
"source_record": "P8772",
"imported_at": "2026-09-08",
"status_as_of": "2026-09-08",
"evidence_label": "Source candidate · not reviewed",
"note": "Full public directory statement. Source lane: backlog; source resolution: open. Source verification claims are attributed only; no local Lean verification has be... | {"created_at":"2026-08-17T19:15:18+00:00","display_standard":{"editorial_review":"submitter_supplied","formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":8772,"public_status":{"evidence":{"hasFormalization":false,"hasInformalProof":false,"hasRecordedWork":false,"solutionRecord":null},"lean":{... | [
"formalization-not-compiled-during-export",
"current-status-unassessed",
"source-full-text-not-present"
] | [] | {
"created_at": "2026-09-08T03:32:15.935Z",
"updated_at": "2026-09-08T03:32:15.935Z",
"published_at": "2026-09-08T03:32:50.188Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"3e2bb531-d5ca-4da0-801f-8c167d1723c2","format_issues":null,"number":3369} |
1.0.0 | problem:theoremdb-p8776 | problem | theoremdb-p8776 | Erdős's problem on prime factors of shifted consecutive integers | {
"main": "math",
"sub": "number theory",
"slug": "number-theory",
"name": "Number theory",
"original_discipline": "mathematics"
} | [] | [
{
"code": "11",
"basis": "community-map",
"evidence": "Inferred from community field: number-theory"
}
] | For each pair of natural numbers $(n,k)$, let $v(n,k)$ denote the number of prime factors of $n+k$ which do not divide $n+i$ for any integer $i$ with $0 \leq i < k$. Define $v_0(n) = \max_{k \geq 0} v(n,k)$. Does $v_0(n) \to \infty$ as $n \to \infty$? | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P8776",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P8776",
"citation": null,
"doi": null,
"arxiv_id": null,
"license": "Selected packets label their content CC0-1.0; original references retain their own terms",
"notes": "Selected problem... | theorem erdos_889 : Tendsto v₀ atTop (𝓝 ⊤) | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem erdos_889 : Tendsto v₀ atTop (𝓝 ⊤)",
"target": null,
"source_url": "https://theoremdb.org/statements/P8776",
"format": "declaration-or-fragment",
"sha256": "4fca0447c8f0bb95ef838cf32e5d12af98ac5dd51cc58443a60b08875fec65bd",
"license": "Selected p... | [] | TheoremDB P8776 · Candidate awaiting source review | null | Source-reported assessment, retrieved 8 September 2026. Frontier Science has not independently reviewed this status.
Open. No signed Lean verification is attached to this problem or its selected resolution.
This record is in the source backlog; its novelty and unresolved status have not passed source review. | null | published | incremental | hard | [
"source-candidate",
"theoremdb"
] | null | null | null | null | [
{
"kind": "website",
"citation": "TheoremDB P8776",
"url": "https://theoremdb.org/statements/P8776",
"note": null,
"sort_order": 0
},
{
"kind": "formalization",
"citation": "Source formal statement (not locally verified)",
"url": "https://theoremdb.org/statements/P8776",
"not... | {
"source_record": "P8776",
"imported_at": "2026-09-08",
"status_as_of": "2026-09-08",
"evidence_label": "Source candidate · not reviewed",
"note": "Full public directory statement. Source lane: backlog; source resolution: open. Source verification claims are attributed only; no local Lean verification has be... | {"created_at":"2026-08-17T19:18:34+00:00","display_standard":{"editorial_review":"submitter_supplied","formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":8776,"public_status":{"evidence":{"hasFormalization":false,"hasInformalProof":false,"hasRecordedWork":false,"solutionRecord":null},"lean":{... | [
"formalization-not-compiled-during-export",
"current-status-unassessed",
"source-full-text-not-present"
] | [] | {
"created_at": "2026-09-08T03:32:15.935Z",
"updated_at": "2026-09-08T03:32:15.935Z",
"published_at": "2026-09-08T03:32:56.523Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"14876064-baac-4f4d-8b02-a90acfe02906","format_issues":null,"number":4759} |
1.0.0 | problem:theoremdb-p8778 | problem | theoremdb-p8778 | Improving the lower bound for residue class coverings of intervals | {
"main": "math",
"sub": "number theory",
"slug": "number-theory",
"name": "Number theory",
"original_discipline": "mathematics"
} | [] | [
{
"code": "11",
"basis": "community-map",
"evidence": "Inferred from community field: number-theory"
}
] | For each natural number $x$, let $\mathcal{P}_{\leq x}$ denote the set of primes $p \leq x$. For natural numbers $x$ and $y$, we say that the interval $[1, y]$ is covered by residue classes modulo primes up to $x$ if there exists a function $a: \mathbb{N} \to \mathbb{N}$ such that for every integer $m$ with $1 \leq m \... | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P8778",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P8778",
"citation": null,
"doi": null,
"arxiv_id": null,
"license": "Selected packets label their content CC0-1.0; original references retain their own terms",
"notes": "Selected problem... | theorem green_46.improve_lower :
let ans := (answer(sorry) : ℕ → ℝ)
(bestLower =o[atTop] ans) ∧ (ans ≪ maxY) | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem green_46.improve_lower :\n let ans := (answer(sorry) : ℕ → ℝ)\n (bestLower =o[atTop] ans) ∧ (ans ≪ maxY)",
"target": null,
"source_url": "https://theoremdb.org/statements/P8778",
"format": "declaration-or-fragment",
"sha256": "359b6044e05be6f1bf... | [] | TheoremDB P8778 · Candidate awaiting source review | null | Source-reported assessment, retrieved 8 September 2026. Frontier Science has not independently reviewed this status.
Open. No signed Lean verification is attached to this problem or its selected resolution.
This record is in the source backlog; its novelty and unresolved status have not passed source review. | null | published | incremental | hard | [
"source-candidate",
"theoremdb"
] | null | null | null | null | [
{
"kind": "website",
"citation": "TheoremDB P8778",
"url": "https://theoremdb.org/statements/P8778",
"note": null,
"sort_order": 0
},
{
"kind": "formalization",
"citation": "Source formal statement (not locally verified)",
"url": "https://theoremdb.org/statements/P8778",
"not... | {
"source_record": "P8778",
"imported_at": "2026-09-08",
"status_as_of": "2026-09-08",
"evidence_label": "Source candidate · not reviewed",
"note": "Full public directory statement. Source lane: backlog; source resolution: open. Source verification claims are attributed only; no local Lean verification has be... | {"created_at":"2026-08-17T19:19:23+00:00","display_standard":{"editorial_review":"submitter_supplied","formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":8778,"public_status":{"evidence":{"hasFormalization":false,"hasInformalProof":false,"hasRecordedWork":false,"solutionRecord":null},"lean":{... | [
"formalization-contains-placeholder",
"formalization-not-compiled-during-export",
"current-status-unassessed",
"source-full-text-not-present"
] | [] | {
"created_at": "2026-09-08T03:32:15.935Z",
"updated_at": "2026-09-08T03:32:15.935Z",
"published_at": "2026-09-08T03:32:59.676Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"da0a04ec-105c-480b-9c11-78b851bb4d8c","format_issues":null,"number":5446} |
1.0.0 | problem:theoremdb-p8780 | problem | theoremdb-p8780 | Independent Domination Number Bound for Odd Maximum Degree | {
"main": "math",
"sub": "combinatorics",
"slug": "combinatorics",
"name": "Combinatorics",
"original_discipline": "mathematics"
} | [] | [
{
"code": "05",
"basis": "community-map",
"evidence": "Inferred from community field: combinatorics"
}
] | Let $G$ be a finite simple graph on $n$ vertices with no isolated vertices. Suppose that the maximum degree $D$ of $G$ is odd. Prove that
$$
(D + 1)(D + 3) \cdot i(G) \leq (D^2 + 3) \cdot n,
$$
where $i(G)$ denotes the independent domination number of $G$. | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P8780",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P8780",
"citation": null,
"doi": null,
"arxiv_id": null,
"license": "Selected packets label their content CC0-1.0; original references retain their own terms",
"notes": "Selected problem... | theorem independentDominationOdd (hIso : 0 < G.minDegree) (hOdd : Odd G.maxDegree) :
let D := G.maxDegree
let i := G.indepDominationNumber
let n := Fintype.card V
(D + 1) * (D + 3) * i ≤ (D^2 + 3) * n | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem independentDominationOdd (hIso : 0 < G.minDegree) (hOdd : Odd G.maxDegree) :\n let D := G.maxDegree\n let i := G.indepDominationNumber\n let n := Fintype.card V\n (D + 1) * (D + 3) * i ≤ (D^2 + 3) * n",
"target": null,
"source_url": "https://theor... | [] | TheoremDB P8780 · Candidate awaiting source review | null | Source-reported assessment, retrieved 8 September 2026. Frontier Science has not independently reviewed this status.
Open. No signed Lean verification is attached to this problem or its selected resolution.
This record is in the source backlog; its novelty and unresolved status have not passed source review. | null | published | incremental | hard | [
"source-candidate",
"theoremdb"
] | null | null | null | null | [
{
"kind": "website",
"citation": "TheoremDB P8780",
"url": "https://theoremdb.org/statements/P8780",
"note": null,
"sort_order": 0
},
{
"kind": "formalization",
"citation": "Source formal statement (not locally verified)",
"url": "https://theoremdb.org/statements/P8780",
"not... | {
"source_record": "P8780",
"imported_at": "2026-09-08",
"status_as_of": "2026-09-08",
"evidence_label": "Source candidate · not reviewed",
"note": "Full public directory statement. Source lane: backlog; source resolution: open. Source verification claims are attributed only; no local Lean verification has be... | {"created_at":"2026-08-17T19:19:43+00:00","display_standard":{"formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":8780,"public_status":{"evidence":{"hasFormalization":false,"hasInformalProof":false,"hasRecordedWork":false,"solutionRecord":null},"lean":{"kernelAccepted":false,"label":"","reaso... | [
"formalization-not-compiled-during-export",
"current-status-unassessed",
"source-full-text-not-present"
] | [] | {
"created_at": "2026-09-08T03:32:15.935Z",
"updated_at": "2026-09-08T03:32:15.935Z",
"published_at": "2026-09-08T03:33:00.793Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"243467d0-008f-4dc2-a8fe-cbd9f80e94e3","format_issues":null,"number":5591} |
1.0.0 | problem:theoremdb-p8782 | problem | theoremdb-p8782 | Upper bound for the corner-density exponent in vector spaces over the field of two elements | {
"main": "math",
"sub": "combinatorics",
"slug": "combinatorics",
"name": "Combinatorics",
"original_discipline": "mathematics"
} | [] | [
{
"code": "05",
"basis": "community-map",
"evidence": "Inferred from community field: combinatorics"
}
] | Let $\mathbb{F}_2^n$ denote the $n$-dimensional vector space over the field with two elements. For a finite set $A \subseteq \mathbb{F}_2^n \times \mathbb{F}_2^n$ and an element $d \in \mathbb{F}_2^n$, define a corner in $A$ with common difference $d$ to be a triple of points $(x, y), (x+d, y), (x, y+d) \in A$ for some... | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P8782",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P8782",
"citation": null,
"doi": null,
"arxiv_id": null,
"license": "Selected packets label their content CC0-1.0; original references retain their own terms",
"notes": "Selected problem... | theorem green_19.upper : C <= 4 | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem green_19.upper : C <= 4",
"target": null,
"source_url": "https://theoremdb.org/statements/P8782",
"format": "declaration-or-fragment",
"sha256": "fb1b1b8e769cdf9cc8b290fa772f32c1680148d6f3ba4f86273340807590d5ef",
"license": "Selected packets label... | [] | TheoremDB P8782 · Candidate awaiting source review | null | Source-reported assessment, retrieved 8 September 2026. Frontier Science has not independently reviewed this status.
Open. No signed Lean verification is attached to this problem or its selected resolution.
This record is in the source backlog; its novelty and unresolved status have not passed source review. | null | published | incremental | hard | [
"source-candidate",
"theoremdb"
] | null | null | null | null | [
{
"kind": "website",
"citation": "TheoremDB P8782",
"url": "https://theoremdb.org/statements/P8782",
"note": null,
"sort_order": 0
},
{
"kind": "formalization",
"citation": "Source formal statement (not locally verified)",
"url": "https://theoremdb.org/statements/P8782",
"not... | {
"source_record": "P8782",
"imported_at": "2026-09-08",
"status_as_of": "2026-09-08",
"evidence_label": "Source candidate · not reviewed",
"note": "Full public directory statement. Source lane: backlog; source resolution: open. Source verification claims are attributed only; no local Lean verification has be... | {"created_at":"2026-08-17T19:19:59+00:00","display_standard":{"editorial_review":"submitter_supplied","formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":8782,"public_status":{"evidence":{"hasFormalization":false,"hasInformalProof":false,"hasRecordedWork":false,"solutionRecord":null},"lean":{... | [
"formalization-not-compiled-during-export",
"current-status-unassessed",
"source-full-text-not-present"
] | [] | {
"created_at": "2026-09-08T03:32:15.935Z",
"updated_at": "2026-09-08T03:32:15.935Z",
"published_at": "2026-09-08T03:33:07.333Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"ebdea2cc-ef5e-486c-9461-3ec235e0ab10","format_issues":null,"number":6937} |
1.0.0 | problem:theoremdb-p8784 | problem | theoremdb-p8784 | Erdős Problem 623 on Choice-Free Independent Sets | {
"main": "math",
"sub": "logic and foundations",
"slug": "logic-and-foundations",
"name": "Logic and foundations",
"original_discipline": "mathematics"
} | [] | [
{
"code": "03",
"basis": "community-map",
"evidence": "Inferred from community field: logic-and-foundations"
}
] | Let $X$ be a set of cardinality $\aleph_\omega$, and let $f$ be a function from the finite subsets of $X$ to $X$ such that $f(A) \notin A$ for every finite subset $A \subseteq X$. Must there exist an infinite subset $Y \subseteq X$ such that $f(B) \notin Y$ for every finite subset $B \subseteq Y$? | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P8784",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P8784",
"citation": null,
"doi": null,
"arxiv_id": null,
"license": "Selected packets label their content CC0-1.0; original references retain their own terms",
"notes": "Selected problem... | theorem erdos_623 : answer(sorry) ↔ ∀ (X : Type u) (hX : #X = ℵ_ ω)
(f : Finset X → X), (∀ A : Finset X, f A ∉ A) →
(∃ Y : Set X, Set.Infinite Y ∧ (∀ (B : Finset X), ↑B ⊆ Y → f B ∉ Y)) | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem erdos_623 : answer(sorry) ↔ ∀ (X : Type u) (hX : #X = ℵ_ ω)\n (f : Finset X → X), (∀ A : Finset X, f A ∉ A) →\n (∃ Y : Set X, Set.Infinite Y ∧ (∀ (B : Finset X), ↑B ⊆ Y → f B ∉ Y))",
"target": null,
"source_url": "https://theoremdb.org/statements/P8784"... | [] | TheoremDB P8784 · Candidate awaiting source review | null | Source-reported assessment, retrieved 8 September 2026. Frontier Science has not independently reviewed this status.
Open. No signed Lean verification is attached to this problem or its selected resolution.
This record is in the source backlog; its novelty and unresolved status have not passed source review. | null | published | incremental | hard | [
"source-candidate",
"theoremdb"
] | null | null | null | null | [
{
"kind": "website",
"citation": "TheoremDB P8784",
"url": "https://theoremdb.org/statements/P8784",
"note": null,
"sort_order": 0
},
{
"kind": "formalization",
"citation": "Source formal statement (not locally verified)",
"url": "https://theoremdb.org/statements/P8784",
"not... | {
"source_record": "P8784",
"imported_at": "2026-09-08",
"status_as_of": "2026-09-08",
"evidence_label": "Source candidate · not reviewed",
"note": "Full public directory statement. Source lane: backlog; source resolution: open. Source verification claims are attributed only; no local Lean verification has be... | {"created_at":"2026-08-17T19:20:25+00:00","display_standard":{"formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":8784,"public_status":{"evidence":{"hasFormalization":false,"hasInformalProof":false,"hasRecordedWork":false,"solutionRecord":null},"lean":{"kernelAccepted":false,"label":"","reaso... | [
"formalization-contains-placeholder",
"formalization-not-compiled-during-export",
"current-status-unassessed",
"source-full-text-not-present"
] | [] | {
"created_at": "2026-09-08T03:32:15.935Z",
"updated_at": "2026-09-08T03:32:15.935Z",
"published_at": "2026-09-08T03:32:55.752Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"9d0d1060-57ad-4cc0-bfe1-870a98a9b9b8","format_issues":null,"number":4589} |
1.0.0 | problem:theoremdb-p8786 | problem | theoremdb-p8786 | Integer Exponents from Integer Powers | {
"main": "math",
"sub": "number theory",
"slug": "number-theory",
"name": "Number theory",
"original_discipline": "mathematics"
} | [] | [
{
"code": "11",
"basis": "community-map",
"evidence": "Inferred from community field: number-theory"
}
] | Let $x$ be a real number such that $2^x$ is a natural number and $3^x$ is a natural number. Must $x$ be a natural number? | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P8786",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P8786",
"citation": null,
"doi": null,
"arxiv_id": null,
"license": "Selected packets label their content CC0-1.0; original references retain their own terms",
"notes": "Selected problem... | theorem mathoverflow_17560 {x : ℝ} (hx : ∃ m : ℕ, (2 : ℝ) ^ x = m) (hx' : ∃ m : ℕ, (3 : ℝ) ^ x = m) :
∃ m : ℕ, x = m | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem mathoverflow_17560 {x : ℝ} (hx : ∃ m : ℕ, (2 : ℝ) ^ x = m) (hx' : ∃ m : ℕ, (3 : ℝ) ^ x = m) :\n ∃ m : ℕ, x = m",
"target": null,
"source_url": "https://theoremdb.org/statements/P8786",
"format": "declaration-or-fragment",
"sha256": "d1f4a07b822b988... | [] | TheoremDB P8786 · Candidate awaiting source review | null | Source-reported assessment, retrieved 8 September 2026. Frontier Science has not independently reviewed this status.
Open. No signed Lean verification is attached to this problem or its selected resolution.
This record is in the source backlog; its novelty and unresolved status have not passed source review. | null | published | incremental | hard | [
"source-candidate",
"theoremdb"
] | null | null | null | null | [
{
"kind": "website",
"citation": "TheoremDB P8786",
"url": "https://theoremdb.org/statements/P8786",
"note": null,
"sort_order": 0
},
{
"kind": "formalization",
"citation": "Source formal statement (not locally verified)",
"url": "https://theoremdb.org/statements/P8786",
"not... | {
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"note": "Full public directory statement. Source lane: backlog; source resolution: open. Source verification claims are attributed only; no local Lean verification has be... | {"created_at":"2026-08-17T19:20:35+00:00","display_standard":{"editorial_review":"submitter_supplied","formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":8786,"public_status":{"evidence":{"hasFormalization":false,"hasInformalProof":false,"hasRecordedWork":false,"solutionRecord":null},"lean":{... | [
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"updated_at": "2026-09-08T03:32:15.935Z",
"published_at": "2026-09-08T03:33:01.295Z",
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} | 2026-10-10T16:07:49.246Z | {"database_id":"f427ae8d-4efc-4a3b-9ad1-3834ac4bc91c","format_issues":null,"number":5699} |
1.0.0 | problem:theoremdb-p8788 | problem | theoremdb-p8788 | Infinitude of sexy primes | {
"main": "math",
"sub": "number theory",
"slug": "number-theory",
"name": "Number theory",
"original_discipline": "mathematics"
} | [] | [
{
"code": "11",
"basis": "community-map",
"evidence": "Inferred from community field: number-theory"
}
] | A pair of prime numbers is called sexy if they differ by exactly $6$. Prove that there are infinitely many sexy prime pairs; equivalently, prove that the set of natural numbers $p$ such that both $p$ and $p+6$ are prime is infinite. | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P8788",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P8788",
"citation": null,
"doi": null,
"arxiv_id": null,
"license": "Selected packets label their content CC0-1.0; original references retain their own terms",
"notes": "Selected problem... | theorem infinite_sexy_primes :
Infinite {p : ℕ | Prime p ∧ Prime (p + 6)} | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem infinite_sexy_primes :\n Infinite {p : ℕ | Prime p ∧ Prime (p + 6)}",
"target": null,
"source_url": "https://theoremdb.org/statements/P8788",
"format": "declaration-or-fragment",
"sha256": "ef8a69e6181a913f93da43db4dae5a18ce83345613e9eacc15707792f0... | [] | TheoremDB P8788 · Candidate awaiting source review | null | Source-reported assessment, retrieved 8 September 2026. Frontier Science has not independently reviewed this status.
Open. No signed Lean verification is attached to this problem or its selected resolution.
This record is in the source backlog; its novelty and unresolved status have not passed source review. | null | published | incremental | hard | [
"source-candidate",
"theoremdb"
] | null | null | null | null | [
{
"kind": "website",
"citation": "TheoremDB P8788",
"url": "https://theoremdb.org/statements/P8788",
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"citation": "Source formal statement (not locally verified)",
"url": "https://theoremdb.org/statements/P8788",
"not... | {
"source_record": "P8788",
"imported_at": "2026-09-08",
"status_as_of": "2026-09-08",
"evidence_label": "Source candidate · not reviewed",
"note": "Full public directory statement. Source lane: backlog; source resolution: open. Source verification claims are attributed only; no local Lean verification has be... | {"created_at":"2026-08-17T19:20:44+00:00","display_standard":{"editorial_review":"submitter_supplied","formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":8788,"public_status":{"evidence":{"hasFormalization":false,"hasInformalProof":false,"hasRecordedWork":false,"solutionRecord":null},"lean":{... | [
"formalization-not-compiled-during-export",
"current-status-unassessed",
"source-full-text-not-present"
] | [] | {
"created_at": "2026-09-08T03:32:15.935Z",
"updated_at": "2026-09-08T03:32:15.935Z",
"published_at": "2026-09-08T03:33:01.261Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"94271691-3e1b-4088-a496-ebe81ed84572","format_issues":null,"number":5692} |
1.0.0 | problem:theoremdb-p8790 | problem | theoremdb-p8790 | Wichmann's Conjecture on Optimal Sparse Rulers | {
"main": "math",
"sub": "combinatorics",
"slug": "combinatorics",
"name": "Combinatorics",
"original_discipline": "mathematics"
} | [] | [
{
"code": "05",
"basis": "community-map",
"evidence": "Inferred from community field: combinatorics"
}
] | A sparse ruler is described by a finite sequence $g = (g_1, \ldots, g_n)$ of positive integers called segment lengths or gaps. Its marks are the partial sums $m_0 = 0 < m_1 < \cdots < m_n = L$ where $m_i = g_1 + \cdots + g_i$, and $L = g_1 + \cdots + g_n$ is the length of the ruler. A distance $k \in \{0, 1, \ldots, L\... | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P8790",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P8790",
"citation": null,
"doi": null,
"arxiv_id": null,
"license": "Selected packets label their content CC0-1.0; original references retain their own terms",
"notes": "Selected problem... | theorem wichmann_conjecture {g : List ℕ} (hopt : IsOptimal g) (hseg : 13 < g.length) :
∃ r s : ℕ, g = wichmannGaps r s ∨ g = (wichmannGaps r s).reverse | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem wichmann_conjecture {g : List ℕ} (hopt : IsOptimal g) (hseg : 13 < g.length) :\n ∃ r s : ℕ, g = wichmannGaps r s ∨ g = (wichmannGaps r s).reverse",
"target": null,
"source_url": "https://theoremdb.org/statements/P8790",
"format": "declaration-or-fragme... | [] | TheoremDB P8790 · Candidate awaiting source review | null | Source-reported assessment, retrieved 8 September 2026. Frontier Science has not independently reviewed this status.
Open. No signed Lean verification is attached to this problem or its selected resolution.
This record is in the source backlog; its novelty and unresolved status have not passed source review. | null | published | incremental | hard | [
"source-candidate",
"theoremdb"
] | null | null | null | null | [
{
"kind": "website",
"citation": "TheoremDB P8790",
"url": "https://theoremdb.org/statements/P8790",
"note": null,
"sort_order": 0
},
{
"kind": "formalization",
"citation": "Source formal statement (not locally verified)",
"url": "https://theoremdb.org/statements/P8790",
"not... | {
"source_record": "P8790",
"imported_at": "2026-09-08",
"status_as_of": "2026-09-08",
"evidence_label": "Source candidate · not reviewed",
"note": "Full public directory statement. Source lane: backlog; source resolution: open. Source verification claims are attributed only; no local Lean verification has be... | {"created_at":"2026-08-17T19:20:59+00:00","display_standard":{"editorial_review":"submitter_supplied","formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":8790,"public_status":{"evidence":{"hasFormalization":false,"hasInformalProof":false,"hasRecordedWork":false,"solutionRecord":null},"lean":{... | [
"formalization-not-compiled-during-export",
"current-status-unassessed",
"source-full-text-not-present"
] | [] | {
"created_at": "2026-09-08T03:32:15.935Z",
"updated_at": "2026-09-08T03:32:15.935Z",
"published_at": "2026-09-08T03:33:07.551Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"ea85440d-a0f9-450d-9603-ec20398314b5","format_issues":null,"number":6984} |
1.0.0 | problem:theoremdb-p8792 | problem | theoremdb-p8792 | Büchi's problem for five consecutive squares | {
"main": "math",
"sub": "number theory",
"slug": "number-theory",
"name": "Number theory",
"original_discipline": "mathematics"
} | [] | [
{
"code": "11",
"basis": "community-map",
"evidence": "Inferred from community field: number-theory"
}
] | For integers $x$ and $a$, suppose that $(x+n)^2 + a$ is a perfect square for each $n \in \{0, 1, 2, 3, 4\}$. Must $a = 0$? In other words, if five consecutive values of the form $(x+n)^2 + a$ are all perfect squares, does it follow that $a = 0$? | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P8792",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P8792",
"citation": null,
"doi": null,
"arxiv_id": null,
"license": "Selected packets label their content CC0-1.0; original references retain their own terms",
"notes": "Selected problem... | theorem buchi_problem_M5 : IsBuchi 5 | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem buchi_problem_M5 : IsBuchi 5",
"target": null,
"source_url": "https://theoremdb.org/statements/P8792",
"format": "declaration-or-fragment",
"sha256": "8958a34225d472c7a671cc3e43f76c1553be48d3ffa5d0d21bb82500c5c2e988",
"license": "Selected packets ... | [] | TheoremDB P8792 · Candidate awaiting source review | null | Source-reported assessment, retrieved 8 September 2026. Frontier Science has not independently reviewed this status.
Open. No signed Lean verification is attached to this problem or its selected resolution.
This record is in the source backlog; its novelty and unresolved status have not passed source review. | null | published | incremental | hard | [
"source-candidate",
"theoremdb"
] | null | null | null | null | [
{
"kind": "website",
"citation": "TheoremDB P8792",
"url": "https://theoremdb.org/statements/P8792",
"note": null,
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},
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"citation": "Source formal statement (not locally verified)",
"url": "https://theoremdb.org/statements/P8792",
"not... | {
"source_record": "P8792",
"imported_at": "2026-09-08",
"status_as_of": "2026-09-08",
"evidence_label": "Source candidate · not reviewed",
"note": "Full public directory statement. Source lane: backlog; source resolution: open. Source verification claims are attributed only; no local Lean verification has be... | {"created_at":"2026-08-17T19:21:08+00:00","display_standard":{"editorial_review":"submitter_supplied","formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":8792,"public_status":{"evidence":{"hasFormalization":false,"hasInformalProof":false,"hasRecordedWork":false,"solutionRecord":null},"lean":{... | [
"formalization-not-compiled-during-export",
"current-status-unassessed",
"source-full-text-not-present"
] | [] | {
"created_at": "2026-09-08T03:32:15.935Z",
"updated_at": "2026-09-08T03:32:15.935Z",
"published_at": "2026-09-08T03:32:50.804Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"a15eb673-cfc6-4261-b939-a485041a44e5","format_issues":null,"number":3496} |
1.0.0 | problem:theoremdb-p8794 | problem | theoremdb-p8794 | Existence of $n$-dimensional Euler hyperbricks for all dimensions greater than three | {
"main": "math",
"sub": "number theory",
"slug": "number-theory",
"name": "Number theory",
"original_discipline": "mathematics"
} | [] | [
{
"code": "11",
"basis": "community-map",
"evidence": "Inferred from community field: number-theory"
}
] | For an integer $n \geq 2$, an Euler hyperbrick in $n$-dimensional space is a collection of $n$ positive integers $(s_1, s_2, \ldots, s_n)$ such that for every pair of distinct indices $i$ and $j$, the sum $s_i^2 + s_j^2$ is a perfect square. This generalizes the classical notion of an Euler brick, which is the case $n ... | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P8794",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P8794",
"citation": null,
"doi": null,
"arxiv_id": null,
"license": "Selected packets label their content CC0-1.0; original references retain their own terms",
"notes": "Selected problem... | theorem n_dim_euler_brick_existence :
answer(sorry) ↔ ∀ n > 3, ∃ sides : Fin n → ℕ+, IsEulerHyperBrick n sides | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem n_dim_euler_brick_existence :\nanswer(sorry) ↔ ∀ n > 3, ∃ sides : Fin n → ℕ+, IsEulerHyperBrick n sides",
"target": null,
"source_url": "https://theoremdb.org/statements/P8794",
"format": "declaration-or-fragment",
"sha256": "7fee77e5ede581c5a7d401583... | [] | TheoremDB P8794 · Candidate awaiting source review | null | Source-reported assessment, retrieved 8 September 2026. Frontier Science has not independently reviewed this status.
Open. No signed Lean verification is attached to this problem or its selected resolution.
This record is in the source backlog; its novelty and unresolved status have not passed source review. | null | published | incremental | hard | [
"source-candidate",
"theoremdb"
] | null | null | null | null | [
{
"kind": "website",
"citation": "TheoremDB P8794",
"url": "https://theoremdb.org/statements/P8794",
"note": null,
"sort_order": 0
},
{
"kind": "formalization",
"citation": "Source formal statement (not locally verified)",
"url": "https://theoremdb.org/statements/P8794",
"not... | {
"source_record": "P8794",
"imported_at": "2026-09-08",
"status_as_of": "2026-09-08",
"evidence_label": "Source candidate · not reviewed",
"note": "Full public directory statement. Source lane: backlog; source resolution: open. Source verification claims are attributed only; no local Lean verification has be... | {"created_at":"2026-08-17T19:21:22+00:00","display_standard":{"editorial_review":"submitter_supplied","formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":8794,"public_status":{"evidence":{"hasFormalization":false,"hasInformalProof":false,"hasRecordedWork":false,"solutionRecord":null},"lean":{... | [
"formalization-contains-placeholder",
"formalization-not-compiled-during-export",
"current-status-unassessed",
"source-full-text-not-present"
] | [] | {
"created_at": "2026-09-08T03:32:15.935Z",
"updated_at": "2026-09-08T03:32:15.935Z",
"published_at": "2026-09-08T03:32:58.385Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"6e6d99b5-c94f-40fd-802e-ae57292bc629","format_issues":null,"number":5172} |
1.0.0 | problem:theoremdb-p8796 | problem | theoremdb-p8796 | Asymptotic growth of the gap between successive triangle-multiplicity thresholds | {
"main": "math",
"sub": "combinatorics",
"slug": "combinatorics",
"name": "Combinatorics",
"original_discipline": "mathematics"
} | [] | [
{
"code": "05",
"basis": "community-map",
"evidence": "Inferred from community field: combinatorics"
}
] | For positive integers $n$ and $r$, let $e(n,r)$ denote the minimum integer $e$ such that every graph on $n$ vertices with at least $e$ edges, each edge contained in at least one triangle, contains some edge that lies in at least $r$ triangles. For each integer $r \geq 2$, determine whether $e(n,r+1) - e(n,r) \to \infty... | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P8796",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P8796",
"citation": null,
"doi": null,
"arxiv_id": null,
"license": "Selected packets label their content CC0-1.0; original references retain their own terms",
"notes": "Selected problem... | theorem erdos_600.parts.i :
answer(sorry) ↔ ∀ r : ℕ, 2 ≤ r →
Tendsto (fun (n : ℕ) ↦ (eFunction n (r + 1) : ℝ) - (eFunction n r : ℝ)) atTop atTop | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem erdos_600.parts.i :\n answer(sorry) ↔ ∀ r : ℕ, 2 ≤ r →\n Tendsto (fun (n : ℕ) ↦ (eFunction n (r + 1) : ℝ) - (eFunction n r : ℝ)) atTop atTop",
"target": null,
"source_url": "https://theoremdb.org/statements/P8796",
"format": "declaration-or-fragme... | [] | TheoremDB P8796 · Candidate awaiting source review | null | Source-reported assessment, retrieved 8 September 2026. Frontier Science has not independently reviewed this status.
Open. No signed Lean verification is attached to this problem or its selected resolution.
This record is in the source backlog; its novelty and unresolved status have not passed source review. | null | published | incremental | hard | [
"source-candidate",
"theoremdb"
] | null | null | null | null | [
{
"kind": "website",
"citation": "TheoremDB P8796",
"url": "https://theoremdb.org/statements/P8796",
"note": null,
"sort_order": 0
},
{
"kind": "formalization",
"citation": "Source formal statement (not locally verified)",
"url": "https://theoremdb.org/statements/P8796",
"not... | {
"source_record": "P8796",
"imported_at": "2026-09-08",
"status_as_of": "2026-09-08",
"evidence_label": "Source candidate · not reviewed",
"note": "Full public directory statement. Source lane: backlog; source resolution: open. Source verification claims are attributed only; no local Lean verification has be... | {"created_at":"2026-08-17T19:21:36+00:00","display_standard":{"editorial_review":"submitter_supplied","formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":8796,"public_status":{"evidence":{"hasFormalization":false,"hasInformalProof":false,"hasRecordedWork":false,"solutionRecord":null},"lean":{... | [
"formalization-contains-placeholder",
"formalization-not-compiled-during-export",
"current-status-unassessed",
"source-full-text-not-present"
] | [] | {
"created_at": "2026-09-08T03:32:15.935Z",
"updated_at": "2026-09-08T03:32:15.935Z",
"published_at": "2026-09-08T03:32:50.176Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"641f2e2f-e22c-4a8c-8c9a-d58b249d1a90","format_issues":null,"number":3367} |
1.0.0 | problem:theoremdb-p8798 | problem | theoremdb-p8798 | Existence of an absolutely maximally entangled state for eleven parties of local dimension six | {
"main": "math",
"sub": "combinatorics",
"slug": "combinatorics",
"name": "Combinatorics",
"original_discipline": "mathematics"
} | [] | [
{
"code": "05",
"basis": "community-map",
"evidence": "Inferred from community field: combinatorics"
}
] | Let $n \geq 2$ be an integer and let $d \geq 2$ be an integer. An $n$-partite pure state $\psi$ with local dimension $d$ is called absolutely maximally entangled (AME) if, for every subset of at most $\lfloor n/2 \rfloor$ of the parties, the reduced density matrix obtained by tracing out the complementary parties is ma... | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P8798",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P8798",
"citation": null,
"doi": null,
"arxiv_id": null,
"license": "Selected packets label their content CC0-1.0; original references retain their own terms",
"notes": "Selected problem... | theorem ame_11_6_open :
answer(sorry) ↔ ExistsAME 11 6 | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem ame_11_6_open :\n answer(sorry) ↔ ExistsAME 11 6",
"target": null,
"source_url": "https://theoremdb.org/statements/P8798",
"format": "declaration-or-fragment",
"sha256": "58a23335cc2342bf7e55002c0f9f79fd037ead39cc8519cbedc84d5b92d6be55",
"licen... | [] | TheoremDB P8798 · Candidate awaiting source review | null | Source-reported assessment, retrieved 8 September 2026. Frontier Science has not independently reviewed this status.
Open. No signed Lean verification is attached to this problem or its selected resolution.
This record is in the source backlog; its novelty and unresolved status have not passed source review. | null | published | incremental | hard | [
"source-candidate",
"theoremdb"
] | null | null | null | null | [
{
"kind": "website",
"citation": "TheoremDB P8798",
"url": "https://theoremdb.org/statements/P8798",
"note": null,
"sort_order": 0
},
{
"kind": "formalization",
"citation": "Source formal statement (not locally verified)",
"url": "https://theoremdb.org/statements/P8798",
"not... | {
"source_record": "P8798",
"imported_at": "2026-09-08",
"status_as_of": "2026-09-08",
"evidence_label": "Source candidate · not reviewed",
"note": "Full public directory statement. Source lane: backlog; source resolution: open. Source verification claims are attributed only; no local Lean verification has be... | {"created_at":"2026-08-17T19:21:49+00:00","display_standard":{"editorial_review":"submitter_supplied","formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":8798,"public_status":{"evidence":{"hasFormalization":false,"hasInformalProof":false,"hasRecordedWork":false,"solutionRecord":null},"lean":{... | [
"formalization-contains-placeholder",
"formalization-not-compiled-during-export",
"current-status-unassessed",
"source-full-text-not-present"
] | [] | {
"created_at": "2026-09-08T03:32:15.935Z",
"updated_at": "2026-09-08T03:32:15.935Z",
"published_at": "2026-09-08T03:32:58.631Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"f40c7106-337b-4634-8855-de44b8aa9f79","format_issues":null,"number":5223} |
1.0.0 | problem:theoremdb-p8800 | problem | theoremdb-p8800 | Schanuel's conjecture on the transcendence degree of exponential field extensions | {
"main": "math",
"sub": "number theory",
"slug": "number-theory",
"name": "Number theory",
"original_discipline": "mathematics"
} | [] | [
{
"code": "11",
"basis": "community-map",
"evidence": "Inferred from community field: number-theory"
}
] | Let $n$ be a natural number and let $z_1, \ldots, z_n$ be complex numbers that are linearly independent over $\mathbb{Q}$. Consider the field $\mathbb{Q}(z_1, \ldots, z_n, e^{z_1}, \ldots, e^{z_n})$ obtained by adjoining these complex numbers and their complex exponentials to the field of rational numbers. Does this fi... | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P8800",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P8800",
"citation": null,
"doi": null,
"arxiv_id": null,
"license": "Selected packets label their content CC0-1.0; original references retain their own terms",
"notes": "Selected problem... | theorem schanuel_conjecture (n : ℕ) (z : Fin n → ℂ) (h : LinearIndependent ℚ z) :
n ≤ Algebra.trdeg ℚ (adjoin ℚ (Set.range z ∪ Set.range (cexp ∘ z))) | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem schanuel_conjecture (n : ℕ) (z : Fin n → ℂ) (h : LinearIndependent ℚ z) :\n n ≤ Algebra.trdeg ℚ (adjoin ℚ (Set.range z ∪ Set.range (cexp ∘ z)))",
"target": null,
"source_url": "https://theoremdb.org/statements/P8800",
"format": "declaration-or-fragment... | [] | TheoremDB P8800 · Candidate awaiting source review | null | Source-reported assessment, retrieved 8 September 2026. Frontier Science has not independently reviewed this status.
Open. No signed Lean verification is attached to this problem or its selected resolution.
This record is in the source backlog; its novelty and unresolved status have not passed source review. | null | published | incremental | hard | [
"source-candidate",
"theoremdb"
] | null | null | null | null | [
{
"kind": "website",
"citation": "TheoremDB P8800",
"url": "https://theoremdb.org/statements/P8800",
"note": null,
"sort_order": 0
},
{
"kind": "formalization",
"citation": "Source formal statement (not locally verified)",
"url": "https://theoremdb.org/statements/P8800",
"not... | {
"source_record": "P8800",
"imported_at": "2026-09-08",
"status_as_of": "2026-09-08",
"evidence_label": "Source candidate · not reviewed",
"note": "Full public directory statement. Source lane: backlog; source resolution: open. Source verification claims are attributed only; no local Lean verification has be... | {"created_at":"2026-08-17T19:22:05+00:00","display_standard":{"editorial_review":"submitter_supplied","formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":8800,"public_status":{"evidence":{"hasFormalization":false,"hasInformalProof":false,"hasRecordedWork":false,"solutionRecord":null},"lean":{... | [
"formalization-not-compiled-during-export",
"current-status-unassessed",
"source-full-text-not-present"
] | [] | {
"created_at": "2026-09-08T03:32:15.935Z",
"updated_at": "2026-09-08T03:32:15.935Z",
"published_at": "2026-09-08T03:33:05.724Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"9e573ca5-661c-4b6a-967e-3a8b5e949503","format_issues":null,"number":6589} |
1.0.0 | problem:theoremdb-p8802 | problem | theoremdb-p8802 | Erdős's Problem on Almost Bipartite Infinite-Chromatic Graphs | {
"main": "math",
"sub": "combinatorics",
"slug": "combinatorics",
"name": "Combinatorics",
"original_discipline": "mathematics"
} | [] | [
{
"code": "05",
"basis": "community-map",
"evidence": "Inferred from community field: combinatorics"
}
] | Let a graph $G$ have infinite chromatic number, meaning its chromatic number $\chi(G)$ is not a finite cardinal. For a finite induced subgraph $A$ of $G$, define the edge distance to bipartite of $A$ to be the minimum number of edges that must be deleted from $A$ to obtain a bipartite graph. For each $n \in \mathbb{N}$... | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P8802",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P8802",
"citation": null,
"doi": null,
"arxiv_id": null,
"license": "Selected packets label their content CC0-1.0; original references retain their own terms",
"notes": "Selected problem... | theorem erdos_74 : answer(sorry) ↔ ∀ f : ℕ → ℕ, Tendsto f atTop atTop →
(∃ (V : Type u) (G : SimpleGraph V), G.chromaticNumber = ⊤ ∧
∀ n, G.maxSubgraphEdgeDistToBipartite n ≤ f n) | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem erdos_74 : answer(sorry) ↔ ∀ f : ℕ → ℕ, Tendsto f atTop atTop →\n (∃ (V : Type u) (G : SimpleGraph V), G.chromaticNumber = ⊤ ∧\n ∀ n, G.maxSubgraphEdgeDistToBipartite n ≤ f n)",
"target": null,
"source_url": "https://theoremdb.org/statements/P8802",
... | [] | TheoremDB P8802 · Candidate awaiting source review | null | Source-reported assessment, retrieved 8 September 2026. Frontier Science has not independently reviewed this status.
Open. No signed Lean verification is attached to this problem or its selected resolution.
This record is in the source backlog; its novelty and unresolved status have not passed source review. | null | published | incremental | hard | [
"source-candidate",
"theoremdb"
] | null | null | null | null | [
{
"kind": "website",
"citation": "TheoremDB P8802",
"url": "https://theoremdb.org/statements/P8802",
"note": null,
"sort_order": 0
},
{
"kind": "formalization",
"citation": "Source formal statement (not locally verified)",
"url": "https://theoremdb.org/statements/P8802",
"not... | {
"source_record": "P8802",
"imported_at": "2026-09-08",
"status_as_of": "2026-09-08",
"evidence_label": "Source candidate · not reviewed",
"note": "Full public directory statement. Source lane: backlog; source resolution: open. Source verification claims are attributed only; no local Lean verification has be... | {"created_at":"2026-08-17T19:22:20+00:00","display_standard":{"editorial_review":"submitter_supplied","formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":8802,"public_status":{"evidence":{"hasFormalization":false,"hasInformalProof":false,"hasRecordedWork":false,"solutionRecord":null},"lean":{... | [
"formalization-contains-placeholder",
"formalization-not-compiled-during-export",
"current-status-unassessed",
"source-full-text-not-present"
] | [] | {
"created_at": "2026-09-08T03:32:15.935Z",
"updated_at": "2026-09-08T03:32:15.935Z",
"published_at": "2026-09-08T03:32:56.442Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"ffcdd3af-0142-46d2-b67d-dd20974a7b2e","format_issues":null,"number":4741} |
1.0.0 | problem:theoremdb-p8804 | problem | theoremdb-p8804 | Algebraic Independence of e and π | {
"main": "math",
"sub": "analysis",
"slug": "analysis",
"name": "Analysis",
"original_discipline": "mathematics"
} | [] | [] | Let $e = \exp(1)$ denote the base of the natural logarithm, and let $\pi$ denote the ratio of the circumference of a circle to its diameter. Determine whether the set $\{e, \pi\}$ is algebraically independent over the field of rational numbers $\mathbb{Q}$. That is, does there exist a nonzero polynomial $P \in \mathbb{... | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P8804",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P8804",
"citation": null,
"doi": null,
"arxiv_id": null,
"license": "Selected packets label their content CC0-1.0; original references retain their own terms",
"notes": "Selected problem... | theorem algebraicIndependent_e_pi :
answer(sorry) ↔ AlgebraicIndependent ℚ ![e, π] | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem algebraicIndependent_e_pi :\n answer(sorry) ↔ AlgebraicIndependent ℚ ![e, π]",
"target": null,
"source_url": "https://theoremdb.org/statements/P8804",
"format": "declaration-or-fragment",
"sha256": "15dc757e88a63773aa3c5b2c1e50b725284d5c2c23e1f57c1... | [] | TheoremDB P8804 · Candidate awaiting source review | null | Source-reported assessment, retrieved 8 September 2026. Frontier Science has not independently reviewed this status.
Open. No signed Lean verification is attached to this problem or its selected resolution.
This record is in the source backlog; its novelty and unresolved status have not passed source review. | null | published | incremental | hard | [
"source-candidate",
"theoremdb"
] | null | null | null | null | [
{
"kind": "website",
"citation": "TheoremDB P8804",
"url": "https://theoremdb.org/statements/P8804",
"note": null,
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},
{
"kind": "formalization",
"citation": "Source formal statement (not locally verified)",
"url": "https://theoremdb.org/statements/P8804",
"not... | {
"source_record": "P8804",
"imported_at": "2026-09-08",
"status_as_of": "2026-09-08",
"evidence_label": "Source candidate · not reviewed",
"note": "Full public directory statement. Source lane: backlog; source resolution: open. Source verification claims are attributed only; no local Lean verification has be... | {"created_at":"2026-08-17T19:22:36+00:00","display_standard":{"editorial_review":"submitter_supplied","formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":8804,"public_status":{"evidence":{"hasFormalization":false,"hasInformalProof":false,"hasRecordedWork":false,"solutionRecord":null},"lean":{... | [
"formalization-contains-placeholder",
"formalization-not-compiled-during-export",
"current-status-unassessed",
"source-full-text-not-present"
] | [] | {
"created_at": "2026-09-08T03:32:15.935Z",
"updated_at": "2026-09-08T03:32:15.935Z",
"published_at": "2026-09-08T03:32:49.506Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"61188ae2-44c9-42f4-ad2d-bd235e4f3644","format_issues":null,"number":3222} |
1.0.0 | problem:theoremdb-p8806 | problem | theoremdb-p8806 | Kaplansky's zero-divisor conjecture for torsion-free groups | {
"main": "math",
"sub": "algebra",
"slug": "algebra",
"name": "Algebra",
"original_discipline": "mathematics"
} | [] | [] | Let $K$ be a field and let $G$ be a group that is multiplicatively torsion-free, meaning every non-identity element of $G$ has infinite order. The group algebra $K[G]$, which is the $K$-vector space with basis $G$ equipped with multiplication extended bilinearly from the group operation, is also known as the monoid alg... | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P8806",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P8806",
"citation": null,
"doi": null,
"arxiv_id": null,
"license": "Selected packets label their content CC0-1.0; original references retain their own terms",
"notes": "Selected problem... | theorem zero_divisor_conjecture : NoZeroDivisors (MonoidAlgebra K G) | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem zero_divisor_conjecture : NoZeroDivisors (MonoidAlgebra K G)",
"target": null,
"source_url": "https://theoremdb.org/statements/P8806",
"format": "declaration-or-fragment",
"sha256": "ee514ee66939ae750e74d0821fdaf6a2c6421ec5a2e17f796e5ed91812971f6d",
... | [] | TheoremDB P8806 · Candidate awaiting source review | null | Source-reported assessment, retrieved 8 September 2026. Frontier Science has not independently reviewed this status.
Open. No signed Lean verification is attached to this problem or its selected resolution.
This record is in the source backlog; its novelty and unresolved status have not passed source review. | null | published | incremental | hard | [
"source-candidate",
"theoremdb"
] | null | null | null | null | [
{
"kind": "website",
"citation": "TheoremDB P8806",
"url": "https://theoremdb.org/statements/P8806",
"note": null,
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},
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"kind": "formalization",
"citation": "Source formal statement (not locally verified)",
"url": "https://theoremdb.org/statements/P8806",
"not... | {
"source_record": "P8806",
"imported_at": "2026-09-08",
"status_as_of": "2026-09-08",
"evidence_label": "Source candidate · not reviewed",
"note": "Full public directory statement. Source lane: backlog; source resolution: open. Source verification claims are attributed only; no local Lean verification has be... | {"created_at":"2026-08-17T19:22:50+00:00","display_standard":{"editorial_review":"submitter_supplied","formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":8806,"public_status":{"evidence":{"hasFormalization":false,"hasInformalProof":false,"hasRecordedWork":false,"solutionRecord":null},"lean":{... | [
"formalization-not-compiled-during-export",
"current-status-unassessed",
"source-full-text-not-present"
] | [] | {
"created_at": "2026-09-08T03:32:15.935Z",
"updated_at": "2026-09-08T03:32:15.935Z",
"published_at": "2026-09-08T03:33:02.179Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"37aa158f-3c9d-4074-842f-a8deb56d283b","format_issues":null,"number":5821} |
1.0.0 | problem:theoremdb-p8808 | problem | theoremdb-p8808 | Asymptotic growth of van der Waerden numbers relative to powers of two | {
"main": "math",
"sub": "number theory",
"slug": "number-theory",
"name": "Number theory",
"original_discipline": "mathematics"
} | [] | [
{
"code": "11",
"basis": "community-map",
"evidence": "Inferred from community field: number-theory"
}
] | For each positive integer $k$, let $W(k)$ denote the van der Waerden number for 2 colors, defined as the smallest integer $N$ such that every 2-coloring of $\{1, \ldots, N\}$ contains a monochromatic arithmetic progression of length $k$. Determine whether
$$
\frac{W(k)}{2^k} \to \infty \quad \text{as} \quad k \to \in... | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P8808",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P8808",
"citation": null,
"doi": null,
"arxiv_id": null,
"license": "Selected packets label their content CC0-1.0; original references retain their own terms",
"notes": "Selected problem... | theorem erdos_138.variants.dvd_two_pow :
answer(sorry) ↔ atTop.Tendsto (fun k => ((W k : ℚ)/ (2 ^ k))) atTop | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem erdos_138.variants.dvd_two_pow :\n answer(sorry) ↔ atTop.Tendsto (fun k => ((W k : ℚ)/ (2 ^ k))) atTop",
"target": null,
"source_url": "https://theoremdb.org/statements/P8808",
"format": "declaration-or-fragment",
"sha256": "cbc6e2e076e735fdcd09fd6... | [] | TheoremDB P8808 · Candidate awaiting source review | null | Source-reported assessment, retrieved 8 September 2026. Frontier Science has not independently reviewed this status.
Open. No signed Lean verification is attached to this problem or its selected resolution.
This record is in the source backlog; its novelty and unresolved status have not passed source review. | null | published | incremental | hard | [
"source-candidate",
"theoremdb"
] | null | null | null | null | [
{
"kind": "website",
"citation": "TheoremDB P8808",
"url": "https://theoremdb.org/statements/P8808",
"note": null,
"sort_order": 0
},
{
"kind": "formalization",
"citation": "Source formal statement (not locally verified)",
"url": "https://theoremdb.org/statements/P8808",
"not... | {
"source_record": "P8808",
"imported_at": "2026-09-08",
"status_as_of": "2026-09-08",
"evidence_label": "Source candidate · not reviewed",
"note": "Full public directory statement. Source lane: backlog; source resolution: open. Source verification claims are attributed only; no local Lean verification has be... | {"created_at":"2026-08-17T19:23:11+00:00","display_standard":{"editorial_review":"submitter_supplied","formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":8808,"public_status":{"evidence":{"hasFormalization":false,"hasInformalProof":false,"hasRecordedWork":false,"solutionRecord":null},"lean":{... | [
"formalization-contains-placeholder",
"formalization-not-compiled-during-export",
"current-status-unassessed",
"source-full-text-not-present"
] | [] | {
"created_at": "2026-09-08T03:32:15.935Z",
"updated_at": "2026-09-08T03:32:15.935Z",
"published_at": "2026-09-08T03:32:50.273Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"a9b325de-307b-4a71-89e1-56bb3985a953","format_issues":null,"number":3386} |
1.0.0 | problem:theoremdb-p8810 | problem | theoremdb-p8810 | Critical exponent for powerful integers in short intervals | {
"main": "math",
"sub": "number theory",
"slug": "number-theory",
"name": "Number theory",
"original_discipline": "mathematics"
} | [] | [
{
"code": "11",
"basis": "community-map",
"evidence": "Inferred from community field: number-theory"
}
] | For each positive integer $n$, let $h(n)$ denote the number of powerful integers in the interval $[n^2, (n+1)^2)$. Determine whether there exists a constant $c > 0$ and a function $o : \mathbb{N} \to \mathbb{R}$ with $o(n) = o(1)$ as $n \to \infty$ such that $h(n) < (\log n)^{c + o(n)}$ for all sufficiently large $n$, ... | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P8810",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P8810",
"citation": null,
"doi": null,
"arxiv_id": null,
"license": "Selected packets label their content CC0-1.0; original references retain their own terms",
"notes": "Selected problem... | theorem erdos_942 : answer(sorry) ↔ ∃ c > 0, ∃ (o : ℕ → ℝ), o =o[atTop] (1 : ℕ → ℝ) ∧
(∀ᶠ n in atTop, erdos_942.h n < (Real.log n) ^ (c + o n)) ∧
{n | erdos_942.h n > (Real.log n) ^ (c - o n)}.Infinite | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem erdos_942 : answer(sorry) ↔ ∃ c > 0, ∃ (o : ℕ → ℝ), o =o[atTop] (1 : ℕ → ℝ) ∧\n (∀ᶠ n in atTop, erdos_942.h n < (Real.log n) ^ (c + o n)) ∧\n {n | erdos_942.h n > (Real.log n) ^ (c - o n)}.Infinite",
"target": null,
"source_url": "https://theoremdb.org/... | [] | TheoremDB P8810 · Candidate awaiting source review | null | Source-reported assessment, retrieved 8 September 2026. Frontier Science has not independently reviewed this status.
Open. No signed Lean verification is attached to this problem or its selected resolution.
This record is in the source backlog; its novelty and unresolved status have not passed source review. | null | published | incremental | hard | [
"source-candidate",
"theoremdb"
] | null | null | null | null | [
{
"kind": "website",
"citation": "TheoremDB P8810",
"url": "https://theoremdb.org/statements/P8810",
"note": null,
"sort_order": 0
},
{
"kind": "formalization",
"citation": "Source formal statement (not locally verified)",
"url": "https://theoremdb.org/statements/P8810",
"not... | {
"source_record": "P8810",
"imported_at": "2026-09-08",
"status_as_of": "2026-09-08",
"evidence_label": "Source candidate · not reviewed",
"note": "Full public directory statement. Source lane: backlog; source resolution: open. Source verification claims are attributed only; no local Lean verification has be... | {"created_at":"2026-08-17T19:23:24+00:00","display_standard":{"editorial_review":"submitter_supplied","formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":8810,"public_status":{"evidence":{"hasFormalization":false,"hasInformalProof":false,"hasRecordedWork":false,"solutionRecord":null},"lean":{... | [
"formalization-contains-placeholder",
"formalization-not-compiled-during-export",
"current-status-unassessed",
"source-full-text-not-present"
] | [] | {
"created_at": "2026-09-08T03:32:15.935Z",
"updated_at": "2026-09-08T03:32:15.935Z",
"published_at": "2026-09-08T03:32:52.865Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"4cf31563-7ac9-44e4-806f-18de39cde87b","format_issues":null,"number":3952} |
1.0.0 | problem:theoremdb-p8812 | problem | theoremdb-p8812 | Lower bound for Zhao's solution-free set problem | {
"main": "math",
"sub": "combinatorics",
"slug": "combinatorics",
"name": "Combinatorics",
"original_discipline": "mathematics"
} | [] | [
{
"code": "05",
"basis": "community-map",
"evidence": "Inferred from community field: combinatorics"
}
] | For a positive integer $N$, let $[N]$ denote the set $\{1, 2, \ldots, N\}$. A subset $A \subseteq [N]$ is called Zhao-solution-free if there do not exist distinct integers $x, y, z, x', y', z' \in A$ such that $x + 2y + 3z = x' + 2y' + 3z'$. Let $g(N)$ denote the maximum cardinality of a Zhao-solution-free subset of $[... | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P8812",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P8812",
"citation": null,
"doi": null,
"arxiv_id": null,
"license": "Selected packets label their content CC0-1.0; original references retain their own terms",
"notes": "Selected problem... | theorem zhao_question :
answer(sorry) ↔ ∃ h : ℕ → ℝ, Tendsto h atTop (𝓝 0) ∧
∀ᶠ N in atTop, (g N : ℝ) ≥ (N : ℝ) ^ (1 / 3 - h N) | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem zhao_question :\n answer(sorry) ↔ ∃ h : ℕ → ℝ, Tendsto h atTop (𝓝 0) ∧\n ∀ᶠ N in atTop, (g N : ℝ) ≥ (N : ℝ) ^ (1 / 3 - h N)",
"target": null,
"source_url": "https://theoremdb.org/statements/P8812",
"format": "declaration-or-fragment",
"sha256... | [] | TheoremDB P8812 · Candidate awaiting source review | null | Source-reported assessment, retrieved 8 September 2026. Frontier Science has not independently reviewed this status.
Open. No signed Lean verification is attached to this problem or its selected resolution.
This record is in the source backlog; its novelty and unresolved status have not passed source review. | null | published | incremental | hard | [
"source-candidate",
"theoremdb"
] | null | null | null | null | [
{
"kind": "website",
"citation": "TheoremDB P8812",
"url": "https://theoremdb.org/statements/P8812",
"note": null,
"sort_order": 0
},
{
"kind": "formalization",
"citation": "Source formal statement (not locally verified)",
"url": "https://theoremdb.org/statements/P8812",
"not... | {
"source_record": "P8812",
"imported_at": "2026-09-08",
"status_as_of": "2026-09-08",
"evidence_label": "Source candidate · not reviewed",
"note": "Full public directory statement. Source lane: backlog; source resolution: open. Source verification claims are attributed only; no local Lean verification has be... | {"created_at":"2026-08-17T19:23:29+00:00","display_standard":{"formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":8812,"public_status":{"evidence":{"hasFormalization":false,"hasInformalProof":false,"hasRecordedWork":false,"solutionRecord":null},"lean":{"kernelAccepted":false,"label":"","reaso... | [
"formalization-contains-placeholder",
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"current-status-unassessed",
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] | [] | {
"created_at": "2026-09-08T03:32:15.935Z",
"updated_at": "2026-09-08T03:32:15.935Z",
"published_at": "2026-09-08T03:33:03.551Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"fab5d1ab-7ea6-40d1-b531-a4916aad7abb","format_issues":null,"number":6119} |
1.0.0 | problem:theoremdb-p8814 | problem | theoremdb-p8814 | Singmaster's Conjecture on Multiplicities in Pascal's Triangle | {
"main": "math",
"sub": "number theory",
"slug": "number-theory",
"name": "Number theory",
"original_discipline": "mathematics"
} | [] | [
{
"code": "11",
"basis": "community-map",
"evidence": "Inferred from community field: number-theory"
}
] | For any integer $t > 1$, consider the set of solutions to the equation $\binom{n}{k} = t$ subject to the constraint $1 \le k < n$, where $\binom{n}{k}$ denotes the binomial coefficient. Does there exist a universal constant $C$ such that for every integer $t > 1$, the number of such solutions is finite and at most $C$? | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P8814",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P8814",
"citation": null,
"doi": null,
"arxiv_id": null,
"license": "Selected packets label their content CC0-1.0; original references retain their own terms",
"notes": "Selected problem... | theorem singmaster: ∃ (C : ℕ), ∀ (t : ℕ), t > 1 →
(Singmaster.solutions t).Finite ∧ (Singmaster.solutions t).ncard ≤ C | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem singmaster: ∃ (C : ℕ), ∀ (t : ℕ), t > 1 →\n (Singmaster.solutions t).Finite ∧ (Singmaster.solutions t).ncard ≤ C",
"target": null,
"source_url": "https://theoremdb.org/statements/P8814",
"format": "declaration-or-fragment",
"sha256": "f848bc126ff33... | [] | TheoremDB P8814 · Candidate awaiting source review | null | Source-reported assessment, retrieved 8 September 2026. Frontier Science has not independently reviewed this status.
Open. No signed Lean verification is attached to this problem or its selected resolution.
This record is in the source backlog; its novelty and unresolved status have not passed source review. | null | published | incremental | hard | [
"source-candidate",
"theoremdb"
] | null | null | null | null | [
{
"kind": "website",
"citation": "TheoremDB P8814",
"url": "https://theoremdb.org/statements/P8814",
"note": null,
"sort_order": 0
},
{
"kind": "formalization",
"citation": "Source formal statement (not locally verified)",
"url": "https://theoremdb.org/statements/P8814",
"not... | {
"source_record": "P8814",
"imported_at": "2026-09-08",
"status_as_of": "2026-09-08",
"evidence_label": "Source candidate · not reviewed",
"note": "Full public directory statement. Source lane: backlog; source resolution: open. Source verification claims are attributed only; no local Lean verification has be... | {"created_at":"2026-08-17T19:24:04+00:00","display_standard":{"editorial_review":"submitter_supplied","formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":8814,"public_status":{"evidence":{"hasFormalization":false,"hasInformalProof":false,"hasRecordedWork":false,"solutionRecord":null},"lean":{... | [
"formalization-not-compiled-during-export",
"current-status-unassessed",
"source-full-text-not-present"
] | [] | {
"created_at": "2026-09-08T03:32:15.935Z",
"updated_at": "2026-09-08T03:32:15.935Z",
"published_at": "2026-09-08T03:33:05.860Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"5357430c-b712-44df-80f0-2518c123f2d3","format_issues":null,"number":6618} |
1.0.0 | problem:theoremdb-p8816 | problem | theoremdb-p8816 | Finiteness of Egyptian fraction representations with finitely many unit-fraction interval sums | {
"main": "math",
"sub": "number theory",
"slug": "number-theory",
"name": "Number theory",
"original_discipline": "mathematics"
} | [] | [
{
"code": "11",
"basis": "community-map",
"evidence": "Inferred from community field: number-theory"
}
] | For an integer $k \ge 1$, a $k$-interval family is a function $I$ assigning to each index $j \in \{1, \dots, k\}$ an interval $I_j = [a_j, b_j]$ of positive integers, where $a_j \le b_j$. For such a family, define the total harmonic sum as
$$
S(I) = \sum_{j=1}^{k} \sum_{n_j = a_j}^{b_j} \frac{1}{n_j}.
$$
Does there... | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P8816",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P8816",
"citation": null,
"doi": null,
"arxiv_id": null,
"license": "Selected packets label their content CC0-1.0; original references retain their own terms",
"notes": "Selected problem... | theorem erdos_288.variants.exists_k_gt_2 : answer(sorry) ↔
∃ k > 2, Set.Finite { I : Fin k → ℕ+ × ℕ+ |
∀ j, (I j).1 ≤ (I j).2 ∧
∃ n : ℕ+, (∑ j : Fin k, ∑ nⱼ ∈ Set.Icc (I j).1 (I j).2, (nⱼ⁻¹ : ℚ)) = n } | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem erdos_288.variants.exists_k_gt_2 : answer(sorry) ↔\n ∃ k > 2, Set.Finite { I : Fin k → ℕ+ × ℕ+ |\n ∀ j, (I j).1 ≤ (I j).2 ∧\n ∃ n : ℕ+, (∑ j : Fin k, ∑ nⱼ ∈ Set.Icc (I j).1 (I j).2, (nⱼ⁻¹ : ℚ)) = n }",
"target": null,
"source_url": "https://the... | [] | TheoremDB P8816 · Candidate awaiting source review | null | Source-reported assessment, retrieved 8 September 2026. Frontier Science has not independently reviewed this status.
Open. No signed Lean verification is attached to this problem or its selected resolution.
This record is in the source backlog; its novelty and unresolved status have not passed source review. | null | published | incremental | hard | [
"source-candidate",
"theoremdb"
] | null | null | null | null | [
{
"kind": "website",
"citation": "TheoremDB P8816",
"url": "https://theoremdb.org/statements/P8816",
"note": null,
"sort_order": 0
},
{
"kind": "formalization",
"citation": "Source formal statement (not locally verified)",
"url": "https://theoremdb.org/statements/P8816",
"not... | {
"source_record": "P8816",
"imported_at": "2026-09-08",
"status_as_of": "2026-09-08",
"evidence_label": "Source candidate · not reviewed",
"note": "Full public directory statement. Source lane: backlog; source resolution: open. Source verification claims are attributed only; no local Lean verification has be... | {"created_at":"2026-08-17T19:24:14+00:00","display_standard":{"editorial_review":"submitter_supplied","formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":8816,"public_status":{"evidence":{"hasFormalization":false,"hasInformalProof":false,"hasRecordedWork":false,"solutionRecord":null},"lean":{... | [
"formalization-contains-placeholder",
"formalization-not-compiled-during-export",
"current-status-unassessed",
"source-full-text-not-present"
] | [] | {
"created_at": "2026-09-08T03:32:15.935Z",
"updated_at": "2026-09-08T03:32:15.935Z",
"published_at": "2026-09-08T03:32:59.034Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"6c81d5ec-184d-4f56-8511-55f6946ff901","format_issues":null,"number":5308} |
1.0.0 | problem:theoremdb-p8818 | problem | theoremdb-p8818 | Density of primitive-root primes for odd perfect powers with squarefree part congruent to 1 modulo 4 | {
"main": "math",
"sub": "number theory",
"slug": "number-theory",
"name": "Number theory",
"original_discipline": "mathematics"
} | [] | [
{
"code": "11",
"basis": "community-map",
"evidence": "Inferred from community field: number-theory"
}
] | Let $a$, $m$, and $b$ be natural numbers such that $a = b^m$. Assume that $m > 1$ is odd, that $b$ is not a perfect power (that is, for all integers $u > 1$ and $v$, we have $b \neq v^u$), and that the squarefree part of $b$ is congruent to $1$ modulo $4$. Let $S(a)$ denote the set of prime numbers $p$ such that $a$ is... | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P8818",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P8818",
"citation": null,
"doi": null,
"arxiv_id": null,
"license": "Selected packets label their content CC0-1.0; original references retain their own terms",
"notes": "Selected problem... | theorem artin_primitive_roots.variants.part_ii_power_squarefreePart_modeq_one
(a m b : ℕ) (ha : a = b ^ m) (hb : ∀ u v, 1 < u → b ≠ v ^ u) (hm₁ : 1 < m)
(hm₂ : Odd m) (hb' : b.squarefreePart ≡ 1 [MOD 4]) :
(S a).HasDensity
(ArtinConstant * powCorrectionFactor m * entanglementFactor b m)
{p | p.P... | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem artin_primitive_roots.variants.part_ii_power_squarefreePart_modeq_one\n (a m b : ℕ) (ha : a = b ^ m) (hb : ∀ u v, 1 < u → b ≠ v ^ u) (hm₁ : 1 < m)\n (hm₂ : Odd m) (hb' : b.squarefreePart ≡ 1 [MOD 4]) :\n (S a).HasDensity\n (ArtinConstant * powCorrection... | [] | TheoremDB P8818 · Candidate awaiting source review | null | Source-reported assessment, retrieved 8 September 2026. Frontier Science has not independently reviewed this status.
Open. No signed Lean verification is attached to this problem or its selected resolution.
This record is in the source backlog; its novelty and unresolved status have not passed source review. | null | published | incremental | hard | [
"source-candidate",
"theoremdb"
] | null | null | null | null | [
{
"kind": "website",
"citation": "TheoremDB P8818",
"url": "https://theoremdb.org/statements/P8818",
"note": null,
"sort_order": 0
},
{
"kind": "formalization",
"citation": "Source formal statement (not locally verified)",
"url": "https://theoremdb.org/statements/P8818",
"not... | {
"source_record": "P8818",
"imported_at": "2026-09-08",
"status_as_of": "2026-09-08",
"evidence_label": "Source candidate · not reviewed",
"note": "Full public directory statement. Source lane: backlog; source resolution: open. Source verification claims are attributed only; no local Lean verification has be... | {"created_at":"2026-08-17T19:24:21+00:00","display_standard":{"editorial_review":"submitter_supplied","formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":8818,"public_status":{"evidence":{"hasFormalization":false,"hasInformalProof":false,"hasRecordedWork":false,"solutionRecord":null},"lean":{... | [
"formalization-not-compiled-during-export",
"current-status-unassessed",
"source-full-text-not-present"
] | [] | {
"created_at": "2026-09-08T03:32:15.935Z",
"updated_at": "2026-09-08T03:32:15.935Z",
"published_at": "2026-09-08T03:32:53.108Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"2f96082f-d674-4a24-b2cb-913840fae606","format_issues":null,"number":4005} |
1.0.0 | problem:theoremdb-p8820 | problem | theoremdb-p8820 | Artin's conjecture on primitive roots for non-power integers with squarefree part not congruent to 1 modulo 4 | {
"main": "math",
"sub": "number theory",
"slug": "number-theory",
"name": "Number theory",
"original_discipline": "mathematics"
} | [] | [
{
"code": "11",
"basis": "community-map",
"evidence": "Inferred from community field: number-theory"
}
] | Let $a$ be an integer. We say that $a$ is a primitive root modulo a prime $p$ if the multiplicative order of $a$ in $(\mathbb{Z}/p\mathbb{Z})^\times$ equals $p-1$. For an integer $a$, let $S(a)$ denote the set of primes $p$ such that $a$ is a primitive root modulo $p$. A positive integer is called squarefree if it is n... | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P8820",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P8820",
"citation": null,
"doi": null,
"arxiv_id": null,
"license": "Selected packets label their content CC0-1.0; original references retain their own terms",
"notes": "Selected problem... | theorem artin_primitive_roots.parts.ii
(a a_0 b : ℤ) (ha : a = a_0 * b ^ 2)
(ha' : ∀ n m, m ≠ 1 → a ≠ n ^ m) (ha_0 : Squarefree a_0)
(ha_0' : ¬a_0 ≡ 1 [ZMOD 4]) :
(S a).HasDensity ArtinConstant {p | p.Prime} | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem artin_primitive_roots.parts.ii\n (a a_0 b : ℤ) (ha : a = a_0 * b ^ 2)\n (ha' : ∀ n m, m ≠ 1 → a ≠ n ^ m) (ha_0 : Squarefree a_0)\n (ha_0' : ¬a_0 ≡ 1 [ZMOD 4]) :\n (S a).HasDensity ArtinConstant {p | p.Prime}",
"target": null,
"source_url": "https:... | [] | TheoremDB P8820 · Candidate awaiting source review | null | Source-reported assessment, retrieved 8 September 2026. Frontier Science has not independently reviewed this status.
Open. No signed Lean verification is attached to this problem or its selected resolution.
This record is in the source backlog; its novelty and unresolved status have not passed source review. | null | published | incremental | hard | [
"source-candidate",
"theoremdb"
] | null | null | null | null | [
{
"kind": "website",
"citation": "TheoremDB P8820",
"url": "https://theoremdb.org/statements/P8820",
"note": null,
"sort_order": 0
},
{
"kind": "formalization",
"citation": "Source formal statement (not locally verified)",
"url": "https://theoremdb.org/statements/P8820",
"not... | {
"source_record": "P8820",
"imported_at": "2026-09-08",
"status_as_of": "2026-09-08",
"evidence_label": "Source candidate · not reviewed",
"note": "Full public directory statement. Source lane: backlog; source resolution: open. Source verification claims are attributed only; no local Lean verification has be... | {"created_at":"2026-08-17T19:24:23+00:00","display_standard":{"editorial_review":"submitter_supplied","formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":8820,"public_status":{"evidence":{"hasFormalization":false,"hasInformalProof":false,"hasRecordedWork":false,"solutionRecord":null},"lean":{... | [
"formalization-not-compiled-during-export",
"current-status-unassessed",
"source-full-text-not-present"
] | [] | {
"created_at": "2026-09-08T03:32:15.935Z",
"updated_at": "2026-09-08T03:32:15.935Z",
"published_at": "2026-09-08T03:32:49.889Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"b7fdb67e-9284-47ff-a0d0-aae9bf84baeb","format_issues":null,"number":3307} |
1.0.0 | problem:theoremdb-p8822 | problem | theoremdb-p8822 | Existence of the asymptotic limit for Bézier–Bernstein operators with shape parameter | {
"main": "math",
"sub": "analysis",
"slug": "analysis",
"name": "Analysis",
"original_discipline": "mathematics"
} | [] | [] | Let $\alpha > 0$ with $\alpha \neq 1$. For each natural number $m$, let $f : \mathbb{R} \to \mathbb{R}$ be a function and let $x \in [0,1]$. Suppose that $f$ is $m$-times continuously differentiable on $[0,1]$. Consider the Bézier-type Bernstein operator $B_{n,\alpha}$ defined for $f : [0,1] \to \mathbb{R}$ by
$$
(B_{... | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P8822",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P8822",
"citation": null,
"doi": null,
"arxiv_id": null,
"license": "Selected packets label their content CC0-1.0; original references retain their own terms",
"notes": "Selected problem... | theorem voronovskaja_theorem.bezier_bernstein_operators.variants.eventually_smooth.limit_exists
(α : ℝ) (hα_pos : 0 < α) (hα : α ≠ 1) :
∀ᶠ m : ℕ in atTop,
∀ (f : ℝ → ℝ) (x : ℝ), x ∈ I → ContDiffOn ℝ m f I →
∃ L : ℝ,
Tendsto (fun n : ℕ => Real.sqrt n * (bezierBernstein n α f x - f x)) atT... | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem voronovskaja_theorem.bezier_bernstein_operators.variants.eventually_smooth.limit_exists\n (α : ℝ) (hα_pos : 0 < α) (hα : α ≠ 1) :\n ∀ᶠ m : ℕ in atTop,\n ∀ (f : ℝ → ℝ) (x : ℝ), x ∈ I → ContDiffOn ℝ m f I →\n ∃ L : ℝ,\n Tendsto (fun n : ℕ => ... | [] | TheoremDB P8822 · Candidate awaiting source review | null | Source-reported assessment, retrieved 8 September 2026. Frontier Science has not independently reviewed this status.
Open. No signed Lean verification is attached to this problem or its selected resolution.
This record is in the source backlog; its novelty and unresolved status have not passed source review. | null | published | incremental | hard | [
"source-candidate",
"theoremdb"
] | null | null | null | null | [
{
"kind": "website",
"citation": "TheoremDB P8822",
"url": "https://theoremdb.org/statements/P8822",
"note": null,
"sort_order": 0
},
{
"kind": "formalization",
"citation": "Source formal statement (not locally verified)",
"url": "https://theoremdb.org/statements/P8822",
"not... | {
"source_record": "P8822",
"imported_at": "2026-09-08",
"status_as_of": "2026-09-08",
"evidence_label": "Source candidate · not reviewed",
"note": "Full public directory statement. Source lane: backlog; source resolution: open. Source verification claims are attributed only; no local Lean verification has be... | {"created_at":"2026-08-17T19:24:31+00:00","display_standard":{"editorial_review":"submitter_supplied","formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":8822,"public_status":{"evidence":{"hasFormalization":false,"hasInformalProof":false,"hasRecordedWork":false,"solutionRecord":null},"lean":{... | [
"formalization-not-compiled-during-export",
"current-status-unassessed",
"source-full-text-not-present"
] | [] | {
"created_at": "2026-09-08T03:32:15.935Z",
"updated_at": "2026-09-08T03:32:15.935Z",
"published_at": "2026-09-08T03:32:58.834Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"b741079b-8eb6-4051-ac0a-3dfb7ebfc610","format_issues":null,"number":5265} |
1.0.0 | problem:theoremdb-p8824 | problem | theoremdb-p8824 | Erdős Problem 385, Part I: The Positivity of F(n) − n | {
"main": "math",
"sub": "number theory",
"slug": "number-theory",
"name": "Number theory",
"original_discipline": "mathematics"
} | [] | [
{
"code": "11",
"basis": "community-map",
"evidence": "Inferred from community field: number-theory"
}
] | For each integer $n \geq 1$, define $F(n)$ to be the maximum value of $m + p(m)$ over all composite integers $m$ with $m < n$, where $p(m)$ denotes the least prime divisor of $m$. Determine whether $F(n) > n$ holds for all sufficiently large $n$. | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P8824",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P8824",
"citation": null,
"doi": null,
"arxiv_id": null,
"license": "Selected packets label their content CC0-1.0; original references retain their own terms",
"notes": "Selected problem... | theorem erdos_385.parts.i : answer(sorry) ↔ ∀ᶠ n in atTop, n < F n | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem erdos_385.parts.i : answer(sorry) ↔ ∀ᶠ n in atTop, n < F n",
"target": null,
"source_url": "https://theoremdb.org/statements/P8824",
"format": "declaration-or-fragment",
"sha256": "751fa6b55e927f3e6b61d7ede01ea97534bdb9fe5c2dfaa621c5957a3ae33091",
... | [] | TheoremDB P8824 · Candidate awaiting source review | null | Source-reported assessment, retrieved 8 September 2026. Frontier Science has not independently reviewed this status.
Open. No signed Lean verification is attached to this problem or its selected resolution.
This record is in the source backlog; its novelty and unresolved status have not passed source review. | null | published | incremental | hard | [
"source-candidate",
"theoremdb"
] | null | null | null | null | [
{
"kind": "website",
"citation": "TheoremDB P8824",
"url": "https://theoremdb.org/statements/P8824",
"note": null,
"sort_order": 0
},
{
"kind": "formalization",
"citation": "Source formal statement (not locally verified)",
"url": "https://theoremdb.org/statements/P8824",
"not... | {
"source_record": "P8824",
"imported_at": "2026-09-08",
"status_as_of": "2026-09-08",
"evidence_label": "Source candidate · not reviewed",
"note": "Full public directory statement. Source lane: backlog; source resolution: open. Source verification claims are attributed only; no local Lean verification has be... | {"created_at":"2026-08-17T19:24:33+00:00","display_standard":{"formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":8824,"public_status":{"evidence":{"hasFormalization":false,"hasInformalProof":false,"hasRecordedWork":false,"solutionRecord":null},"lean":{"kernelAccepted":false,"label":"","reaso... | [
"formalization-contains-placeholder",
"formalization-not-compiled-during-export",
"current-status-unassessed",
"source-full-text-not-present"
] | [] | {
"created_at": "2026-09-08T03:32:15.935Z",
"updated_at": "2026-09-08T03:32:15.935Z",
"published_at": "2026-09-08T03:32:55.401Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"ba0761ca-01bf-4016-840f-80f53fd3be7e","format_issues":null,"number":4512} |
1.0.0 | problem:theoremdb-p8826 | problem | theoremdb-p8826 | Factorization of 0,1-polynomials into monic polynomials with non-negative coefficients | {
"main": "math",
"sub": "algebra",
"slug": "algebra",
"name": "Algebra",
"original_discipline": "mathematics"
} | [] | [] | A real polynomial is called a 0,1-polynomial if every nonzero coefficient is equal to 1. Let $P(x), Q(x) \in \mathbb{R}[x]$ be monic polynomials such that all coefficients of $P$ and all coefficients of $Q$ are non-negative. Suppose that the product $R(x) = P(x)Q(x)$ is a 0,1-polynomial. Prove that $P(x)$ and $Q(x)$ ar... | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P8826",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P8826",
"citation": null,
"doi": null,
"arxiv_id": null,
"license": "Selected packets label their content CC0-1.0; original references retain their own terms",
"notes": "Selected problem... | theorem mathoverflow_339137 (P Q R : ℝ[X]) (hP: P.Monic) (hQ : Q.Monic)
(hp : ∀ c ∈ P.coeffs, 0 ≤ c) (hq : ∀ c ∈ Q.coeffs, 0 ≤ c)
(h : R = P * Q) (hR : IsZeroOne R) :
IsZeroOne P ∧ IsZeroOne Q | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem mathoverflow_339137 (P Q R : ℝ[X]) (hP: P.Monic) (hQ : Q.Monic)\n (hp : ∀ c ∈ P.coeffs, 0 ≤ c) (hq : ∀ c ∈ Q.coeffs, 0 ≤ c)\n (h : R = P * Q) (hR : IsZeroOne R) :\n IsZeroOne P ∧ IsZeroOne Q",
"target": null,
"source_url": "https://theoremdb.org/stat... | [] | TheoremDB P8826 · Candidate awaiting source review | null | Source-reported assessment, retrieved 8 September 2026. Frontier Science has not independently reviewed this status.
Open. No signed Lean verification is attached to this problem or its selected resolution.
This record is in the source backlog; its novelty and unresolved status have not passed source review. | null | published | incremental | hard | [
"source-candidate",
"theoremdb"
] | null | null | null | null | [
{
"kind": "website",
"citation": "TheoremDB P8826",
"url": "https://theoremdb.org/statements/P8826",
"note": null,
"sort_order": 0
},
{
"kind": "formalization",
"citation": "Source formal statement (not locally verified)",
"url": "https://theoremdb.org/statements/P8826",
"not... | {
"source_record": "P8826",
"imported_at": "2026-09-08",
"status_as_of": "2026-09-08",
"evidence_label": "Source candidate · not reviewed",
"note": "Full public directory statement. Source lane: backlog; source resolution: open. Source verification claims are attributed only; no local Lean verification has be... | {"created_at":"2026-08-17T19:24:40+00:00","display_standard":{"editorial_review":"submitter_supplied","formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":8826,"public_status":{"evidence":{"hasFormalization":false,"hasInformalProof":false,"hasRecordedWork":false,"solutionRecord":null},"lean":{... | [
"formalization-not-compiled-during-export",
"current-status-unassessed",
"source-full-text-not-present"
] | [] | {
"created_at": "2026-09-08T03:32:15.935Z",
"updated_at": "2026-09-08T03:32:15.935Z",
"published_at": "2026-09-08T03:32:58.925Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"7e9f457e-4c79-47c8-b32f-a238f5e117c1","format_issues":null,"number":5284} |
1.0.0 | problem:theoremdb-p8828 | problem | theoremdb-p8828 | Existence of distinct integers with matching consecutive prime factor sets | {
"main": "math",
"sub": "number theory",
"slug": "number-theory",
"name": "Number theory",
"original_discipline": "mathematics"
} | [] | [
{
"code": "11",
"basis": "community-map",
"evidence": "Inferred from community field: number-theory"
}
] | Can there exist two distinct positive integers $x$ and $y$ such that $x$ and $y$ have the same set of prime factors, $x+1$ and $y+1$ have the same set of prime factors, and $x+2$ and $y+2$ also have the same set of prime factors? | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P8828",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P8828",
"citation": null,
"doi": null,
"arxiv_id": null,
"license": "Selected packets label their content CC0-1.0; original references retain their own terms",
"notes": "Selected problem... | theorem erdos_850 :
answer(sorry) ↔ ∃ x y : ℕ, x ≠ y ∧ x.primeFactors = y.primeFactors
∧ (x + 1).primeFactors = (y + 1).primeFactors
∧ (x + 2).primeFactors = (y + 2).primeFactors | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem erdos_850 :\n answer(sorry) ↔ ∃ x y : ℕ, x ≠ y ∧ x.primeFactors = y.primeFactors\n ∧ (x + 1).primeFactors = (y + 1).primeFactors\n ∧ (x + 2).primeFactors = (y + 2).primeFactors",
"target": null,
"source_url": "https://theoremdb.org/statements/P88... | [] | TheoremDB P8828 · Candidate awaiting source review | null | Source-reported assessment, retrieved 8 September 2026. Frontier Science has not independently reviewed this status.
Open. No signed Lean verification is attached to this problem or its selected resolution.
This record is in the source backlog; its novelty and unresolved status have not passed source review. | null | published | incremental | hard | [
"source-candidate",
"theoremdb"
] | null | null | null | null | [
{
"kind": "website",
"citation": "TheoremDB P8828",
"url": "https://theoremdb.org/statements/P8828",
"note": null,
"sort_order": 0
},
{
"kind": "formalization",
"citation": "Source formal statement (not locally verified)",
"url": "https://theoremdb.org/statements/P8828",
"not... | {
"source_record": "P8828",
"imported_at": "2026-09-08",
"status_as_of": "2026-09-08",
"evidence_label": "Source candidate · not reviewed",
"note": "Full public directory statement. Source lane: backlog; source resolution: open. Source verification claims are attributed only; no local Lean verification has be... | {"created_at":"2026-08-17T19:24:43+00:00","display_standard":{"editorial_review":"submitter_supplied","formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":8828,"public_status":{"evidence":{"hasFormalization":false,"hasInformalProof":false,"hasRecordedWork":false,"solutionRecord":null},"lean":{... | [
"formalization-contains-placeholder",
"formalization-not-compiled-during-export",
"current-status-unassessed",
"source-full-text-not-present"
] | [] | {
"created_at": "2026-09-08T03:32:15.935Z",
"updated_at": "2026-09-08T03:32:15.935Z",
"published_at": "2026-09-08T03:32:58.738Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"c9798950-3e32-43c4-90cf-8e0e8b2bef23","format_issues":null,"number":5245} |
1.0.0 | problem:theoremdb-p8830 | problem | theoremdb-p8830 | Existence of a positive proportion of large least primes in arithmetic progressions | {
"main": "math",
"sub": "number theory",
"slug": "number-theory",
"name": "Number theory",
"original_discipline": "mathematics"
} | [] | [
{
"code": "11",
"basis": "community-map",
"evidence": "Inferred from community field: number-theory"
}
] | For positive integers $a$ and $d$ with $\gcd(a,d) = 1$, let $p(a,d)$ denote the least prime congruent to $a$ modulo $d$. Let $\varphi(d)$ denote Euler's totient function, which counts the number of integers $1 \leq a \leq d$ with $\gcd(a,d) = 1$. Determine whether there exist constants $c > 0$ and $C > 0$ such that for... | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P8830",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P8830",
"citation": null,
"doi": null,
"arxiv_id": null,
"license": "Selected packets label their content CC0-1.0; original references retain their own terms",
"notes": "Selected problem... | theorem erdos_971 : answer(sorry) ↔
∃ c > (0 : ℝ), ∃ C > (0 : ℝ), ∀ᶠ d in atTop,
C * (d.totient : ℝ) ≤
#{a < d | a.Coprime d ∧ (leastCongruentPrime a d : ℝ) > (1 + c) * d.totient * log d} | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem erdos_971 : answer(sorry) ↔\n ∃ c > (0 : ℝ), ∃ C > (0 : ℝ), ∀ᶠ d in atTop,\n C * (d.totient : ℝ) ≤\n #{a < d | a.Coprime d ∧ (leastCongruentPrime a d : ℝ) > (1 + c) * d.totient * log d}",
"target": null,
"source_url": "https://theoremdb.org/sta... | [] | TheoremDB P8830 · Candidate awaiting source review | null | Source-reported assessment, retrieved 8 September 2026. Frontier Science has not independently reviewed this status.
Open. No signed Lean verification is attached to this problem or its selected resolution.
This record is in the source backlog; its novelty and unresolved status have not passed source review. | null | published | incremental | hard | [
"source-candidate",
"theoremdb"
] | null | null | null | null | [
{
"kind": "website",
"citation": "TheoremDB P8830",
"url": "https://theoremdb.org/statements/P8830",
"note": null,
"sort_order": 0
},
{
"kind": "formalization",
"citation": "Source formal statement (not locally verified)",
"url": "https://theoremdb.org/statements/P8830",
"not... | {
"source_record": "P8830",
"imported_at": "2026-09-08",
"status_as_of": "2026-09-08",
"evidence_label": "Source candidate · not reviewed",
"note": "Full public directory statement. Source lane: backlog; source resolution: open. Source verification claims are attributed only; no local Lean verification has be... | {"created_at":"2026-08-17T19:24:51+00:00","display_standard":{"editorial_review":"submitter_supplied","formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":8830,"public_status":{"evidence":{"hasFormalization":false,"hasInformalProof":false,"hasRecordedWork":false,"solutionRecord":null},"lean":{... | [
"formalization-contains-placeholder",
"formalization-not-compiled-during-export",
"current-status-unassessed",
"source-full-text-not-present"
] | [] | {
"created_at": "2026-09-08T03:32:15.935Z",
"updated_at": "2026-09-08T03:32:15.935Z",
"published_at": "2026-09-08T03:32:58.477Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"e95aa90a-e776-4b22-a212-c1cfa61f16a1","format_issues":null,"number":5191} |
1.0.0 | problem:theoremdb-p8832 | problem | theoremdb-p8832 | Infinitude of Wolstenholme primes | {
"main": "math",
"sub": "number theory",
"slug": "number-theory",
"name": "Number theory",
"original_discipline": "mathematics"
} | [] | [
{
"code": "11",
"basis": "community-map",
"evidence": "Inferred from community field: number-theory"
}
] | A prime number $p > 7$ is called a Wolstenholme prime if the binomial coefficient $\binom{2p-1}{p-1}$ is congruent to $1$ modulo $p^4$. Equivalently, a prime $p > 7$ is a Wolstenholme prime if $p$ divides the numerator of the Bernoulli number $B_{p-3}$, or if $p^3$ divides the numerator of the harmonic number $H_{p-1}$... | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P8832",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P8832",
"citation": null,
"doi": null,
"arxiv_id": null,
"license": "Selected packets label their content CC0-1.0; original references retain their own terms",
"notes": "Selected problem... | theorem wolstenholme_prime_infinite :
{p : ℕ | IsWolstenholmePrime p}.Infinite | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem wolstenholme_prime_infinite :\n {p : ℕ | IsWolstenholmePrime p}.Infinite",
"target": null,
"source_url": "https://theoremdb.org/statements/P8832",
"format": "declaration-or-fragment",
"sha256": "f9a607e82eb6cf16197980a5153467947dd5508c8c0503cd5e775... | [] | TheoremDB P8832 · Candidate awaiting source review | null | Source-reported assessment, retrieved 8 September 2026. Frontier Science has not independently reviewed this status.
Open. No signed Lean verification is attached to this problem or its selected resolution.
This record is in the source backlog; its novelty and unresolved status have not passed source review. | null | published | incremental | hard | [
"source-candidate",
"theoremdb"
] | null | null | null | null | [
{
"kind": "website",
"citation": "TheoremDB P8832",
"url": "https://theoremdb.org/statements/P8832",
"note": null,
"sort_order": 0
},
{
"kind": "formalization",
"citation": "Source formal statement (not locally verified)",
"url": "https://theoremdb.org/statements/P8832",
"not... | {
"source_record": "P8832",
"imported_at": "2026-09-08",
"status_as_of": "2026-09-08",
"evidence_label": "Source candidate · not reviewed",
"note": "Full public directory statement. Source lane: backlog; source resolution: open. Source verification claims are attributed only; no local Lean verification has be... | {"created_at":"2026-08-17T19:24:55+00:00","display_standard":{"editorial_review":"submitter_supplied","formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":8832,"public_status":{"evidence":{"hasFormalization":false,"hasInformalProof":false,"hasRecordedWork":false,"solutionRecord":null},"lean":{... | [
"formalization-not-compiled-during-export",
"current-status-unassessed",
"source-full-text-not-present"
] | [] | {
"created_at": "2026-09-08T03:32:15.935Z",
"updated_at": "2026-09-08T03:32:15.935Z",
"published_at": "2026-09-08T03:33:01.281Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"ad11c2a2-9048-4e45-a587-b646db28f378","format_issues":null,"number":5696} |
1.0.0 | problem:theoremdb-p8834 | problem | theoremdb-p8834 | Collinear Triples in Infinite S-Walks in Integer Lattices | {
"main": "math",
"sub": "combinatorics",
"slug": "combinatorics",
"name": "Combinatorics",
"original_discipline": "mathematics"
} | [] | [
{
"code": "05",
"basis": "community-map",
"evidence": "Inferred from community field: combinatorics"
}
] | Let $S$ be a finite subset of $\mathbb{Z}^3$. An $S$-walk is a sequence $a = (a_1, a_2, \ldots)$ of points in $\mathbb{Z}^3$ such that $a_{n+1} - a_n \in S$ for all $n \in \mathbb{N}$. A set $A$ of points in $\mathbb{Q}^3$ is said to contain a collinear triple if there exist three distinct points $x, y, z \in A$ that a... | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P8834",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P8834",
"citation": null,
"doi": null,
"arxiv_id": null,
"license": "Selected packets label their content CC0-1.0; original references retain their own terms",
"notes": "Selected problem... | theorem erdos_193 :
answer(sorry) ↔ ∀ S : Set (Fin 3 → ℤ), S.Finite →
/- The statement's $A = \lbrace a_1, a_2, \ldots \rbrace$ is an infinite set.
If the sequence only takes finitely many values, one value has to repeat infinitely many
times, which would yield a trivial collinear triple (x, x, x... | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem erdos_193 :\n answer(sorry) ↔ ∀ S : Set (Fin 3 → ℤ), S.Finite →\n /- The statement's $A = \\lbrace a_1, a_2, \\ldots \\rbrace$ is an infinite set.\n\n If the sequence only takes finitely many values, one value has to repeat infinitely many\n times, ... | [] | TheoremDB P8834 · Candidate awaiting source review | null | Source-reported assessment, retrieved 8 September 2026. Frontier Science has not independently reviewed this status.
Open. No signed Lean verification is attached to this problem or its selected resolution.
This record is in the source backlog; its novelty and unresolved status have not passed source review. | null | published | incremental | hard | [
"source-candidate",
"theoremdb"
] | null | null | null | null | [
{
"kind": "website",
"citation": "TheoremDB P8834",
"url": "https://theoremdb.org/statements/P8834",
"note": null,
"sort_order": 0
},
{
"kind": "formalization",
"citation": "Source formal statement (not locally verified)",
"url": "https://theoremdb.org/statements/P8834",
"not... | {
"source_record": "P8834",
"imported_at": "2026-09-08",
"status_as_of": "2026-09-08",
"evidence_label": "Source candidate · not reviewed",
"note": "Full public directory statement. Source lane: backlog; source resolution: open. Source verification claims are attributed only; no local Lean verification has be... | {"created_at":"2026-08-17T19:25:03+00:00","display_standard":{"editorial_review":"submitter_supplied","formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":8834,"public_status":{"evidence":{"hasFormalization":false,"hasInformalProof":false,"hasRecordedWork":false,"solutionRecord":null},"lean":{... | [
"formalization-contains-placeholder",
"formalization-not-compiled-during-export",
"current-status-unassessed",
"source-full-text-not-present"
] | [] | {
"created_at": "2026-09-08T03:32:15.935Z",
"updated_at": "2026-09-08T03:32:15.935Z",
"published_at": "2026-09-08T03:32:52.430Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"21bf42aa-7ea5-4496-948f-1aeadc8edb1f","format_issues":null,"number":3859} |
1.0.0 | problem:theoremdb-p8836 | problem | theoremdb-p8836 | Common Factor Differences in Families of Integers | {
"main": "math",
"sub": "number theory",
"slug": "number-theory",
"name": "Number theory",
"original_discipline": "mathematics"
} | [] | [
{
"code": "11",
"basis": "community-map",
"evidence": "Inferred from community field: number-theory"
}
] | For an integer $n \geq 1$, define the factor difference set of $n$ by
$$
D(n) = \{|a-b| : a,b \in \mathbb{N},\ n = ab\}.
$$
Is it true that for every integer $k \geq 1$, there exists a set of $k$ positive integers $N_1, \dots, N_k$ such that the intersection $\bigcap_{i=1}^{k} D(N_i)$ contains at least $k$ elements? | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P8836",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P8836",
"citation": null,
"doi": null,
"arxiv_id": null,
"license": "Selected packets label their content CC0-1.0; original references retain their own terms",
"notes": "Selected problem... | theorem erdos_885 : answer(sorry) ↔ ∀ k ≥ 1,
∃ Ns : Finset ℕ,
(∀ n ∈ Ns, 1 ≤ n) ∧
Ns.card = k ∧
(⋂ n ∈ Ns, factorDifferenceSet n).ncard ≥ k | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem erdos_885 : answer(sorry) ↔ ∀ k ≥ 1,\n ∃ Ns : Finset ℕ,\n (∀ n ∈ Ns, 1 ≤ n) ∧\n Ns.card = k ∧\n (⋂ n ∈ Ns, factorDifferenceSet n).ncard ≥ k",
"target": null,
"source_url": "https://theoremdb.org/statements/P8836",
"format": "declaration-... | [] | TheoremDB P8836 · Candidate awaiting source review | null | Source-reported assessment, retrieved 8 September 2026. Frontier Science has not independently reviewed this status.
Open. No signed Lean verification is attached to this problem or its selected resolution.
This record is in the source backlog; its novelty and unresolved status have not passed source review. | null | published | incremental | hard | [
"source-candidate",
"theoremdb"
] | null | null | null | null | [
{
"kind": "website",
"citation": "TheoremDB P8836",
"url": "https://theoremdb.org/statements/P8836",
"note": null,
"sort_order": 0
},
{
"kind": "formalization",
"citation": "Source formal statement (not locally verified)",
"url": "https://theoremdb.org/statements/P8836",
"not... | {
"source_record": "P8836",
"imported_at": "2026-09-08",
"status_as_of": "2026-09-08",
"evidence_label": "Source candidate · not reviewed",
"note": "Full public directory statement. Source lane: backlog; source resolution: open. Source verification claims are attributed only; no local Lean verification has be... | {"created_at":"2026-08-17T19:25:08+00:00","display_standard":{"editorial_review":"submitter_supplied","formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":8836,"public_status":{"evidence":{"hasFormalization":false,"hasInformalProof":false,"hasRecordedWork":false,"solutionRecord":null},"lean":{... | [
"formalization-contains-placeholder",
"formalization-not-compiled-during-export",
"current-status-unassessed",
"source-full-text-not-present"
] | [] | {
"created_at": "2026-09-08T03:32:15.935Z",
"updated_at": "2026-09-08T03:32:15.935Z",
"published_at": "2026-09-08T03:32:52.442Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"86447ee2-18a6-462e-9ddb-e69782d8ca98","format_issues":null,"number":3861} |
1.0.0 | problem:theoremdb-p8838 | problem | theoremdb-p8838 | Infinitely many increasing triples of normalized primes | {
"main": "math",
"sub": "number theory",
"slug": "number-theory",
"name": "Number theory",
"original_discipline": "mathematics"
} | [] | [
{
"code": "11",
"basis": "community-map",
"evidence": "Inferred from community field: number-theory"
}
] | For each natural number $n$, define the normalized prime
$$
u_n = \frac{p_n}{n+1},
$$
where $p_n$ denotes the $n$th prime number (so that $p_0 = 2$, $p_1 = 3$, $p_2 = 5$, and so on). Determine whether the set of indices
$$
\{n \in \mathbb{N} \mid \nu_n < \nu_{n+1} \text{ and } \nu_{n+1} < \nu_{n+2}\}
$$
is infi... | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P8838",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P8838",
"citation": null,
"doi": null,
"arxiv_id": null,
"license": "Selected packets label their content CC0-1.0; original references retain their own terms",
"notes": "Selected problem... | theorem erdos_968.variants.infinite_increasingTriples :
answer(sorry) ↔ {n : ℕ | u n < u (n + 1) ∧ u (n + 1) < u (n + 2)}.Infinite | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem erdos_968.variants.infinite_increasingTriples :\n answer(sorry) ↔ {n : ℕ | u n < u (n + 1) ∧ u (n + 1) < u (n + 2)}.Infinite",
"target": null,
"source_url": "https://theoremdb.org/statements/P8838",
"format": "declaration-or-fragment",
"sha256": "4... | [] | TheoremDB P8838 · Candidate awaiting source review | null | Source-reported assessment, retrieved 8 September 2026. Frontier Science has not independently reviewed this status.
Open. No signed Lean verification is attached to this problem or its selected resolution.
This record is in the source backlog; its novelty and unresolved status have not passed source review. | null | published | incremental | hard | [
"source-candidate",
"theoremdb"
] | null | null | null | null | [
{
"kind": "website",
"citation": "TheoremDB P8838",
"url": "https://theoremdb.org/statements/P8838",
"note": null,
"sort_order": 0
},
{
"kind": "formalization",
"citation": "Source formal statement (not locally verified)",
"url": "https://theoremdb.org/statements/P8838",
"not... | {
"source_record": "P8838",
"imported_at": "2026-09-08",
"status_as_of": "2026-09-08",
"evidence_label": "Source candidate · not reviewed",
"note": "Full public directory statement. Source lane: backlog; source resolution: open. Source verification claims are attributed only; no local Lean verification has be... | {"created_at":"2026-08-17T19:25:24+00:00","display_standard":{"editorial_review":"submitter_supplied","formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":8838,"public_status":{"evidence":{"hasFormalization":false,"hasInformalProof":false,"hasRecordedWork":false,"solutionRecord":null},"lean":{... | [
"formalization-contains-placeholder",
"formalization-not-compiled-during-export",
"current-status-unassessed",
"source-full-text-not-present"
] | [] | {
"created_at": "2026-09-08T03:32:15.935Z",
"updated_at": "2026-09-08T03:32:15.935Z",
"published_at": "2026-09-08T03:33:00.908Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"78d6d10c-6d48-4904-8db5-90343bab6c42","format_issues":null,"number":5616} |
1.0.0 | problem:theoremdb-p8840 | problem | theoremdb-p8840 | The Fermat–Catalan Conjecture | {
"main": "math",
"sub": "number theory",
"slug": "number-theory",
"name": "Number theory",
"original_discipline": "mathematics"
} | [] | [
{
"code": "11",
"basis": "community-map",
"evidence": "Inferred from community field: number-theory"
}
] | Consider the Diophantine equation $a^m + b^n = c^k$ where $a, b, c, m, n, k$ are positive integers. A solution is called a Fermat–Catalan solution if $a, b, c$ are pairwise coprime and the exponents satisfy $\frac{1}{m} + \frac{1}{n} + \frac{1}{k} < 1$. Two solutions are regarded as yielding the same distinct triplet o... | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P8840",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P8840",
"citation": null,
"doi": null,
"arxiv_id": null,
"license": "Selected packets label their content CC0-1.0; original references retain their own terms",
"notes": "Selected problem... | theorem fermat_catalan : fermatCatalanConjecture | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem fermat_catalan : fermatCatalanConjecture",
"target": null,
"source_url": "https://theoremdb.org/statements/P8840",
"format": "declaration-or-fragment",
"sha256": "06c422fed98ea127caf8e2f1b11a27f99a824d80a8525035ba067069b466596d",
"license": "Selec... | [] | TheoremDB P8840 · Candidate awaiting source review | null | Source-reported assessment, retrieved 8 September 2026. Frontier Science has not independently reviewed this status.
Open. No signed Lean verification is attached to this problem or its selected resolution.
This record is in the source backlog; its novelty and unresolved status have not passed source review. | null | published | incremental | hard | [
"source-candidate",
"theoremdb"
] | null | null | null | null | [
{
"kind": "website",
"citation": "TheoremDB P8840",
"url": "https://theoremdb.org/statements/P8840",
"note": null,
"sort_order": 0
},
{
"kind": "formalization",
"citation": "Source formal statement (not locally verified)",
"url": "https://theoremdb.org/statements/P8840",
"not... | {
"source_record": "P8840",
"imported_at": "2026-09-08",
"status_as_of": "2026-09-08",
"evidence_label": "Source candidate · not reviewed",
"note": "Full public directory statement. Source lane: backlog; source resolution: open. Source verification claims are attributed only; no local Lean verification has be... | {"created_at":"2026-08-17T19:25:30+00:00","display_standard":{"formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":8840,"public_status":{"evidence":{"hasFormalization":false,"hasInformalProof":false,"hasRecordedWork":false,"solutionRecord":null},"lean":{"kernelAccepted":false,"label":"","reaso... | [
"formalization-not-compiled-during-export",
"current-status-unassessed",
"source-full-text-not-present"
] | [] | {
"created_at": "2026-09-08T03:32:15.935Z",
"updated_at": "2026-09-08T03:32:15.935Z",
"published_at": "2026-09-08T03:33:06.494Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"3c1dee48-4f0f-46c9-b105-72a9b90f7277","format_issues":null,"number":6754} |
1.0.0 | problem:theoremdb-p8842 | problem | theoremdb-p8842 | Erdős Problem 774 on Covering Proportionately Dissociated Sets | {
"main": "math",
"sub": "combinatorics",
"slug": "combinatorics",
"name": "Combinatorics",
"original_discipline": "mathematics"
} | [] | [
{
"code": "05",
"basis": "community-map",
"evidence": "Inferred from community field: combinatorics"
}
] | A set $A \subseteq \mathbb{N}$ is called dissociated if for all finite subsets $X, Y \subseteq A$ with $X \neq Y$, we have $\sum_{n \in X} n \neq \sum_{m \in Y} m$.
A set $A \subseteq \mathbb{N}$ is called proportionately dissociated if there exists a constant $c > 0$ such that every finite subset $B \subseteq A$ cont... | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P8842",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P8842",
"citation": null,
"doi": null,
"arxiv_id": null,
"license": "Selected packets label their content CC0-1.0; original references retain their own terms",
"notes": "Selected problem... | theorem erdos_774 : answer(sorry) ↔ ∀ (A : Set ℕ), A.Infinite → A.IsProportionatelyDissociated →
∃ (T : Set (Set ℕ)), (∀ S ∈ T, S.IsDissociated) ∧ T.Finite ∧ A = ⋃₀ T | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem erdos_774 : answer(sorry) ↔ ∀ (A : Set ℕ), A.Infinite → A.IsProportionatelyDissociated →\n ∃ (T : Set (Set ℕ)), (∀ S ∈ T, S.IsDissociated) ∧ T.Finite ∧ A = ⋃₀ T",
"target": null,
"source_url": "https://theoremdb.org/statements/P8842",
"format": "declar... | [] | TheoremDB P8842 · Candidate awaiting source review | null | Source-reported assessment, retrieved 8 September 2026. Frontier Science has not independently reviewed this status.
Open. No signed Lean verification is attached to this problem or its selected resolution.
This record is in the source backlog; its novelty and unresolved status have not passed source review. | null | published | incremental | hard | [
"source-candidate",
"theoremdb"
] | null | null | null | null | [
{
"kind": "website",
"citation": "TheoremDB P8842",
"url": "https://theoremdb.org/statements/P8842",
"note": null,
"sort_order": 0
},
{
"kind": "formalization",
"citation": "Source formal statement (not locally verified)",
"url": "https://theoremdb.org/statements/P8842",
"not... | {
"source_record": "P8842",
"imported_at": "2026-09-08",
"status_as_of": "2026-09-08",
"evidence_label": "Source candidate · not reviewed",
"note": "Full public directory statement. Source lane: backlog; source resolution: open. Source verification claims are attributed only; no local Lean verification has be... | {"created_at":"2026-08-17T19:25:45+00:00","display_standard":{"formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":8842,"public_status":{"evidence":{"hasFormalization":false,"hasInformalProof":false,"hasRecordedWork":false,"solutionRecord":null},"lean":{"kernelAccepted":false,"label":"","reaso... | [
"formalization-contains-placeholder",
"formalization-not-compiled-during-export",
"current-status-unassessed",
"source-full-text-not-present"
] | [] | {
"created_at": "2026-09-08T03:32:15.935Z",
"updated_at": "2026-09-08T03:32:15.935Z",
"published_at": "2026-09-08T03:32:56.000Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"563e89bd-edba-4959-99d4-b9254aad2d61","format_issues":null,"number":4644} |
1.0.0 | problem:theoremdb-p8844 | problem | theoremdb-p8844 | Consecutive Powerful Numbers | {
"main": "math",
"sub": "number theory",
"slug": "number-theory",
"name": "Number theory",
"original_discipline": "mathematics"
} | [] | [
{
"code": "11",
"basis": "community-map",
"evidence": "Inferred from community field: number-theory"
}
] | A positive integer is called powerful if for every prime $p$ dividing it, $p^2$ also divides it; equivalently, in its prime factorization every prime occurs with exponent at least $2$. Prove that there do not exist three consecutive powerful numbers. That is, prove that there is no natural number $n$ such that $n$, $n+... | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P8844",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P8844",
"citation": null,
"doi": null,
"arxiv_id": null,
"license": "Selected packets label their content CC0-1.0; original references retain their own terms",
"notes": "Selected problem... | theorem erdos_364 :
¬ ∃ (n : ℕ), Powerful n ∧ Powerful (n + 1) ∧ Powerful (n + 2) | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem erdos_364 :\n ¬ ∃ (n : ℕ), Powerful n ∧ Powerful (n + 1) ∧ Powerful (n + 2)",
"target": null,
"source_url": "https://theoremdb.org/statements/P8844",
"format": "declaration-or-fragment",
"sha256": "190e1fc5753d879e32da936cdec5313637f9cdf0822c52d480... | [] | TheoremDB P8844 · Candidate awaiting source review | null | Source-reported assessment, retrieved 8 September 2026. Frontier Science has not independently reviewed this status.
Open. No signed Lean verification is attached to this problem or its selected resolution.
This record is in the source backlog; its novelty and unresolved status have not passed source review. | null | published | incremental | hard | [
"source-candidate",
"theoremdb"
] | null | null | null | null | [
{
"kind": "website",
"citation": "TheoremDB P8844",
"url": "https://theoremdb.org/statements/P8844",
"note": null,
"sort_order": 0
},
{
"kind": "formalization",
"citation": "Source formal statement (not locally verified)",
"url": "https://theoremdb.org/statements/P8844",
"not... | {
"source_record": "P8844",
"imported_at": "2026-09-08",
"status_as_of": "2026-09-08",
"evidence_label": "Source candidate · not reviewed",
"note": "Full public directory statement. Source lane: backlog; source resolution: open. Source verification claims are attributed only; no local Lean verification has be... | {"created_at":"2026-08-17T19:25:56+00:00","display_standard":{"editorial_review":"submitter_supplied","formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":8844,"public_status":{"evidence":{"hasFormalization":false,"hasInformalProof":false,"hasRecordedWork":false,"solutionRecord":null},"lean":{... | [
"formalization-not-compiled-during-export",
"current-status-unassessed",
"source-full-text-not-present"
] | [] | {
"created_at": "2026-09-08T03:32:15.935Z",
"updated_at": "2026-09-08T03:32:15.935Z",
"published_at": "2026-09-08T03:32:52.605Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"cb7e0c96-9bf3-4159-a59e-d619f5b6f05b","format_issues":null,"number":3896} |
1.0.0 | problem:theoremdb-p8846 | problem | theoremdb-p8846 | Existence of a Composite Lucas Sequence with Trivial Common Divisors | {
"main": "math",
"sub": "number theory",
"slug": "number-theory",
"name": "Number theory",
"original_discipline": "mathematics"
} | [] | [
{
"code": "11",
"basis": "community-map",
"evidence": "Inferred from community field: number-theory"
}
] | A Lucas sequence is a sequence of natural numbers $a_0, a_1, a_2, \ldots$ satisfying the recurrence relation $a_{n+2} = a_{n+1} + a_n$ for all $n \geq 0$. A natural number is composite if it is greater than 1 and not prime. Determine whether there exists a Lucas sequence $(a_k)_{k \geq 0}$ such that every term $a_k$ is... | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P8846",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P8846",
"citation": null,
"doi": null,
"arxiv_id": null,
"license": "Selected packets label their content CC0-1.0; original references retain their own terms",
"notes": "Selected problem... | theorem erdos_276 : answer(sorry) ↔
∃ (a : ℕ → ℕ),
IsLucasSequence a ∧ (∀ k, (a k).Composite) ∧ (∀ n > 1, ∃ k, Nat.gcd n (a k) = 1) | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem erdos_276 : answer(sorry) ↔\n ∃ (a : ℕ → ℕ),\n IsLucasSequence a ∧ (∀ k, (a k).Composite) ∧ (∀ n > 1, ∃ k, Nat.gcd n (a k) = 1)",
"target": null,
"source_url": "https://theoremdb.org/statements/P8846",
"format": "declaration-or-fragment",
"sha25... | [] | TheoremDB P8846 · Candidate awaiting source review | null | Source-reported assessment, retrieved 8 September 2026. Frontier Science has not independently reviewed this status.
Open. No signed Lean verification is attached to this problem or its selected resolution.
This record is in the source backlog; its novelty and unresolved status have not passed source review. | null | published | incremental | hard | [
"source-candidate",
"theoremdb"
] | null | null | null | null | [
{
"kind": "website",
"citation": "TheoremDB P8846",
"url": "https://theoremdb.org/statements/P8846",
"note": null,
"sort_order": 0
},
{
"kind": "formalization",
"citation": "Source formal statement (not locally verified)",
"url": "https://theoremdb.org/statements/P8846",
"not... | {
"source_record": "P8846",
"imported_at": "2026-09-08",
"status_as_of": "2026-09-08",
"evidence_label": "Source candidate · not reviewed",
"note": "Full public directory statement. Source lane: backlog; source resolution: open. Source verification claims are attributed only; no local Lean verification has be... | {"created_at":"2026-08-17T19:26:00+00:00","display_standard":{"editorial_review":"submitter_supplied","formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":8846,"public_status":{"evidence":{"hasFormalization":false,"hasInformalProof":false,"hasRecordedWork":false,"solutionRecord":null},"lean":{... | [
"formalization-contains-placeholder",
"formalization-not-compiled-during-export",
"current-status-unassessed",
"source-full-text-not-present"
] | [] | {
"created_at": "2026-09-08T03:32:15.935Z",
"updated_at": "2026-09-08T03:32:15.935Z",
"published_at": "2026-09-08T03:32:58.408Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"3e12fc92-e979-49b4-bd00-049a873fb289","format_issues":null,"number":5176} |
1.0.0 | problem:theoremdb-p8848 | problem | theoremdb-p8848 | Growth Rate of the Cumulative Divisor Count of Mersenne Numbers | {
"main": "math",
"sub": "combinatorics",
"slug": "combinatorics",
"name": "Combinatorics",
"original_discipline": "mathematics"
} | [] | [
{
"code": "05",
"basis": "community-map",
"evidence": "Inferred from community field: combinatorics"
}
] | For each positive integer $k$, let $M_k = 2^k - 1$ denote the $k$th Mersenne number, and let $\tau(M_k)$ denote the number of positive divisors of $M_k$. Define the cumulative function
$$
f(n) = \sum_{k=1}^{n} \tau(2^k - 1)
$$
for each positive integer $n$. Determine whether
$$
\lim_{n \to \infty} \frac{f(2n)}{f(n)}... | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P8848",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P8848",
"citation": null,
"doi": null,
"arxiv_id": null,
"license": "Selected packets label their content CC0-1.0; original references retain their own terms",
"notes": "Selected problem... | theorem erdos_893 :
answer(sorry) ↔ Tendsto (fun n : ℕ => (f (2 * n) : ℝ) / (f n : ℝ)) atTop atTop | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem erdos_893 :\n answer(sorry) ↔ Tendsto (fun n : ℕ => (f (2 * n) : ℝ) / (f n : ℝ)) atTop atTop",
"target": null,
"source_url": "https://theoremdb.org/statements/P8848",
"format": "declaration-or-fragment",
"sha256": "89accc2e0287af96dd70501ad3d25b441... | [] | TheoremDB P8848 · Candidate awaiting source review | null | Source-reported assessment, retrieved 8 September 2026. Frontier Science has not independently reviewed this status.
Open. No signed Lean verification is attached to this problem or its selected resolution.
This record is in the source backlog; its novelty and unresolved status have not passed source review. | null | published | incremental | hard | [
"source-candidate",
"theoremdb"
] | null | null | null | null | [
{
"kind": "website",
"citation": "TheoremDB P8848",
"url": "https://theoremdb.org/statements/P8848",
"note": null,
"sort_order": 0
},
{
"kind": "formalization",
"citation": "Source formal statement (not locally verified)",
"url": "https://theoremdb.org/statements/P8848",
"not... | {
"source_record": "P8848",
"imported_at": "2026-09-08",
"status_as_of": "2026-09-08",
"evidence_label": "Source candidate · not reviewed",
"note": "Full public directory statement. Source lane: backlog; source resolution: open. Source verification claims are attributed only; no local Lean verification has be... | {"created_at":"2026-08-17T19:26:13+00:00","display_standard":{"editorial_review":"submitter_supplied","formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":8848,"public_status":{"evidence":{"hasFormalization":false,"hasInformalProof":false,"hasRecordedWork":false,"solutionRecord":null},"lean":{... | [
"formalization-contains-placeholder",
"formalization-not-compiled-during-export",
"current-status-unassessed",
"source-full-text-not-present"
] | [] | {
"created_at": "2026-09-08T03:32:15.935Z",
"updated_at": "2026-09-08T03:32:15.935Z",
"published_at": "2026-09-08T03:32:59.511Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"478cd115-996d-401c-afe5-f63b40e38542","format_issues":null,"number":5411} |
1.0.0 | problem:theoremdb-p8850 | problem | theoremdb-p8850 | Erdős's problem on uncountable graphs with large independent sets in all large finite subgraphs | {
"main": "math",
"sub": "combinatorics",
"slug": "combinatorics",
"name": "Combinatorics",
"original_discipline": "mathematics"
} | [] | [
{
"code": "05",
"basis": "community-map",
"evidence": "Inferred from community field: combinatorics"
}
] | Does there exist a graph with chromatic number $\aleph_1$ and exactly $\aleph_1$ vertices such that for every real number $\varepsilon > 0$, there exists a natural number $N$ with the following property: for every natural number $n \geq N$ and every subgraph $H$ on exactly $n$ vertices, there exists an independent set ... | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P8850",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P8850",
"citation": null,
"doi": null,
"arxiv_id": null,
"license": "Selected packets label their content CC0-1.0; original references retain their own terms",
"notes": "Selected problem... | theorem erdos_75 :
answer(sorry) ↔
∃ (V : Type) (G : SimpleGraph V),
G.chromaticCardinal = ℵ_ 1 ∧
#V = ℵ_ 1 ∧
∀ ε > (0 : ℝ),
∀ᶠ (n : ℕ) in Filter.atTop, ∀ (H : G.Subgraph),
H.verts.ncard = n →
∃ (I : Finset V),
(I : Set V) ⊆ H.verts ∧
... | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem erdos_75 :\n answer(sorry) ↔\n ∃ (V : Type) (G : SimpleGraph V),\n G.chromaticCardinal = ℵ_ 1 ∧\n #V = ℵ_ 1 ∧\n ∀ ε > (0 : ℝ),\n ∀ᶠ (n : ℕ) in Filter.atTop, ∀ (H : G.Subgraph),\n H.verts.ncard = n →\n ∃ (I : Finset V)... | [] | TheoremDB P8850 · Candidate awaiting source review | null | Source-reported assessment, retrieved 8 September 2026. Frontier Science has not independently reviewed this status.
Open. No signed Lean verification is attached to this problem or its selected resolution.
This record is in the source backlog; its novelty and unresolved status have not passed source review. | null | published | incremental | hard | [
"source-candidate",
"theoremdb"
] | null | null | null | null | [
{
"kind": "website",
"citation": "TheoremDB P8850",
"url": "https://theoremdb.org/statements/P8850",
"note": null,
"sort_order": 0
},
{
"kind": "formalization",
"citation": "Source formal statement (not locally verified)",
"url": "https://theoremdb.org/statements/P8850",
"not... | {
"source_record": "P8850",
"imported_at": "2026-09-08",
"status_as_of": "2026-09-08",
"evidence_label": "Source candidate · not reviewed",
"note": "Full public directory statement. Source lane: backlog; source resolution: open. Source verification claims are attributed only; no local Lean verification has be... | {"created_at":"2026-08-17T19:26:19+00:00","display_standard":{"editorial_review":"submitter_supplied","formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":8850,"public_status":{"evidence":{"hasFormalization":false,"hasInformalProof":false,"hasRecordedWork":false,"solutionRecord":null},"lean":{... | [
"formalization-contains-placeholder",
"formalization-not-compiled-during-export",
"current-status-unassessed",
"source-full-text-not-present"
] | [] | {
"created_at": "2026-09-08T03:32:15.935Z",
"updated_at": "2026-09-08T03:32:15.935Z",
"published_at": "2026-09-08T03:32:56.568Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"029b64b7-6369-447b-b0fd-b8429bb3858f","format_issues":null,"number":4768} |
1.0.0 | problem:theoremdb-p8852 | problem | theoremdb-p8852 | De Giorgi's conjecture in dimension eight | {
"main": "math",
"sub": "partial differential equations",
"slug": "partial-differential-equations",
"name": "Partial differential equations",
"original_discipline": "mathematics"
} | [] | [
{
"code": "35",
"basis": "community-map",
"evidence": "Inferred from community field: partial-differential-equations"
}
] | Let $u : \mathbb{R}^8 \to \mathbb{R}$ be a function. We say that $u$ is a bounded classical solution to the Allen–Cahn equation if $u$ is twice continuously differentiable, there exists $C > 0$ such that $|u(x)| \leq C$ for all $x \in \mathbb{R}^8$, and $\Delta u(x) + u(x) - u(x)^3 = 0$ for all $x \in \mathbb{R}^8$, wh... | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P8852",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P8852",
"citation": null,
"doi": null,
"arxiv_id": null,
"license": "Selected packets label their content CC0-1.0; original references retain their own terms",
"notes": "Selected problem... | theorem DeGiorgi_eight : (DeGiorgi_conclusion 8) | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem DeGiorgi_eight : (DeGiorgi_conclusion 8)",
"target": null,
"source_url": "https://theoremdb.org/statements/P8852",
"format": "declaration-or-fragment",
"sha256": "ed85d6d95fc73b1955d6c802fec7105477b38b9ec6f94ab1e04bd3c59940b028",
"license": "Selec... | [] | TheoremDB P8852 · Candidate awaiting source review | null | Source-reported assessment, retrieved 8 September 2026. Frontier Science has not independently reviewed this status.
Open. No signed Lean verification is attached to this problem or its selected resolution.
This record is in the source backlog; its novelty and unresolved status have not passed source review. | null | published | incremental | hard | [
"source-candidate",
"theoremdb"
] | null | null | null | null | [
{
"kind": "website",
"citation": "TheoremDB P8852",
"url": "https://theoremdb.org/statements/P8852",
"note": null,
"sort_order": 0
},
{
"kind": "formalization",
"citation": "Source formal statement (not locally verified)",
"url": "https://theoremdb.org/statements/P8852",
"not... | {
"source_record": "P8852",
"imported_at": "2026-09-08",
"status_as_of": "2026-09-08",
"evidence_label": "Source candidate · not reviewed",
"note": "Full public directory statement. Source lane: backlog; source resolution: open. Source verification claims are attributed only; no local Lean verification has be... | {"created_at":"2026-08-17T19:26:37+00:00","display_standard":{"editorial_review":"submitter_supplied","formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":8852,"public_status":{"evidence":{"hasFormalization":false,"hasInformalProof":false,"hasRecordedWork":false,"solutionRecord":null},"lean":{... | [
"formalization-not-compiled-during-export",
"current-status-unassessed",
"source-full-text-not-present"
] | [] | {
"created_at": "2026-09-08T03:32:15.935Z",
"updated_at": "2026-09-08T03:32:15.935Z",
"published_at": "2026-09-08T03:32:52.900Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"d9e0187d-4cf2-4ab7-b412-8c2dc9a09e2e","format_issues":null,"number":3960} |
1.0.0 | problem:theoremdb-p8854 | problem | theoremdb-p8854 | The Extended Sierpiński Problem | {
"main": "math",
"sub": "number theory",
"slug": "number-theory",
"name": "Number theory",
"original_discipline": "mathematics"
} | [] | [
{
"code": "11",
"basis": "community-map",
"evidence": "Inferred from community field: number-theory"
}
] | A positive odd integer $k$ is called a *Sierpiński number* if $k \cdot 2^n + 1$ is composite for every natural number $n$. Let $S$ denote the set of all Sierpiński numbers. Determine whether $271129$ is the least element of the set $\{k \in S \mid \exists k' \in S,\; k' < k\}$, i.e., whether $271129$ is the second-smal... | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P8854",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P8854",
"citation": null,
"doi": null,
"arxiv_id": null,
"license": "Selected packets label their content CC0-1.0; original references retain their own terms",
"notes": "Selected problem... | theorem extended_sierpinski_problem :
answer(sorry) ↔
IsLeast {k | k.IsSierpinskiNumber ∧
∃ k', k'.IsSierpinskiNumber ∧ k' < k} 271129 | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem extended_sierpinski_problem :\n answer(sorry) ↔\n IsLeast {k | k.IsSierpinskiNumber ∧\n ∃ k', k'.IsSierpinskiNumber ∧ k' < k} 271129",
"target": null,
"source_url": "https://theoremdb.org/statements/P8854",
"format": "declaration-or-fragmen... | [] | TheoremDB P8854 · Candidate awaiting source review | null | Source-reported assessment, retrieved 8 September 2026. Frontier Science has not independently reviewed this status.
Open. No signed Lean verification is attached to this problem or its selected resolution.
This record is in the source backlog; its novelty and unresolved status have not passed source review. | null | published | incremental | hard | [
"source-candidate",
"theoremdb"
] | null | null | null | null | [
{
"kind": "website",
"citation": "TheoremDB P8854",
"url": "https://theoremdb.org/statements/P8854",
"note": null,
"sort_order": 0
},
{
"kind": "formalization",
"citation": "Source formal statement (not locally verified)",
"url": "https://theoremdb.org/statements/P8854",
"not... | {
"source_record": "P8854",
"imported_at": "2026-09-08",
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"evidence_label": "Source candidate · not reviewed",
"note": "Full public directory statement. Source lane: backlog; source resolution: open. Source verification claims are attributed only; no local Lean verification has be... | {"created_at":"2026-08-17T19:26:43+00:00","display_standard":{"formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":8854,"public_status":{"evidence":{"hasFormalization":false,"hasInformalProof":false,"hasRecordedWork":false,"solutionRecord":null},"lean":{"kernelAccepted":false,"label":"","reaso... | [
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"updated_at": "2026-09-08T03:32:15.935Z",
"published_at": "2026-09-08T03:33:06.484Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"69458cd0-1c72-4a73-9f27-95986b995162","format_issues":null,"number":6752} |
1.0.0 | problem:theoremdb-p8856 | problem | theoremdb-p8856 | Monotone 4-Term Arithmetic Progressions in Permutations of the Natural Numbers | {
"main": "math",
"sub": "combinatorics",
"slug": "combinatorics",
"name": "Combinatorics",
"original_discipline": "mathematics"
} | [] | [
{
"code": "05",
"basis": "community-map",
"evidence": "Inferred from community field: combinatorics"
}
] | A permutation of the natural numbers is a bijection from $\mathbb{N}$ to $\mathbb{N}$. For a permutation $f$ and a positive integer $k$, we say that $f$ contains a monotone $k$-term arithmetic progression if there exist indices $i_1 < i_2 < \cdots < i_k$ such that either $f(i_1) < f(i_2) < \cdots < f(i_k)$ or $f(i_1) >... | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P8856",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P8856",
"citation": null,
"doi": null,
"arxiv_id": null,
"license": "Selected packets label their content CC0-1.0; original references retain their own terms",
"notes": "Selected problem... | theorem erdos_196 : answer(sorry) ↔ ∀ (f : ℕ ≃ ℕ), HasMonotoneAP f 4 | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem erdos_196 : answer(sorry) ↔ ∀ (f : ℕ ≃ ℕ), HasMonotoneAP f 4",
"target": null,
"source_url": "https://theoremdb.org/statements/P8856",
"format": "declaration-or-fragment",
"sha256": "90b3793ec7c608945b4cf0a0cc2d7052c3d9bd6e6cf8e5349a66e3ff169279d5",
... | [] | TheoremDB P8856 · Candidate awaiting source review | null | Source-reported assessment, retrieved 8 September 2026. Frontier Science has not independently reviewed this status.
Open. No signed Lean verification is attached to this problem or its selected resolution.
This record is in the source backlog; its novelty and unresolved status have not passed source review. | null | published | incremental | hard | [
"source-candidate",
"theoremdb"
] | null | null | null | null | [
{
"kind": "website",
"citation": "TheoremDB P8856",
"url": "https://theoremdb.org/statements/P8856",
"note": null,
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},
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"citation": "Source formal statement (not locally verified)",
"url": "https://theoremdb.org/statements/P8856",
"not... | {
"source_record": "P8856",
"imported_at": "2026-09-08",
"status_as_of": "2026-09-08",
"evidence_label": "Source candidate · not reviewed",
"note": "Full public directory statement. Source lane: backlog; source resolution: open. Source verification claims are attributed only; no local Lean verification has be... | {"created_at":"2026-08-17T19:26:55+00:00","display_standard":{"editorial_review":"submitter_supplied","formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":8856,"public_status":{"evidence":{"hasFormalization":false,"hasInformalProof":false,"hasRecordedWork":false,"solutionRecord":null},"lean":{... | [
"formalization-contains-placeholder",
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"current-status-unassessed",
"source-full-text-not-present"
] | [] | {
"created_at": "2026-09-08T03:32:15.935Z",
"updated_at": "2026-09-08T03:32:15.935Z",
"published_at": "2026-09-08T03:33:04.086Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"3030cd69-c3a7-4f27-b8bd-f7c50ce0dcd7","format_issues":null,"number":6235} |
1.0.0 | problem:theoremdb-p8860 | problem | theoremdb-p8860 | Asymptotic growth rate of the maximal length of a distinct consecutive-sums sequence | {
"main": "math",
"sub": "number theory",
"slug": "number-theory",
"name": "Number theory",
"original_discipline": "mathematics"
} | [] | [
{
"code": "11",
"basis": "community-map",
"evidence": "Inferred from community field: number-theory"
}
] | For each positive integer $n$, let $f(n)$ denote the largest integer $k$ for which there exist integers $1 \le a_1 < a_2 < \dots < a_k \le n$ such that all sums of the form $\sum_{i=u}^{v} a_i$ with $1 \le u \le v \le k$ are distinct. Determine an explicit function $g : \mathbb{N} \to \mathbb{R}$ such that $f(n) = \The... | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P8860",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P8860",
"citation": null,
"doi": null,
"arxiv_id": null,
"license": "Selected packets label their content CC0-1.0; original references retain their own terms",
"notes": "Selected problem... | theorem erdos_357.parts.ii.bigTheta_version :
(fun n ↦ (f n : ℝ)) =Θ[atTop] (answer(sorry) : ℕ → ℝ) | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem erdos_357.parts.ii.bigTheta_version :\n (fun n ↦ (f n : ℝ)) =Θ[atTop] (answer(sorry) : ℕ → ℝ)",
"target": null,
"source_url": "https://theoremdb.org/statements/P8860",
"format": "declaration-or-fragment",
"sha256": "1419280b27cfa8a146fc9b2a28bee1f8... | [] | TheoremDB P8860 · Candidate awaiting source review | null | Source-reported assessment, retrieved 8 September 2026. Frontier Science has not independently reviewed this status.
Open. No signed Lean verification is attached to this problem or its selected resolution.
This record is in the source backlog; its novelty and unresolved status have not passed source review. | null | published | incremental | hard | [
"source-candidate",
"theoremdb"
] | null | null | null | null | [
{
"kind": "website",
"citation": "TheoremDB P8860",
"url": "https://theoremdb.org/statements/P8860",
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},
{
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"citation": "Source formal statement (not locally verified)",
"url": "https://theoremdb.org/statements/P8860",
"not... | {
"source_record": "P8860",
"imported_at": "2026-09-08",
"status_as_of": "2026-09-08",
"evidence_label": "Source candidate · not reviewed",
"note": "Full public directory statement. Source lane: backlog; source resolution: open. Source verification claims are attributed only; no local Lean verification has be... | {"created_at":"2026-08-17T19:35:14+00:00","display_standard":{"editorial_review":"submitter_supplied","formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":8860,"public_status":{"evidence":{"hasFormalization":false,"hasInformalProof":false,"hasRecordedWork":false,"solutionRecord":null},"lean":{... | [
"formalization-contains-placeholder",
"formalization-not-compiled-during-export",
"current-status-unassessed",
"source-full-text-not-present"
] | [] | {
"created_at": "2026-09-08T03:32:15.935Z",
"updated_at": "2026-09-08T03:32:15.935Z",
"published_at": "2026-09-08T03:32:50.278Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"e7c878ea-a1a7-4e6a-80ff-4e834090d45f","format_issues":null,"number":3387} |
1.0.0 | problem:theoremdb-p8862 | problem | theoremdb-p8862 | Strict lower bound for the off-diagonal van der Waerden number W(3, 32) | {
"main": "math",
"sub": "combinatorics",
"slug": "combinatorics",
"name": "Combinatorics",
"original_discipline": "mathematics"
} | [] | [
{
"code": "05",
"basis": "community-map",
"evidence": "Inferred from community field: combinatorics"
}
] | For positive integers $k$ and $r$, the 2-color van der Waerden number $W(k, r)$ is defined as the least positive integer $N$ such that every coloring of $\{1, 2, \ldots, N\}$ with two colors contains either a monochromatic arithmetic progression of length $k$ in the first color or a monochromatic arithmetic progression... | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P8862",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P8862",
"citation": null,
"doi": null,
"arxiv_id": null,
"license": "Selected packets label their content CC0-1.0; original references retain their own terms",
"notes": "Selected problem... | theorem W_3_32_lower : answer(sorry) ↔ W 3 32 > 1006 | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem W_3_32_lower : answer(sorry) ↔ W 3 32 > 1006",
"target": null,
"source_url": "https://theoremdb.org/statements/P8862",
"format": "declaration-or-fragment",
"sha256": "a65cf856d34abdb82c4706c7e75526152d1b51ea51f39f23842f54581cad1cda",
"license": "S... | [] | TheoremDB P8862 · Candidate awaiting source review | null | Source-reported assessment, retrieved 8 September 2026. Frontier Science has not independently reviewed this status.
Open. No signed Lean verification is attached to this problem or its selected resolution.
This record is in the source backlog; its novelty and unresolved status have not passed source review. | null | published | incremental | hard | [
"source-candidate",
"theoremdb"
] | null | null | null | null | [
{
"kind": "website",
"citation": "TheoremDB P8862",
"url": "https://theoremdb.org/statements/P8862",
"note": null,
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},
{
"kind": "formalization",
"citation": "Source formal statement (not locally verified)",
"url": "https://theoremdb.org/statements/P8862",
"not... | {
"source_record": "P8862",
"imported_at": "2026-09-08",
"status_as_of": "2026-09-08",
"evidence_label": "Source candidate · not reviewed",
"note": "Full public directory statement. Source lane: backlog; source resolution: open. Source verification claims are attributed only; no local Lean verification has be... | {"created_at":"2026-08-17T19:35:23+00:00","display_standard":{"editorial_review":"submitter_supplied","formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":8862,"public_status":{"evidence":{"hasFormalization":false,"hasInformalProof":false,"hasRecordedWork":false,"solutionRecord":null},"lean":{... | [
"formalization-contains-placeholder",
"formalization-not-compiled-during-export",
"current-status-unassessed",
"source-full-text-not-present"
] | [] | {
"created_at": "2026-09-08T03:32:15.935Z",
"updated_at": "2026-09-08T03:32:15.935Z",
"published_at": "2026-09-08T03:33:06.088Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"2d89af97-bb12-42f5-9cc9-73eedb7361ab","format_issues":null,"number":6668} |
1.0.0 | problem:theoremdb-p8864 | problem | theoremdb-p8864 | Existence of a SIC-POVM in dimension 70 | {
"main": "math",
"sub": "algebra",
"slug": "algebra",
"name": "Algebra",
"original_discipline": "mathematics"
} | [] | [] | For a positive integer $d$, a SIC family in dimension $d$ is a collection of $d^2$ vectors $\Phi_0, \Phi_1, \dots, \Phi_{d^2-1}$ in the $d$-dimensional complex Hilbert space $\mathbb{C}^d$ such that each vector has unit $L^2$ norm and every pair of distinct vectors has squared overlap equal to $(d+1)^{-1}$. Here the sq... | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P8864",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P8864",
"citation": null,
"doi": null,
"arxiv_id": null,
"license": "Selected packets label their content CC0-1.0; original references retain their own terms",
"notes": "Selected problem... | theorem hasSICPOVM_70 : answer(sorry) ↔ HasSICPOVM 70 | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem hasSICPOVM_70 : answer(sorry) ↔ HasSICPOVM 70",
"target": null,
"source_url": "https://theoremdb.org/statements/P8864",
"format": "declaration-or-fragment",
"sha256": "670af735b6874b7a1b2de23494577ef3a3c23e5c413438a4f679b9ae5a36588a",
"license": "... | [] | TheoremDB P8864 · Candidate awaiting source review | null | Source-reported assessment, retrieved 8 September 2026. Frontier Science has not independently reviewed this status.
Open. No signed Lean verification is attached to this problem or its selected resolution.
This record is in the source backlog; its novelty and unresolved status have not passed source review. | null | published | incremental | hard | [
"source-candidate",
"theoremdb"
] | null | null | null | null | [
{
"kind": "website",
"citation": "TheoremDB P8864",
"url": "https://theoremdb.org/statements/P8864",
"note": null,
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},
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"kind": "formalization",
"citation": "Source formal statement (not locally verified)",
"url": "https://theoremdb.org/statements/P8864",
"not... | {
"source_record": "P8864",
"imported_at": "2026-09-08",
"status_as_of": "2026-09-08",
"evidence_label": "Source candidate · not reviewed",
"note": "Full public directory statement. Source lane: backlog; source resolution: open. Source verification claims are attributed only; no local Lean verification has be... | {"created_at":"2026-08-17T19:35:31+00:00","display_standard":{"editorial_review":"submitter_supplied","formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":8864,"public_status":{"evidence":{"hasFormalization":false,"hasInformalProof":false,"hasRecordedWork":false,"solutionRecord":null},"lean":{... | [
"formalization-contains-placeholder",
"formalization-not-compiled-during-export",
"current-status-unassessed",
"source-full-text-not-present"
] | [] | {
"created_at": "2026-09-08T03:32:15.935Z",
"updated_at": "2026-09-08T03:32:15.935Z",
"published_at": "2026-09-08T03:32:58.522Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"7822a5e0-a201-4ba4-a259-4d4e73875483","format_issues":null,"number":5200} |
1.0.0 | problem:theoremdb-p8866 | problem | theoremdb-p8866 | Unboundedness of the number of representations of integers as sums of like prime powers | {
"main": "math",
"sub": "number theory",
"slug": "number-theory",
"name": "Number theory",
"original_discipline": "mathematics"
} | [] | [
{
"code": "11",
"basis": "community-map",
"evidence": "Inferred from community field: number-theory"
}
] | For integers $n \geq 1$ and $k \geq 2$, let $f_k(n)$ denote the number of multisets $\{p_1, \ldots, p_k\}$ of prime numbers such that $n = p_1^k + p_2^k + \cdots + p_k^k$, where the primes in a multiset need not be distinct and two multisets giving the same sum are considered the same solution if and only if they conta... | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P8866",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P8866",
"citation": null,
"doi": null,
"arxiv_id": null,
"license": "Selected packets label their content CC0-1.0; original references retain their own terms",
"notes": "Selected problem... | theorem erdos_979 : answer(sorry) ↔
∀ k ≥ 2, Filter.limsup (fun n => (solutionSet n k).encard) Filter.atTop = ⊤ | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem erdos_979 : answer(sorry) ↔\n ∀ k ≥ 2, Filter.limsup (fun n => (solutionSet n k).encard) Filter.atTop = ⊤",
"target": null,
"source_url": "https://theoremdb.org/statements/P8866",
"format": "declaration-or-fragment",
"sha256": "6cfbe13bb6a2c2eb91ce... | [] | TheoremDB P8866 · Candidate awaiting source review | null | Source-reported assessment, retrieved 8 September 2026. Frontier Science has not independently reviewed this status.
Open. No signed Lean verification is attached to this problem or its selected resolution.
This record is in the source backlog; its novelty and unresolved status have not passed source review. | null | published | incremental | hard | [
"source-candidate",
"theoremdb"
] | null | null | null | null | [
{
"kind": "website",
"citation": "TheoremDB P8866",
"url": "https://theoremdb.org/statements/P8866",
"note": null,
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},
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"kind": "formalization",
"citation": "Source formal statement (not locally verified)",
"url": "https://theoremdb.org/statements/P8866",
"not... | {
"source_record": "P8866",
"imported_at": "2026-09-08",
"status_as_of": "2026-09-08",
"evidence_label": "Source candidate · not reviewed",
"note": "Full public directory statement. Source lane: backlog; source resolution: open. Source verification claims are attributed only; no local Lean verification has be... | {"created_at":"2026-08-17T19:35:38+00:00","display_standard":{"editorial_review":"submitter_supplied","formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":8866,"public_status":{"evidence":{"hasFormalization":false,"hasInformalProof":false,"hasRecordedWork":false,"solutionRecord":null},"lean":{... | [
"formalization-contains-placeholder",
"formalization-not-compiled-during-export",
"current-status-unassessed",
"source-full-text-not-present"
] | [] | {
"created_at": "2026-09-08T03:32:15.935Z",
"updated_at": "2026-09-08T03:32:15.935Z",
"published_at": "2026-09-08T03:33:07.180Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"99f32697-53e7-45fc-9d2e-08079f59e9a7","format_issues":null,"number":6904} |
1.0.0 | problem:theoremdb-p8868 | problem | theoremdb-p8868 | Infinitely many arithmetic progressions of three consecutive primes | {
"main": "math",
"sub": "combinatorics",
"slug": "combinatorics",
"name": "Combinatorics",
"original_discipline": "mathematics"
} | [] | [
{
"code": "05",
"basis": "community-map",
"evidence": "Inferred from community field: combinatorics"
}
] | For a positive integer $k$, a set $s$ of natural numbers is called an arithmetic progression of length $k$ if there exist natural numbers $a$ and $d$ such that $s = \{a + nd : n \in \mathbb{N}, n < k\}$. A set $s$ is called a prime progression of length $k$ if there exists a natural number $a$ such that $s$ has cardina... | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P8868",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P8868",
"citation": null,
"doi": null,
"arxiv_id": null,
"license": "Selected packets label their content CC0-1.0; original references retain their own terms",
"notes": "Selected problem... | theorem erdos_141.variants.infinite_three : answer(sorry) ↔
(consecutivePrimeArithmeticProgressions 3).Infinite | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem erdos_141.variants.infinite_three : answer(sorry) ↔\n (consecutivePrimeArithmeticProgressions 3).Infinite",
"target": null,
"source_url": "https://theoremdb.org/statements/P8868",
"format": "declaration-or-fragment",
"sha256": "258433892d84c8074721... | [] | TheoremDB P8868 · Candidate awaiting source review | null | Source-reported assessment, retrieved 8 September 2026. Frontier Science has not independently reviewed this status.
Open. No signed Lean verification is attached to this problem or its selected resolution.
This record is in the source backlog; its novelty and unresolved status have not passed source review. | null | published | incremental | hard | [
"source-candidate",
"theoremdb"
] | null | null | null | null | [
{
"kind": "website",
"citation": "TheoremDB P8868",
"url": "https://theoremdb.org/statements/P8868",
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"citation": "Source formal statement (not locally verified)",
"url": "https://theoremdb.org/statements/P8868",
"not... | {
"source_record": "P8868",
"imported_at": "2026-09-08",
"status_as_of": "2026-09-08",
"evidence_label": "Source candidate · not reviewed",
"note": "Full public directory statement. Source lane: backlog; source resolution: open. Source verification claims are attributed only; no local Lean verification has be... | {"created_at":"2026-08-17T19:35:45+00:00","display_standard":{"formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":8868,"public_status":{"evidence":{"hasFormalization":false,"hasInformalProof":false,"hasRecordedWork":false,"solutionRecord":null},"lean":{"kernelAccepted":false,"label":"","reaso... | [
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"current-status-unassessed",
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] | [] | {
"created_at": "2026-09-08T03:32:15.935Z",
"updated_at": "2026-09-08T03:32:15.935Z",
"published_at": "2026-09-08T03:33:00.816Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"75dba38b-a957-4b61-aad7-f31ce6e5d7ea","format_issues":null,"number":5596} |
1.0.0 | problem:theoremdb-p8870 | problem | theoremdb-p8870 | Asymptotic upper bound for the maximal length of a sequence with distinct consecutive sums | {
"main": "math",
"sub": "number theory",
"slug": "number-theory",
"name": "Number theory",
"original_discipline": "mathematics"
} | [] | [
{
"code": "11",
"basis": "community-map",
"evidence": "Inferred from community field: number-theory"
}
] | For each positive integer $n$, let $f(n)$ denote the largest integer $k$ for which there exist integers $1 \le a_1 < a_2 < \dots < a_k \le n$ such that all sums of the form $\sum_{i=u}^{v} a_i$ with $1 \le u \le v \le k$ are distinct. Determine an explicit function $g: \mathbb{N} \to \mathbb{R}$ such that $f(n) = o(g(n... | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P8870",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P8870",
"citation": null,
"doi": null,
"arxiv_id": null,
"license": "Selected packets label their content CC0-1.0; original references retain their own terms",
"notes": "Selected problem... | theorem erdos_357.parts.ii.littleO_version_symm :
(fun n ↦ (f n : ℝ)) =o[atTop] (answer(sorry) : ℕ → ℝ) | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem erdos_357.parts.ii.littleO_version_symm :\n (fun n ↦ (f n : ℝ)) =o[atTop] (answer(sorry) : ℕ → ℝ)",
"target": null,
"source_url": "https://theoremdb.org/statements/P8870",
"format": "declaration-or-fragment",
"sha256": "6264be593f9be8a69fa0ace7e201... | [] | TheoremDB P8870 · Candidate awaiting source review | null | Source-reported assessment, retrieved 8 September 2026. Frontier Science has not independently reviewed this status.
Open. No signed Lean verification is attached to this problem or its selected resolution.
This record is in the source backlog; its novelty and unresolved status have not passed source review. | null | published | incremental | hard | [
"source-candidate",
"theoremdb"
] | null | null | null | null | [
{
"kind": "website",
"citation": "TheoremDB P8870",
"url": "https://theoremdb.org/statements/P8870",
"note": null,
"sort_order": 0
},
{
"kind": "formalization",
"citation": "Source formal statement (not locally verified)",
"url": "https://theoremdb.org/statements/P8870",
"not... | {
"source_record": "P8870",
"imported_at": "2026-09-08",
"status_as_of": "2026-09-08",
"evidence_label": "Source candidate · not reviewed",
"note": "Full public directory statement. Source lane: backlog; source resolution: open. Source verification claims are attributed only; no local Lean verification has be... | {"created_at":"2026-08-17T19:35:55+00:00","display_standard":{"editorial_review":"submitter_supplied","formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":8870,"public_status":{"evidence":{"hasFormalization":false,"hasInformalProof":false,"hasRecordedWork":false,"solutionRecord":null},"lean":{... | [
"formalization-contains-placeholder",
"formalization-not-compiled-during-export",
"current-status-unassessed",
"source-full-text-not-present"
] | [] | {
"created_at": "2026-09-08T03:32:15.935Z",
"updated_at": "2026-09-08T03:32:15.935Z",
"published_at": "2026-09-08T03:32:50.458Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"ec1901cc-f3d4-48b5-bcda-aee2ff01a23f","format_issues":null,"number":3423} |
1.0.0 | problem:theoremdb-p8872 | problem | theoremdb-p8872 | Upper bound for counts of short least common multiples in strictly increasing sequences of natural numbers | {
"main": "math",
"sub": "number theory",
"slug": "number-theory",
"name": "Number theory",
"original_discipline": "mathematics"
} | [] | [
{
"code": "11",
"basis": "community-map",
"evidence": "Inferred from community field: number-theory"
}
] | Let $a : \mathbb{N} \to \mathbb{N}$ be a sequence of natural numbers with $a_0 > 0$ that is strictly monotone increasing. For positive real $X$ and natural $k$, define $F(a, X, k)$ to be the number of indices $i \in \mathbb{N}$ such that the least common multiple $[a_i, a_{i+1}, \ldots, a_{i+k-1}]$ is strictly less tha... | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P8872",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P8872",
"citation": null,
"doi": null,
"arxiv_id": null,
"license": "Selected packets label their content CC0-1.0; original references retain their own terms",
"notes": "Selected problem... | theorem erdos_873 : answer(sorry) ↔ ∀ᵉ (a : ℕ → ℕ) (ε > (0 : ℝ)), 0 < a 0 → StrictMono a →
∃ k, ∀ X > 0, F a X k < (X^ε).toEReal | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem erdos_873 : answer(sorry) ↔ ∀ᵉ (a : ℕ → ℕ) (ε > (0 : ℝ)), 0 < a 0 → StrictMono a →\n ∃ k, ∀ X > 0, F a X k < (X^ε).toEReal",
"target": null,
"source_url": "https://theoremdb.org/statements/P8872",
"format": "declaration-or-fragment",
"sha256": "4ed... | [] | TheoremDB P8872 · Candidate awaiting source review | null | Source-reported assessment, retrieved 8 September 2026. Frontier Science has not independently reviewed this status.
Open. No signed Lean verification is attached to this problem or its selected resolution.
This record is in the source backlog; its novelty and unresolved status have not passed source review. | null | published | incremental | hard | [
"source-candidate",
"theoremdb"
] | null | null | null | null | [
{
"kind": "website",
"citation": "TheoremDB P8872",
"url": "https://theoremdb.org/statements/P8872",
"note": null,
"sort_order": 0
},
{
"kind": "formalization",
"citation": "Source formal statement (not locally verified)",
"url": "https://theoremdb.org/statements/P8872",
"not... | {
"source_record": "P8872",
"imported_at": "2026-09-08",
"status_as_of": "2026-09-08",
"evidence_label": "Source candidate · not reviewed",
"note": "Full public directory statement. Source lane: backlog; source resolution: open. Source verification claims are attributed only; no local Lean verification has be... | {"created_at":"2026-08-17T19:36:02+00:00","display_standard":{"editorial_review":"submitter_supplied","formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":8872,"public_status":{"evidence":{"hasFormalization":false,"hasInformalProof":false,"hasRecordedWork":false,"solutionRecord":null},"lean":{... | [
"formalization-contains-placeholder",
"formalization-not-compiled-during-export",
"current-status-unassessed",
"source-full-text-not-present"
] | [] | {
"created_at": "2026-09-08T03:32:15.935Z",
"updated_at": "2026-09-08T03:32:15.935Z",
"published_at": "2026-09-08T03:33:07.319Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"10ae8e8c-5d6c-4564-9c02-265675d20b80","format_issues":null,"number":6934} |
1.0.0 | problem:theoremdb-p8874 | problem | theoremdb-p8874 | Asymptotic limit of the Bézier-type Bernstein operators | {
"main": "math",
"sub": "analysis",
"slug": "analysis",
"name": "Analysis",
"original_discipline": "mathematics"
} | [] | [] | Let $I = [0,1]$. For $n \in \mathbb{N}$, $k \in \{0, 1, \ldots, n\}$, and $x \in [0,1]$, define the Bernstein basis polynomial
$$
p_{n,k}(x) = \binom{n}{k} x^k (1-x)^{n-k}.
$$
For $k \in \{0, 1, \ldots, n\}$, define the cumulative sum
$$
J_{n,k}(x) = \sum_{j=k}^{n} p_{n,j}(x),
$$
and set $J_{n,n+1}(x) = 0$. For... | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P8874",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P8874",
"citation": null,
"doi": null,
"arxiv_id": null,
"license": "Selected packets label their content CC0-1.0; original references retain their own terms",
"notes": "Selected problem... | theorem voronovskaja_theorem.bezier_bernstein_operators
(α : ℝ) (hα_pos : 0 < α) (hα : α ≠ 1)
(f : ℝ → ℝ) (x : ℝ) (hx : x ∈ I)
(hf : ContDiffOn ℝ 2 f I) :
Tendsto (fun n : ℕ => Real.sqrt n * (bezierBernstein n α f x - f x)) atTop
(𝓝 answer(sorry)) | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem voronovskaja_theorem.bezier_bernstein_operators\n (α : ℝ) (hα_pos : 0 < α) (hα : α ≠ 1)\n (f : ℝ → ℝ) (x : ℝ) (hx : x ∈ I)\n (hf : ContDiffOn ℝ 2 f I) :\n Tendsto (fun n : ℕ => Real.sqrt n * (bezierBernstein n α f x - f x)) atTop\n (𝓝 answer(sorry))... | [] | TheoremDB P8874 · Candidate awaiting source review | null | Source-reported assessment, retrieved 8 September 2026. Frontier Science has not independently reviewed this status.
Open. No signed Lean verification is attached to this problem or its selected resolution.
This record is in the source backlog; its novelty and unresolved status have not passed source review. | null | published | incremental | hard | [
"source-candidate",
"theoremdb"
] | null | null | null | null | [
{
"kind": "website",
"citation": "TheoremDB P8874",
"url": "https://theoremdb.org/statements/P8874",
"note": null,
"sort_order": 0
},
{
"kind": "formalization",
"citation": "Source formal statement (not locally verified)",
"url": "https://theoremdb.org/statements/P8874",
"not... | {
"source_record": "P8874",
"imported_at": "2026-09-08",
"status_as_of": "2026-09-08",
"evidence_label": "Source candidate · not reviewed",
"note": "Full public directory statement. Source lane: backlog; source resolution: open. Source verification claims are attributed only; no local Lean verification has be... | {"created_at":"2026-08-17T19:36:07+00:00","display_standard":{"editorial_review":"submitter_supplied","formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":8874,"public_status":{"evidence":{"hasFormalization":false,"hasInformalProof":false,"hasRecordedWork":false,"solutionRecord":null},"lean":{... | [
"formalization-contains-placeholder",
"formalization-not-compiled-during-export",
"current-status-unassessed",
"source-full-text-not-present"
] | [] | {
"created_at": "2026-09-08T03:32:15.935Z",
"updated_at": "2026-09-08T03:32:15.935Z",
"published_at": "2026-09-08T03:32:50.296Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"aa62eff7-e6e7-4914-8d64-ac9f18c0d6f6","format_issues":null,"number":3391} |
1.0.0 | problem:theoremdb-p8876 | problem | theoremdb-p8876 | Asymptotic behavior of a restricted prime reciprocal sum | {
"main": "math",
"sub": "number theory",
"slug": "number-theory",
"name": "Number theory",
"original_discipline": "mathematics"
} | [] | [
{
"code": "11",
"basis": "community-map",
"evidence": "Inferred from community field: number-theory"
}
] | For each positive integer $n$, define
$$
S(n)=\sum_{\substack{p\leq n\\p\text{ prime}\\p/2\,\lt\,n\bmod p\,\lt\,p}}\frac{1}{p},
$$
where the condition $p/2\lt n\bmod p\lt p$ means that when $n$ is divided by $p$, the remainder $r$ satisfies $p/2\lt r\lt p$. Determine whether
$$
S(n)\sim\frac{\log\log n}{2}
$$
as $n... | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P8876",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P8876",
"citation": null,
"doi": null,
"arxiv_id": null,
"license": "Selected packets label their content CC0-1.0; original references retain their own terms",
"notes": "Selected problem... | theorem erdos_726 :
answer(sorry) ↔
(fun n : ℕ ↦ ∑ p ∈ (range (n + 1)).filter
(fun p : ℕ ↦ p.Prime ∧ (p : ℝ) / 2 < (n % p : ℝ)),
(1 : ℝ) / (p : ℝ))
~[atTop] (fun n : ℕ ↦ Real.log (Real.log (n : ℝ)) / 2) | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem erdos_726 :\n answer(sorry) ↔\n (fun n : ℕ ↦ ∑ p ∈ (range (n + 1)).filter\n (fun p : ℕ ↦ p.Prime ∧ (p : ℝ) / 2 < (n % p : ℝ)),\n (1 : ℝ) / (p : ℝ))\n ~[atTop] (fun n : ℕ ↦ Real.log (Real.log (n : ℝ)) / 2)",
"target": null,
"source... | [] | TheoremDB P8876 · Candidate awaiting source review | null | Source-reported assessment, retrieved 8 September 2026. Frontier Science has not independently reviewed this status.
Open. No signed Lean verification is attached to this problem or its selected resolution.
This record is in the source backlog; its novelty and unresolved status have not passed source review. | null | published | incremental | hard | [
"source-candidate",
"theoremdb"
] | null | null | null | null | [
{
"kind": "website",
"citation": "TheoremDB P8876",
"url": "https://theoremdb.org/statements/P8876",
"note": null,
"sort_order": 0
},
{
"kind": "formalization",
"citation": "Source formal statement (not locally verified)",
"url": "https://theoremdb.org/statements/P8876",
"not... | {
"source_record": "P8876",
"imported_at": "2026-09-08",
"status_as_of": "2026-09-08",
"evidence_label": "Source candidate · not reviewed",
"note": "Full public directory statement. Source lane: backlog; source resolution: open. Source verification claims are attributed only; no local Lean verification has be... | {"created_at":"2026-08-17T19:36:08+00:00","display_standard":{"editorial_review":"submitter_supplied","formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":8876,"public_status":{"evidence":{"hasFormalization":false,"hasInformalProof":false,"hasRecordedWork":false,"solutionRecord":null},"lean":{... | [
"formalization-contains-placeholder",
"formalization-not-compiled-during-export",
"current-status-unassessed",
"source-full-text-not-present"
] | [] | {
"created_at": "2026-09-08T03:32:15.935Z",
"updated_at": "2026-09-08T03:32:15.935Z",
"published_at": "2026-09-08T03:32:49.902Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"f0b5162d-f3f1-4be1-a4a1-c11bd77ab5b3","format_issues":null,"number":3310} |
1.0.0 | problem:theoremdb-p8878 | problem | theoremdb-p8878 | Asymptotic growth rate of the maximal length of a weakly increasing sequence with distinct consecutive sums | {
"main": "math",
"sub": "number theory",
"slug": "number-theory",
"name": "Number theory",
"original_discipline": "mathematics"
} | [] | [
{
"code": "11",
"basis": "community-map",
"evidence": "Inferred from community field: number-theory"
}
] | For each positive integer $n$, let $h(n)$ denote the largest integer $k$ for which there exist integers $1 \le a_1 \le a_2 \le \dots \le a_k \le n$ such that all sums of the form $\sum_{i=u}^{v} a_i$ with $1 \le u \le v \le k$ are distinct. Determine an explicit function $g: \mathbb{N} \to \mathbb{R}$ such that $h(n) =... | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P8878",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P8878",
"citation": null,
"doi": null,
"arxiv_id": null,
"license": "Selected packets label their content CC0-1.0; original references retain their own terms",
"notes": "Selected problem... | theorem erdos_357.variants.monotone.parts.ii.bigTheta_version :
(fun n ↦ (h n : ℝ)) =Θ[atTop] (answer(sorry) : ℕ → ℝ) | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem erdos_357.variants.monotone.parts.ii.bigTheta_version :\n (fun n ↦ (h n : ℝ)) =Θ[atTop] (answer(sorry) : ℕ → ℝ)",
"target": null,
"source_url": "https://theoremdb.org/statements/P8878",
"format": "declaration-or-fragment",
"sha256": "566d006b8e38bc... | [] | TheoremDB P8878 · Candidate awaiting source review | null | Source-reported assessment, retrieved 8 September 2026. Frontier Science has not independently reviewed this status.
Open. No signed Lean verification is attached to this problem or its selected resolution.
This record is in the source backlog; its novelty and unresolved status have not passed source review. | null | published | incremental | hard | [
"source-candidate",
"theoremdb"
] | null | null | null | null | [
{
"kind": "website",
"citation": "TheoremDB P8878",
"url": "https://theoremdb.org/statements/P8878",
"note": null,
"sort_order": 0
},
{
"kind": "formalization",
"citation": "Source formal statement (not locally verified)",
"url": "https://theoremdb.org/statements/P8878",
"not... | {
"source_record": "P8878",
"imported_at": "2026-09-08",
"status_as_of": "2026-09-08",
"evidence_label": "Source candidate · not reviewed",
"note": "Full public directory statement. Source lane: backlog; source resolution: open. Source verification claims are attributed only; no local Lean verification has be... | {"created_at":"2026-08-17T19:36:11+00:00","display_standard":{"editorial_review":"submitter_supplied","formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":8878,"public_status":{"evidence":{"hasFormalization":false,"hasInformalProof":false,"hasRecordedWork":false,"solutionRecord":null},"lean":{... | [
"formalization-contains-placeholder",
"formalization-not-compiled-during-export",
"current-status-unassessed",
"source-full-text-not-present"
] | [] | {
"created_at": "2026-09-08T03:32:15.935Z",
"updated_at": "2026-09-08T03:32:15.935Z",
"published_at": "2026-09-08T03:32:50.283Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"8d61792b-e0cf-4b32-8eb7-b9bb9d15556e","format_issues":null,"number":3388} |
1.0.0 | problem:theoremdb-p8880 | problem | theoremdb-p8880 | The pebbling number conjecture for Cartesian products of connected graphs | {
"main": "math",
"sub": "combinatorics",
"slug": "combinatorics",
"name": "Combinatorics",
"original_discipline": "mathematics"
} | [] | [
{
"code": "05",
"basis": "community-map",
"evidence": "Inferred from community field: combinatorics"
}
] | Let $G$ be a connected simple graph with finite vertex set $V$, and let $H$ be a connected simple graph with finite vertex set $W$. A pebble distribution on a graph is an assignment of a non-negative integer number of pebbles to each vertex. The total number of pebbles in a distribution is the sum of pebbles over all v... | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P8880",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P8880",
"citation": null,
"doi": null,
"arxiv_id": null,
"license": "Selected packets label their content CC0-1.0; original references retain their own terms",
"notes": "Selected problem... | theorem pebbling_number_conjecture {W : Type} [Fintype V] [Fintype W] [DecidableEq W]
(G : SimpleGraph V) (H : SimpleGraph W) (hG : G.Connected) (hH : H.Connected) :
PebblingNumber (G □ H) ≤ PebblingNumber G * PebblingNumber H | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem pebbling_number_conjecture {W : Type} [Fintype V] [Fintype W] [DecidableEq W]\n (G : SimpleGraph V) (H : SimpleGraph W) (hG : G.Connected) (hH : H.Connected) :\n PebblingNumber (G □ H) ≤ PebblingNumber G * PebblingNumber H",
"target": null,
"source_url"... | [] | TheoremDB P8880 · Candidate awaiting source review | null | Source-reported assessment, retrieved 8 September 2026. Frontier Science has not independently reviewed this status.
Open. No signed Lean verification is attached to this problem or its selected resolution.
This record is in the source backlog; its novelty and unresolved status have not passed source review. | null | published | incremental | hard | [
"source-candidate",
"theoremdb"
] | null | null | null | null | [
{
"kind": "website",
"citation": "TheoremDB P8880",
"url": "https://theoremdb.org/statements/P8880",
"note": null,
"sort_order": 0
},
{
"kind": "formalization",
"citation": "Source formal statement (not locally verified)",
"url": "https://theoremdb.org/statements/P8880",
"not... | {
"source_record": "P8880",
"imported_at": "2026-09-08",
"status_as_of": "2026-09-08",
"evidence_label": "Source candidate · not reviewed",
"note": "Full public directory statement. Source lane: backlog; source resolution: open. Source verification claims are attributed only; no local Lean verification has be... | {"created_at":"2026-08-17T19:36:15+00:00","display_standard":{"formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":8880,"public_status":{"evidence":{"hasFormalization":false,"hasInformalProof":false,"hasRecordedWork":false,"solutionRecord":null},"lean":{"kernelAccepted":false,"label":"","reaso... | [
"formalization-not-compiled-during-export",
"current-status-unassessed",
"source-full-text-not-present"
] | [] | {
"created_at": "2026-09-08T03:32:15.935Z",
"updated_at": "2026-09-08T03:32:15.935Z",
"published_at": "2026-09-08T03:33:06.699Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"f8b2fd60-b007-4aef-a15d-67c044d73e26","format_issues":null,"number":6799} |
1.0.0 | problem:theoremdb-p8882 | problem | theoremdb-p8882 | Infinitude of primes of the form n² + 1 | {
"main": "math",
"sub": "number theory",
"slug": "number-theory",
"name": "Number theory",
"original_discipline": "mathematics"
} | [] | [
{
"code": "11",
"basis": "community-map",
"evidence": "Inferred from community field: number-theory"
}
] | A natural number is called prime if it is greater than 1 and has no positive divisors other than 1 and itself. Determine whether the set $\{n \in \mathbb{N} \mid n^2 + 1 \text{ is prime}\}$ is infinite. In other words, are there infinitely many primes of the form $n^2 + 1$? | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P8882",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P8882",
"citation": null,
"doi": null,
"arxiv_id": null,
"license": "Selected packets label their content CC0-1.0; original references retain their own terms",
"notes": "Selected problem... | theorem infinite_prime_sq_add_one :
answer(sorry) ↔ {n : ℕ | Prime (n^2 + 1)}.Infinite | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem infinite_prime_sq_add_one :\n answer(sorry) ↔ {n : ℕ | Prime (n^2 + 1)}.Infinite",
"target": null,
"source_url": "https://theoremdb.org/statements/P8882",
"format": "declaration-or-fragment",
"sha256": "8a05350f70a07ad6040bc0ef8d4804b4584a0ad61ff99... | [] | TheoremDB P8882 · Candidate awaiting source review | null | Source-reported assessment, retrieved 8 September 2026. Frontier Science has not independently reviewed this status.
Open. No signed Lean verification is attached to this problem or its selected resolution.
This record is in the source backlog; its novelty and unresolved status have not passed source review. | null | published | incremental | hard | [
"source-candidate",
"theoremdb"
] | null | null | null | null | [
{
"kind": "website",
"citation": "TheoremDB P8882",
"url": "https://theoremdb.org/statements/P8882",
"note": null,
"sort_order": 0
},
{
"kind": "formalization",
"citation": "Source formal statement (not locally verified)",
"url": "https://theoremdb.org/statements/P8882",
"not... | {
"source_record": "P8882",
"imported_at": "2026-09-08",
"status_as_of": "2026-09-08",
"evidence_label": "Source candidate · not reviewed",
"note": "Full public directory statement. Source lane: backlog; source resolution: open. Source verification claims are attributed only; no local Lean verification has be... | {"created_at":"2026-08-17T19:36:19+00:00","display_standard":{"editorial_review":"submitter_supplied","formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":8882,"public_status":{"evidence":{"hasFormalization":false,"hasInformalProof":false,"hasRecordedWork":false,"solutionRecord":null},"lean":{... | [
"formalization-contains-placeholder",
"formalization-not-compiled-during-export",
"current-status-unassessed",
"source-full-text-not-present"
] | [] | {
"created_at": "2026-09-08T03:32:15.935Z",
"updated_at": "2026-09-08T03:32:15.935Z",
"published_at": "2026-09-08T03:33:01.221Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"bba900a7-42ef-4e99-a369-f5063c8766bd","format_issues":null,"number":5684} |
1.0.0 | problem:theoremdb-p8884 | problem | theoremdb-p8884 | Existence of an Equal Pair in a Recursively Defined Integer Sequence | {
"main": "math",
"sub": "combinatorics",
"slug": "combinatorics",
"name": "Combinatorics",
"original_discipline": "mathematics"
} | [] | [
{
"code": "05",
"basis": "community-map",
"evidence": "Inferred from community field: combinatorics"
}
] | Let $(a_n)_{n \ge 0}$ and $(b_n)_{n \ge 0}$ be two sequences of natural numbers defined by the initial values $a_0 = 1$ and $b_0 = 2$, and for all $n \ge 0$ by the recurrence relations
$$
a_{n+1} = \begin{cases}
a_n - b_n & \text{if } b_n \le a_n, \\
2a_n + 1 & \text{if } b_n > a_n,
\end{cases}
$$
and
$$
b_{n+1} ... | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P8884",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P8884",
"citation": null,
"doi": null,
"arxiv_id": null,
"license": "Selected packets label their content CC0-1.0; original references retain their own terms",
"notes": "Selected problem... | theorem beaver_math_olympiad_problem_1 :
answer(sorry) ↔ ∀ᵉ (a : ℕ → ℕ) (b : ℕ → ℕ)
(a_ini : a 0 = 1)
(a_rec : ∀ n, a (n + 1) = if b n ≤ a n then a n - b n else 2 * a n + 1)
(b_ini : b 0 = 2)
(b_rec : ∀ n, b (n + 1) = if b n ≤ a n then 4 * b n + 2 else b n - a n),
∃ i, a i = b i | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem beaver_math_olympiad_problem_1 :\n answer(sorry) ↔ ∀ᵉ (a : ℕ → ℕ) (b : ℕ → ℕ)\n (a_ini : a 0 = 1)\n (a_rec : ∀ n, a (n + 1) = if b n ≤ a n then a n - b n else 2 * a n + 1)\n (b_ini : b 0 = 2)\n (b_rec : ∀ n, b (n + 1) = if b n ≤ a n then 4 * b n + 2 el... | [] | TheoremDB P8884 · Candidate awaiting source review | null | Source-reported assessment, retrieved 8 September 2026. Frontier Science has not independently reviewed this status.
Open. No signed Lean verification is attached to this problem or its selected resolution.
This record is in the source backlog; its novelty and unresolved status have not passed source review. | null | published | incremental | hard | [
"source-candidate",
"theoremdb"
] | null | null | null | null | [
{
"kind": "website",
"citation": "TheoremDB P8884",
"url": "https://theoremdb.org/statements/P8884",
"note": null,
"sort_order": 0
},
{
"kind": "formalization",
"citation": "Source formal statement (not locally verified)",
"url": "https://theoremdb.org/statements/P8884",
"not... | {
"source_record": "P8884",
"imported_at": "2026-09-08",
"status_as_of": "2026-09-08",
"evidence_label": "Source candidate · not reviewed",
"note": "Full public directory statement. Source lane: backlog; source resolution: open. Source verification claims are attributed only; no local Lean verification has be... | {"created_at":"2026-08-17T19:36:22+00:00","display_standard":{"editorial_review":"submitter_supplied","formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":8884,"public_status":{"evidence":{"hasFormalization":false,"hasInformalProof":false,"hasRecordedWork":false,"solutionRecord":null},"lean":{... | [
"formalization-contains-placeholder",
"formalization-not-compiled-during-export",
"current-status-unassessed",
"source-full-text-not-present"
] | [] | {
"created_at": "2026-09-08T03:32:15.935Z",
"updated_at": "2026-09-08T03:32:15.935Z",
"published_at": "2026-09-08T03:32:58.678Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"0166d55b-4f8f-43e9-bd99-88738c01c7c6","format_issues":null,"number":5232} |
1.0.0 | problem:theoremdb-p8886 | problem | theoremdb-p8886 | Improving the upper bound in the minimum overlap problem | {
"main": "math",
"sub": "combinatorics",
"slug": "combinatorics",
"name": "Combinatorics",
"original_discipline": "mathematics"
} | [] | [
{
"code": "05",
"basis": "community-map",
"evidence": "Inferred from community field: combinatorics"
}
] | For each integer $n\ge 1$, partition $\{1,\ldots,2n\}$ into disjoint sets $A$ and $B$ with $|A|=|B|=n$. Define
$$
r_{A,B}(k)=\#\{(a,b)\in A\times B:a-b=k\},\qquad M(n)=\min_{A,B}\max_{k\in\mathbb Z}r_{A,B}(k).
$$
Find an explicit real constant $c<0.380926853433087$ and prove that
$$
\limsup_{n\to\infty}\frac{M(n)... | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P8886",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P8886",
"citation": null,
"doi": null,
"arxiv_id": null,
"license": "Selected packets label their content CC0-1.0; original references retain their own terms",
"notes": "Selected problem... | theorem erdos_36.variants.upper :
∃ (c : ℝ), c < 0.380926853433087 ∧ atTop.limsup MinOverlapQuotient ≤ c ∧ c = answer(sorry) | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem erdos_36.variants.upper :\n ∃ (c : ℝ), c < 0.380926853433087 ∧ atTop.limsup MinOverlapQuotient ≤ c ∧ c = answer(sorry)",
"target": null,
"source_url": "https://theoremdb.org/statements/P8886",
"format": "declaration-or-fragment",
"sha256": "d635ac4... | [] | TheoremDB P8886 · Candidate awaiting source review | null | Source-reported assessment, retrieved 8 September 2026. Frontier Science has not independently reviewed this status.
Open. No signed Lean verification is attached to this problem or its selected resolution.
This record is in the source backlog; its novelty and unresolved status have not passed source review. | null | published | incremental | hard | [
"source-candidate",
"theoremdb"
] | null | null | null | null | [
{
"kind": "website",
"citation": "TheoremDB P8886",
"url": "https://theoremdb.org/statements/P8886",
"note": null,
"sort_order": 0
},
{
"kind": "formalization",
"citation": "Source formal statement (not locally verified)",
"url": "https://theoremdb.org/statements/P8886",
"not... | {
"source_record": "P8886",
"imported_at": "2026-09-08",
"status_as_of": "2026-09-08",
"evidence_label": "Source candidate · not reviewed",
"note": "Full public directory statement. Source lane: backlog; source resolution: open. Source verification claims are attributed only; no local Lean verification has be... | {"created_at":"2026-08-17T19:36:26+00:00","display_standard":{"editorial_review":"submitter_supplied","formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":8886,"public_status":{"evidence":{"hasFormalization":false,"hasInformalProof":false,"hasRecordedWork":false,"solutionRecord":null},"lean":{... | [
"formalization-contains-placeholder",
"formalization-not-compiled-during-export",
"current-status-unassessed",
"source-full-text-not-present"
] | [] | {
"created_at": "2026-09-08T03:32:15.935Z",
"updated_at": "2026-09-08T03:32:15.935Z",
"published_at": "2026-09-08T03:32:59.685Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"3535c8fd-e604-4a05-a2b8-7a4d96f969dc","format_issues":null,"number":5448} |
1.0.0 | problem:theoremdb-p8888 | problem | theoremdb-p8888 | Divisibility of the central binomial coefficient by descending factorials | {
"main": "math",
"sub": "number theory",
"slug": "number-theory",
"name": "Number theory",
"original_discipline": "mathematics"
} | [] | [
{
"code": "11",
"basis": "community-map",
"evidence": "Inferred from community field: number-theory"
}
] | For a positive integer $n$, the central binomial coefficient is defined as $\binom{2n}{n}$. For non-negative integers $n$ and $k$, the descending factorial $\prod_{0 \leq i \leq k}(n-i)$ is denoted by $n^{\underline{k+1}}$. Determine whether for every non-negative integer $k$, there exists a non-negative integer $n$ su... | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P8888",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P8888",
"citation": null,
"doi": null,
"arxiv_id": null,
"license": "Selected packets label their content CC0-1.0; original references retain their own terms",
"notes": "Selected problem... | theorem erdos_396 : answer(sorry) ↔ ∀ k : ℕ, ∃ n : ℕ, descFactorial n (k + 1) ∣ centralBinom n | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem erdos_396 : answer(sorry) ↔ ∀ k : ℕ, ∃ n : ℕ, descFactorial n (k + 1) ∣ centralBinom n",
"target": null,
"source_url": "https://theoremdb.org/statements/P8888",
"format": "declaration-or-fragment",
"sha256": "65ab9f96888da4e3df0d549236057aee6b2a132642... | [] | TheoremDB P8888 · Candidate awaiting source review | null | Source-reported assessment, retrieved 8 September 2026. Frontier Science has not independently reviewed this status.
Open. No signed Lean verification is attached to this problem or its selected resolution.
This record is in the source backlog; its novelty and unresolved status have not passed source review. | null | published | incremental | hard | [
"source-candidate",
"theoremdb"
] | null | null | null | null | [
{
"kind": "website",
"citation": "TheoremDB P8888",
"url": "https://theoremdb.org/statements/P8888",
"note": null,
"sort_order": 0
},
{
"kind": "formalization",
"citation": "Source formal statement (not locally verified)",
"url": "https://theoremdb.org/statements/P8888",
"not... | {
"source_record": "P8888",
"imported_at": "2026-09-08",
"status_as_of": "2026-09-08",
"evidence_label": "Source candidate · not reviewed",
"note": "Full public directory statement. Source lane: backlog; source resolution: open. Source verification claims are attributed only; no local Lean verification has be... | {"created_at":"2026-08-17T19:36:26+00:00","display_standard":{"editorial_review":"submitter_supplied","formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":8888,"public_status":{"evidence":{"hasFormalization":false,"hasInformalProof":false,"hasRecordedWork":false,"solutionRecord":null},"lean":{... | [
"formalization-contains-placeholder",
"formalization-not-compiled-during-export",
"current-status-unassessed",
"source-full-text-not-present"
] | [] | {
"created_at": "2026-09-08T03:32:15.935Z",
"updated_at": "2026-09-08T03:32:15.935Z",
"published_at": "2026-09-08T03:32:53.297Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"3e0ed096-b509-4af1-96f0-6a51f110d5bd","format_issues":null,"number":4046} |
1.0.0 | problem:theoremdb-p8890 | problem | theoremdb-p8890 | The Four Exponentials Conjecture | {
"main": "math",
"sub": "analysis",
"slug": "analysis",
"name": "Analysis",
"original_discipline": "mathematics"
} | [] | [] | Let $x_0, x_1$ and $y_0, y_1$ be complex numbers such that the set $\{x_0, x_1\}$ is linearly independent over $\mathbb{Q}$ and the set $\{y_0, y_1\}$ is linearly independent over $\mathbb{Q}$. Does there exist indices $i, j \in \{0, 1\}$ such that the complex number $e^{x_i y_j}$ is transcendental over $\mathbb{Q}$? | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P8890",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P8890",
"citation": null,
"doi": null,
"arxiv_id": null,
"license": "Selected packets label their content CC0-1.0; original references retain their own terms",
"notes": "Selected problem... | theorem four_exponentials_conjecture (x : Fin 2 → ℂ) (y : Fin 2 → ℂ)
(h1 : LinearIndependent ℚ x) (h2 : LinearIndependent ℚ y) :
∃ i j : Fin 2, Transcendental ℚ (exp (x i * y j)) | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem four_exponentials_conjecture (x : Fin 2 → ℂ) (y : Fin 2 → ℂ)\n (h1 : LinearIndependent ℚ x) (h2 : LinearIndependent ℚ y) :\n ∃ i j : Fin 2, Transcendental ℚ (exp (x i * y j))",
"target": null,
"source_url": "https://theoremdb.org/statements/P8890",
... | [] | TheoremDB P8890 · Candidate awaiting source review | null | Source-reported assessment, retrieved 8 September 2026. Frontier Science has not independently reviewed this status.
Open. No signed Lean verification is attached to this problem or its selected resolution.
This record is in the source backlog; its novelty and unresolved status have not passed source review. | null | published | incremental | hard | [
"source-candidate",
"theoremdb"
] | null | null | null | null | [
{
"kind": "website",
"citation": "TheoremDB P8890",
"url": "https://theoremdb.org/statements/P8890",
"note": null,
"sort_order": 0
},
{
"kind": "formalization",
"citation": "Source formal statement (not locally verified)",
"url": "https://theoremdb.org/statements/P8890",
"not... | {
"source_record": "P8890",
"imported_at": "2026-09-08",
"status_as_of": "2026-09-08",
"evidence_label": "Source candidate · not reviewed",
"note": "Full public directory statement. Source lane: backlog; source resolution: open. Source verification claims are attributed only; no local Lean verification has be... | {"created_at":"2026-08-17T19:36:36+00:00","display_standard":{"formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":8890,"public_status":{"evidence":{"hasFormalization":false,"hasInformalProof":false,"hasRecordedWork":false,"solutionRecord":null},"lean":{"kernelAccepted":false,"label":"","reaso... | [
"formalization-not-compiled-during-export",
"current-status-unassessed",
"source-full-text-not-present"
] | [] | {
"created_at": "2026-09-08T03:32:15.935Z",
"updated_at": "2026-09-08T03:32:15.935Z",
"published_at": "2026-09-08T03:33:06.502Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"c2b5d887-4c5e-466d-8029-c4218e7b4265","format_issues":null,"number":6756} |
1.0.0 | problem:theoremdb-p8894 | problem | theoremdb-p8894 | Positive Lower Density of Prime-Powers-of-C Sums | {
"main": "math",
"sub": "number theory",
"slug": "number-theory",
"name": "Number theory",
"original_discipline": "mathematics"
} | [] | [
{
"code": "11",
"basis": "community-map",
"evidence": "Inferred from community field: number-theory"
}
] | Let $C > 1$ be a real number. Consider the set
$$
S_C = \{ p + \lfloor C^k \rfloor : p \text{ is prime and } k \geq 0 \},
$$
where $\lfloor x \rfloor$ denotes the greatest integer less than or equal to $x$, and the powers $C^k$ are taken with $k$ a non-negative integer. The lower density of a set $A \subseteq \math... | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P8894",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P8894",
"citation": null,
"doi": null,
"arxiv_id": null,
"license": "Selected packets label their content CC0-1.0; original references retain their own terms",
"notes": "Selected problem... | theorem erdos_244 : answer(sorry) ↔
∀ C > (1 : ℝ), 0 < { p + ⌊C ^ k⌋₊ | (p) (k) (_ : p.Prime) }.lowerDensity | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem erdos_244 : answer(sorry) ↔\n ∀ C > (1 : ℝ), 0 < { p + ⌊C ^ k⌋₊ | (p) (k) (_ : p.Prime) }.lowerDensity",
"target": null,
"source_url": "https://theoremdb.org/statements/P8894",
"format": "declaration-or-fragment",
"sha256": "92ebb8a0059ed990d4a739e... | [] | TheoremDB P8894 · Candidate awaiting source review | null | Source-reported assessment, retrieved 8 September 2026. Frontier Science has not independently reviewed this status.
Open. No signed Lean verification is attached to this problem or its selected resolution.
This record is in the source backlog; its novelty and unresolved status have not passed source review. | null | published | incremental | hard | [
"source-candidate",
"theoremdb"
] | null | null | null | null | [
{
"kind": "website",
"citation": "TheoremDB P8894",
"url": "https://theoremdb.org/statements/P8894",
"note": null,
"sort_order": 0
},
{
"kind": "formalization",
"citation": "Source formal statement (not locally verified)",
"url": "https://theoremdb.org/statements/P8894",
"not... | {
"source_record": "P8894",
"imported_at": "2026-09-08",
"status_as_of": "2026-09-08",
"evidence_label": "Source candidate · not reviewed",
"note": "Full public directory statement. Source lane: backlog; source resolution: open. Source verification claims are attributed only; no local Lean verification has be... | {"created_at":"2026-08-17T19:39:55+00:00","display_standard":{"formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":8894,"public_status":{"evidence":{"hasFormalization":false,"hasInformalProof":false,"hasRecordedWork":false,"solutionRecord":null},"lean":{"kernelAccepted":false,"label":"","reaso... | [
"formalization-contains-placeholder",
"formalization-not-compiled-during-export",
"current-status-unassessed",
"source-full-text-not-present"
] | [] | {
"created_at": "2026-09-08T03:32:15.935Z",
"updated_at": "2026-09-08T03:32:15.935Z",
"published_at": "2026-09-08T03:33:04.949Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"08e0e9ff-028f-44b8-9a65-c9624ce6e2a5","format_issues":null,"number":6421} |
1.0.0 | problem:theoremdb-p8896 | problem | theoremdb-p8896 | Irrationality of $e^e$ | {
"main": "math",
"sub": "analysis",
"slug": "analysis",
"name": "Analysis",
"original_discipline": "mathematics"
} | [] | [] | Let $e = \exp(1)$ denote Euler's number, the base of the natural logarithm. Determine whether $e^e$ is irrational, that is, whether $e^e$ cannot be expressed as a ratio of two integers. | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P8896",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P8896",
"citation": null,
"doi": null,
"arxiv_id": null,
"license": "Selected packets label their content CC0-1.0; original references retain their own terms",
"notes": "Selected problem... | theorem irrational_e_to_e :
answer(sorry) ↔ Irrational (e ^ e) | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem irrational_e_to_e :\n answer(sorry) ↔ Irrational (e ^ e)",
"target": null,
"source_url": "https://theoremdb.org/statements/P8896",
"format": "declaration-or-fragment",
"sha256": "6b22a23c5d2c77b70403918cecb58751d5179db22b883fc7f37c4f1f2df0d81c",
... | [] | TheoremDB P8896 · Candidate awaiting source review | null | Source-reported assessment, retrieved 8 September 2026. Frontier Science has not independently reviewed this status.
Open. No signed Lean verification is attached to this problem or its selected resolution.
This record is in the source backlog; its novelty and unresolved status have not passed source review. | null | published | incremental | hard | [
"source-candidate",
"theoremdb"
] | null | null | null | null | [
{
"kind": "website",
"citation": "TheoremDB P8896",
"url": "https://theoremdb.org/statements/P8896",
"note": null,
"sort_order": 0
},
{
"kind": "formalization",
"citation": "Source formal statement (not locally verified)",
"url": "https://theoremdb.org/statements/P8896",
"not... | {
"source_record": "P8896",
"imported_at": "2026-09-08",
"status_as_of": "2026-09-08",
"evidence_label": "Source candidate · not reviewed",
"note": "Full public directory statement. Source lane: backlog; source resolution: open. Source verification claims are attributed only; no local Lean verification has be... | {"created_at":"2026-08-17T19:40:07+00:00","display_standard":{"editorial_review":"submitter_supplied","formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":8896,"public_status":{"evidence":{"hasFormalization":false,"hasInformalProof":false,"hasRecordedWork":false,"solutionRecord":null},"lean":{... | [
"formalization-contains-placeholder",
"formalization-not-compiled-during-export",
"current-status-unassessed",
"source-full-text-not-present"
] | [] | {
"created_at": "2026-09-08T03:32:15.935Z",
"updated_at": "2026-09-08T03:32:15.935Z",
"published_at": "2026-09-08T03:33:01.386Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"05db5d86-14fe-47b7-bb71-2aee95707ac3","format_issues":null,"number":5719} |
1.0.0 | problem:theoremdb-p8898 | problem | theoremdb-p8898 | Linear lower bound for the Erdős primitive-set saturation game | {
"main": "math",
"sub": "combinatorics",
"slug": "combinatorics",
"name": "Combinatorics",
"original_discipline": "mathematics"
} | [] | [
{
"code": "05",
"basis": "community-map",
"evidence": "Inferred from community field: combinatorics"
}
] | Consider the following two-player game on the set $\{2, 3, \dots, n\}$. A subset $A \subseteq \{2, 3, \dots, n\}$ is called primitive if no element of $A$ divides another element of $A$; that is, for all distinct $a, b \in A$, we have $a \nmid b$. The game proceeds as a saturation game: starting from the empty set, pla... | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P8898",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P8898",
"citation": null,
"doi": null,
"arxiv_id": null,
"license": "Selected packets label their content CC0-1.0; original references retain their own terms",
"notes": "Selected problem... | theorem erdos_872.parts.i : answer(sorry) ↔
∃ ε > (0 : ℝ), ∀ᶠ n in atTop, (L n : ℝ) ≥ ε * n | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem erdos_872.parts.i : answer(sorry) ↔\n ∃ ε > (0 : ℝ), ∀ᶠ n in atTop, (L n : ℝ) ≥ ε * n",
"target": null,
"source_url": "https://theoremdb.org/statements/P8898",
"format": "declaration-or-fragment",
"sha256": "6c2413cafe4f158c6c66d0d888574ab1d6ad615d... | [] | TheoremDB P8898 · Candidate awaiting source review | null | Source-reported assessment, retrieved 8 September 2026. Frontier Science has not independently reviewed this status.
Open. No signed Lean verification is attached to this problem or its selected resolution.
This record is in the source backlog; its novelty and unresolved status have not passed source review. | null | published | incremental | hard | [
"source-candidate",
"theoremdb"
] | null | null | null | null | [
{
"kind": "website",
"citation": "TheoremDB P8898",
"url": "https://theoremdb.org/statements/P8898",
"note": null,
"sort_order": 0
},
{
"kind": "formalization",
"citation": "Source formal statement (not locally verified)",
"url": "https://theoremdb.org/statements/P8898",
"not... | {
"source_record": "P8898",
"imported_at": "2026-09-08",
"status_as_of": "2026-09-08",
"evidence_label": "Source candidate · not reviewed",
"note": "Full public directory statement. Source lane: backlog; source resolution: open. Source verification claims are attributed only; no local Lean verification has be... | {"created_at":"2026-08-17T19:40:07+00:00","display_standard":{"editorial_review":"submitter_supplied","formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":8898,"public_status":{"evidence":{"hasFormalization":false,"hasInformalProof":false,"hasRecordedWork":false,"solutionRecord":null},"lean":{... | [
"formalization-contains-placeholder",
"formalization-not-compiled-during-export",
"current-status-unassessed",
"source-full-text-not-present"
] | [] | {
"created_at": "2026-09-08T03:32:15.935Z",
"updated_at": "2026-09-08T03:32:15.935Z",
"published_at": "2026-09-08T03:33:03.246Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"9e7d08fc-c207-4237-8d34-ce907dfb110a","format_issues":null,"number":6054} |
1.0.0 | problem:theoremdb-p8900 | problem | theoremdb-p8900 | Gottschalk's Surjunctivity Conjecture | {
"main": "math",
"sub": "algebra",
"slug": "algebra",
"name": "Algebra",
"original_discipline": "mathematics"
} | [] | [] | Let $G$ be a group. For any set $A$, let $A^G$ denote the set of all functions $G \to A$, equipped with the product topology where $A$ carries the discrete topology. The left shift action of $G$ on $A^G$ is defined by $(g \cdot x)(h) = x(g^{-1}h)$ for all $g, h \in G$ and $x \in A^G$. A map $\tau : A^G \to A^G$ is call... | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P8900",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P8900",
"citation": null,
"doi": null,
"arxiv_id": null,
"license": "Selected packets label their content CC0-1.0; original references retain their own terms",
"notes": "Selected problem... | theorem gottschalk_surjunctivity_conjecture (G : Type) [Group G] :
IsSurjunctive G | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem gottschalk_surjunctivity_conjecture (G : Type) [Group G] :\n IsSurjunctive G",
"target": null,
"source_url": "https://theoremdb.org/statements/P8900",
"format": "declaration-or-fragment",
"sha256": "596f1fb337e71af3e9f9bcae6de07bd9ad6d00283cc2f2890... | [] | TheoremDB P8900 · Candidate awaiting source review | null | Source-reported assessment, retrieved 8 September 2026. Frontier Science has not independently reviewed this status.
Open. No signed Lean verification is attached to this problem or its selected resolution.
This record is in the source backlog; its novelty and unresolved status have not passed source review. | null | published | incremental | hard | [
"source-candidate",
"theoremdb"
] | null | null | null | null | [
{
"kind": "website",
"citation": "TheoremDB P8900",
"url": "https://theoremdb.org/statements/P8900",
"note": null,
"sort_order": 0
},
{
"kind": "formalization",
"citation": "Source formal statement (not locally verified)",
"url": "https://theoremdb.org/statements/P8900",
"not... | {
"source_record": "P8900",
"imported_at": "2026-09-08",
"status_as_of": "2026-09-08",
"evidence_label": "Source candidate · not reviewed",
"note": "Full public directory statement. Source lane: backlog; source resolution: open. Source verification claims are attributed only; no local Lean verification has be... | {"created_at":"2026-08-17T19:40:08+00:00","display_standard":{"editorial_review":"submitter_supplied","formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":8900,"public_status":{"evidence":{"hasFormalization":false,"hasInformalProof":false,"hasRecordedWork":false,"solutionRecord":null},"lean":{... | [
"formalization-contains-placeholder",
"formalization-not-compiled-during-export",
"current-status-unassessed",
"source-full-text-not-present"
] | [] | {
"created_at": "2026-09-08T03:32:15.935Z",
"updated_at": "2026-09-08T03:32:15.935Z",
"published_at": "2026-09-08T03:32:59.380Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"ee36d14e-1e53-4445-b639-9640aafa0812","format_issues":null,"number":5383} |
1.0.0 | problem:theoremdb-p8902 | problem | theoremdb-p8902 | Universal sums of a triangular, a generalized pentagonal, and a generalized heptagonal number | {
"main": "math",
"sub": "number theory",
"slug": "number-theory",
"name": "Number theory",
"original_discipline": "mathematics"
} | [] | [
{
"code": "11",
"basis": "community-map",
"evidence": "Inferred from community field: number-theory"
}
] | For every nonnegative integer $n$, there exist nonnegative integers $x$, $y$, and $z$ such that
$$
n = \frac{x(x+1)}{2} + \frac{y(3y+1)}{2} + \frac{z(5z+1)}{2}.
$$
Prove that this holds for all nonnegative integers $n$. | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P8902",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P8902",
"citation": null,
"doi": null,
"arxiv_id": null,
"license": "Selected packets label their content CC0-1.0; original references retain their own terms",
"notes": "Selected problem... | theorem conjecture (n : ℕ) : A n | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem conjecture (n : ℕ) : A n",
"target": null,
"source_url": "https://theoremdb.org/statements/P8902",
"format": "declaration-or-fragment",
"sha256": "79122c62c8b920d59aac24d0de97415fd020e2a20cec64f4a2ce6f643313560f",
"license": "Selected packets labe... | [] | TheoremDB P8902 · Candidate awaiting source review | null | Source-reported assessment, retrieved 8 September 2026. Frontier Science has not independently reviewed this status.
Open. No signed Lean verification is attached to this problem or its selected resolution.
This record is in the source backlog; its novelty and unresolved status have not passed source review. | null | published | incremental | hard | [
"source-candidate",
"theoremdb"
] | null | null | null | null | [
{
"kind": "website",
"citation": "TheoremDB P8902",
"url": "https://theoremdb.org/statements/P8902",
"note": null,
"sort_order": 0
},
{
"kind": "formalization",
"citation": "Source formal statement (not locally verified)",
"url": "https://theoremdb.org/statements/P8902",
"not... | {
"source_record": "P8902",
"imported_at": "2026-09-08",
"status_as_of": "2026-09-08",
"evidence_label": "Source candidate · not reviewed",
"note": "Full public directory statement. Source lane: backlog; source resolution: open. Source verification claims are attributed only; no local Lean verification has be... | {"created_at":"2026-08-17T19:40:09+00:00","display_standard":{"editorial_review":"submitter_supplied","formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":8902,"public_status":{"evidence":{"hasFormalization":false,"hasInformalProof":false,"hasRecordedWork":false,"solutionRecord":null},"lean":{... | [
"formalization-not-compiled-during-export",
"current-status-unassessed",
"source-full-text-not-present"
] | [] | {
"created_at": "2026-09-08T03:32:15.935Z",
"updated_at": "2026-09-08T03:32:15.935Z",
"published_at": "2026-09-08T03:33:07.286Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"475f35d8-c5d9-477d-8c2c-9c6375997e6d","format_issues":null,"number":6927} |
1.0.0 | problem:theoremdb-p8904 | problem | theoremdb-p8904 | Nonexistence of a Fifth-Power Equation with Three Terms on One Side and One on the Other | {
"main": "math",
"sub": "number theory",
"slug": "number-theory",
"name": "Number theory",
"original_discipline": "mathematics"
} | [] | [
{
"code": "11",
"basis": "community-map",
"evidence": "Inferred from community field: number-theory"
}
] | Does there exist a solution in positive integers to the equation $x_1^5 + x_2^5 + x_3^5 = y^5$? That is, for all positive integers $x_1, x_2, x_3, y$, must it be that $x_1^5 + x_2^5 + x_3^5 \neq y^5$? | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P8904",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P8904",
"citation": null,
"doi": null,
"arxiv_id": null,
"license": "Selected packets label their content CC0-1.0; original references retain their own terms",
"notes": "Selected problem... | theorem lander_parkin_selfridge.variants.five_three :
∀ x₁ x₂ x₃ y : ℕ,
0 < x₁ → 0 < x₂ → 0 < x₃ → 0 < y →
x₁ ^ 5 + x₂ ^ 5 + x₃ ^ 5 ≠ y ^ 5 | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem lander_parkin_selfridge.variants.five_three :\n ∀ x₁ x₂ x₃ y : ℕ,\n 0 < x₁ → 0 < x₂ → 0 < x₃ → 0 < y →\n x₁ ^ 5 + x₂ ^ 5 + x₃ ^ 5 ≠ y ^ 5",
"target": null,
"source_url": "https://theoremdb.org/statements/P8904",
"format": "declaration-or-frag... | [] | TheoremDB P8904 · Candidate awaiting source review | null | Source-reported assessment, retrieved 8 September 2026. Frontier Science has not independently reviewed this status.
Open. No signed Lean verification is attached to this problem or its selected resolution.
This record is in the source backlog; its novelty and unresolved status have not passed source review. | null | published | incremental | hard | [
"source-candidate",
"theoremdb"
] | null | null | null | null | [
{
"kind": "website",
"citation": "TheoremDB P8904",
"url": "https://theoremdb.org/statements/P8904",
"note": null,
"sort_order": 0
},
{
"kind": "formalization",
"citation": "Source formal statement (not locally verified)",
"url": "https://theoremdb.org/statements/P8904",
"not... | {
"source_record": "P8904",
"imported_at": "2026-09-08",
"status_as_of": "2026-09-08",
"evidence_label": "Source candidate · not reviewed",
"note": "Full public directory statement. Source lane: backlog; source resolution: open. Source verification claims are attributed only; no local Lean verification has be... | {"created_at":"2026-08-17T19:40:10+00:00","display_standard":{"editorial_review":"submitter_supplied","formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":8904,"public_status":{"evidence":{"hasFormalization":false,"hasInformalProof":false,"hasRecordedWork":false,"solutionRecord":null},"lean":{... | [
"formalization-not-compiled-during-export",
"current-status-unassessed",
"source-full-text-not-present"
] | [] | {
"created_at": "2026-09-08T03:32:15.935Z",
"updated_at": "2026-09-08T03:32:15.935Z",
"published_at": "2026-09-08T03:33:04.474Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"a7277bde-7558-439a-996f-2dbcd78e51a9","format_issues":null,"number":6317} |
1.0.0 | problem:theoremdb-p8906 | problem | theoremdb-p8906 | Irrationality of the sequence $2^{2^n}$ | {
"main": "math",
"sub": "number theory",
"slug": "number-theory",
"name": "Number theory",
"original_discipline": "mathematics"
} | [] | [
{
"code": "11",
"basis": "community-map",
"evidence": "Inferred from community field: number-theory"
}
] | A sequence $(a_n)$ of positive integers is called an irrationality sequence if for every sequence $(b_n)$ of positive integers such that $\frac{a_n}{b_n} \to 1$ as $n \to \infty$, the series $\sum_{n=0}^{\infty} \frac{1}{b_n}$ converges to an irrational number. Determine whether the sequence $a_n = 2^{2^n}$ is an irrat... | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P8906",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P8906",
"citation": null,
"doi": null,
"arxiv_id": null,
"license": "Selected packets label their content CC0-1.0; original references retain their own terms",
"notes": "Selected problem... | theorem erdos_263.parts.i : answer(sorry) ↔ IsIrrationalitySequence (fun n : ℕ => 2 ^ 2 ^ n) | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem erdos_263.parts.i : answer(sorry) ↔ IsIrrationalitySequence (fun n : ℕ => 2 ^ 2 ^ n)",
"target": null,
"source_url": "https://theoremdb.org/statements/P8906",
"format": "declaration-or-fragment",
"sha256": "f49febb8c95faa8c5b027e1e4fdb6926653c0cfb30b4... | [] | TheoremDB P8906 · Candidate awaiting source review | null | Source-reported assessment, retrieved 8 September 2026. Frontier Science has not independently reviewed this status.
Open. No signed Lean verification is attached to this problem or its selected resolution.
This record is in the source backlog; its novelty and unresolved status have not passed source review. | null | published | incremental | hard | [
"source-candidate",
"theoremdb"
] | null | null | null | null | [
{
"kind": "website",
"citation": "TheoremDB P8906",
"url": "https://theoremdb.org/statements/P8906",
"note": null,
"sort_order": 0
},
{
"kind": "formalization",
"citation": "Source formal statement (not locally verified)",
"url": "https://theoremdb.org/statements/P8906",
"not... | {
"source_record": "P8906",
"imported_at": "2026-09-08",
"status_as_of": "2026-09-08",
"evidence_label": "Source candidate · not reviewed",
"note": "Full public directory statement. Source lane: backlog; source resolution: open. Source verification claims are attributed only; no local Lean verification has be... | {"created_at":"2026-08-17T19:40:11+00:00","display_standard":{"editorial_review":"submitter_supplied","formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":8906,"public_status":{"evidence":{"hasFormalization":false,"hasInformalProof":false,"hasRecordedWork":false,"solutionRecord":null},"lean":{... | [
"formalization-contains-placeholder",
"formalization-not-compiled-during-export",
"current-status-unassessed",
"source-full-text-not-present"
] | [] | {
"created_at": "2026-09-08T03:32:15.935Z",
"updated_at": "2026-09-08T03:32:15.935Z",
"published_at": "2026-09-08T03:33:01.830Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"8bfb3427-07b3-48b6-81eb-15b570bf10ae","format_issues":null,"number":5744} |
1.0.0 | problem:theoremdb-p8908 | problem | theoremdb-p8908 | Existence of large finite multiplicative cardinality sets | {
"main": "math",
"sub": "number theory",
"slug": "number-theory",
"name": "Number theory",
"original_discipline": "mathematics"
} | [] | [
{
"code": "11",
"basis": "community-map",
"evidence": "Inferred from community field: number-theory"
}
] | A set $A$ of natural numbers is called a multiplicative cardinality set if whenever $a_1 \cdots a_r = b_1 \cdots b_s$ with $a_i, b_j \in A$, it follows that $r = s$. Here the products are taken over finite multisets of elements from $A$, and the equality of products forces the equality of the cardinalities of these mul... | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P8908",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P8908",
"citation": null,
"doi": null,
"arxiv_id": null,
"license": "Selected packets label their content CC0-1.0; original references retain their own terms",
"notes": "Selected problem... | theorem erdos_786.parts.ii : answer(sorry) ↔
∃ (A : ℕ → Set ℕ) (f : ℕ → ℝ) (_ : f =o[atTop] (1 : ℕ → ℝ)),
∀ N, A N ⊆ Set.Icc 1 (N + 1) ∧ (1 - f N) * N ≤ (A N).ncard ∧ (A N).IsMulCardSet | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem erdos_786.parts.ii : answer(sorry) ↔\n ∃ (A : ℕ → Set ℕ) (f : ℕ → ℝ) (_ : f =o[atTop] (1 : ℕ → ℝ)),\n ∀ N, A N ⊆ Set.Icc 1 (N + 1) ∧ (1 - f N) * N ≤ (A N).ncard ∧ (A N).IsMulCardSet",
"target": null,
"source_url": "https://theoremdb.org/statements/P8908... | [] | TheoremDB P8908 · Candidate awaiting source review | null | Source-reported assessment, retrieved 8 September 2026. Frontier Science has not independently reviewed this status.
Open. No signed Lean verification is attached to this problem or its selected resolution.
This record is in the source backlog; its novelty and unresolved status have not passed source review. | null | published | incremental | hard | [
"source-candidate",
"theoremdb"
] | null | null | null | null | [
{
"kind": "website",
"citation": "TheoremDB P8908",
"url": "https://theoremdb.org/statements/P8908",
"note": null,
"sort_order": 0
},
{
"kind": "formalization",
"citation": "Source formal statement (not locally verified)",
"url": "https://theoremdb.org/statements/P8908",
"not... | {
"source_record": "P8908",
"imported_at": "2026-09-08",
"status_as_of": "2026-09-08",
"evidence_label": "Source candidate · not reviewed",
"note": "Full public directory statement. Source lane: backlog; source resolution: open. Source verification claims are attributed only; no local Lean verification has be... | {"created_at":"2026-08-17T19:40:13+00:00","display_standard":{"editorial_review":"submitter_supplied","formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":8908,"public_status":{"evidence":{"hasFormalization":false,"hasInformalProof":false,"hasRecordedWork":false,"solutionRecord":null},"lean":{... | [
"formalization-contains-placeholder",
"formalization-not-compiled-during-export",
"current-status-unassessed",
"source-full-text-not-present"
] | [] | {
"created_at": "2026-09-08T03:32:15.935Z",
"updated_at": "2026-09-08T03:32:15.935Z",
"published_at": "2026-09-08T03:32:58.761Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"c9f06e68-5853-4559-b298-1fa66db34b1e","format_issues":null,"number":5250} |
1.0.0 | problem:theoremdb-p8910 | problem | theoremdb-p8910 | Erdős Problem 933 on the Growth of the 2-3 Part of Consecutive Integers | {
"main": "math",
"sub": "number theory",
"slug": "number-theory",
"name": "Number theory",
"original_discipline": "mathematics"
} | [] | [
{
"code": "11",
"basis": "community-map",
"evidence": "Inferred from community field: number-theory"
}
] | For each positive integer $n$, write $n(n+1) = 2^{k(n)} 3^{l(n)} m$ where $m$ is coprime to $6$; that is, let $k(n)$ be the $2$-adic valuation of $n(n+1)$ and let $l(n)$ be the $3$-adic valuation of $n(n+1)$. Does
$$
\limsup_{n \to \infty} \frac{2^{k(n)} 3^{l(n)}}{n \log n} = \infty
$$
hold? | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P8910",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P8910",
"citation": null,
"doi": null,
"arxiv_id": null,
"license": "Selected packets label their content CC0-1.0; original references retain their own terms",
"notes": "Selected problem... | theorem erdos_933 :
answer(sorry) ↔
atTop.limsup (fun n : ℕ ↦
((((2 ^ k n * 3 ^ l n : ℕ) : ℝ) / ((n : ℝ) * Real.log (n : ℝ))) : EReal)) = ⊤ | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem erdos_933 :\n answer(sorry) ↔\n atTop.limsup (fun n : ℕ ↦\n ((((2 ^ k n * 3 ^ l n : ℕ) : ℝ) / ((n : ℝ) * Real.log (n : ℝ))) : EReal)) = ⊤",
"target": null,
"source_url": "https://theoremdb.org/statements/P8910",
"format": "declaration-or-fr... | [] | TheoremDB P8910 · Candidate awaiting source review | null | Source-reported assessment, retrieved 8 September 2026. Frontier Science has not independently reviewed this status.
Open. No signed Lean verification is attached to this problem or its selected resolution.
This record is in the source backlog; its novelty and unresolved status have not passed source review. | null | published | incremental | hard | [
"source-candidate",
"theoremdb"
] | null | null | null | null | [
{
"kind": "website",
"citation": "TheoremDB P8910",
"url": "https://theoremdb.org/statements/P8910",
"note": null,
"sort_order": 0
},
{
"kind": "formalization",
"citation": "Source formal statement (not locally verified)",
"url": "https://theoremdb.org/statements/P8910",
"not... | {
"source_record": "P8910",
"imported_at": "2026-09-08",
"status_as_of": "2026-09-08",
"evidence_label": "Source candidate · not reviewed",
"note": "Full public directory statement. Source lane: backlog; source resolution: open. Source verification claims are attributed only; no local Lean verification has be... | {"created_at":"2026-08-17T19:40:16+00:00","display_standard":{"formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":8910,"public_status":{"evidence":{"hasFormalization":false,"hasInformalProof":false,"hasRecordedWork":false,"solutionRecord":null},"lean":{"kernelAccepted":false,"label":"","reaso... | [
"formalization-contains-placeholder",
"formalization-not-compiled-during-export",
"current-status-unassessed",
"source-full-text-not-present"
] | [] | {
"created_at": "2026-09-08T03:32:15.935Z",
"updated_at": "2026-09-08T03:32:15.935Z",
"published_at": "2026-09-08T03:32:56.228Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"3bdaf866-78cd-4b80-a28b-7eef37437b3a","format_issues":null,"number":4695} |
1.0.0 | problem:theoremdb-p8912 | problem | theoremdb-p8912 | Existence of a perfect Euler brick | {
"main": "math",
"sub": "number theory",
"slug": "number-theory",
"name": "Number theory",
"original_discipline": "mathematics"
} | [] | [
{
"code": "11",
"basis": "community-map",
"evidence": "Inferred from community field: number-theory"
}
] | A positive integer is a perfect square if it is the square of a positive integer. An Euler brick is a rectangular cuboid whose three edge lengths $a$, $b$, $c$ are positive integers and whose three face diagonals $\sqrt{a^2+b^2}$, $\sqrt{a^2+c^2}$, $\sqrt{b^2+c^2}$ are also positive integers; equivalently, $a^2+b^2$, $... | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P8912",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P8912",
"citation": null,
"doi": null,
"arxiv_id": null,
"license": "Selected packets label their content CC0-1.0; original references retain their own terms",
"notes": "Selected problem... | theorem perfect_euler_brick_existence :
answer(sorry) ↔ ∃ a b c : ℕ+, IsPerfectCuboid a b c | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem perfect_euler_brick_existence :\n answer(sorry) ↔ ∃ a b c : ℕ+, IsPerfectCuboid a b c",
"target": null,
"source_url": "https://theoremdb.org/statements/P8912",
"format": "declaration-or-fragment",
"sha256": "dc35f432f8b6d5c103df5c6e086da8d9bdc0e6f7... | [] | TheoremDB P8912 · Candidate awaiting source review | null | Source-reported assessment, retrieved 8 September 2026. Frontier Science has not independently reviewed this status.
Open. No signed Lean verification is attached to this problem or its selected resolution.
This record is in the source backlog; its novelty and unresolved status have not passed source review. | null | published | incremental | hard | [
"source-candidate",
"theoremdb"
] | null | null | null | null | [
{
"kind": "website",
"citation": "TheoremDB P8912",
"url": "https://theoremdb.org/statements/P8912",
"note": null,
"sort_order": 0
},
{
"kind": "formalization",
"citation": "Source formal statement (not locally verified)",
"url": "https://theoremdb.org/statements/P8912",
"not... | {
"source_record": "P8912",
"imported_at": "2026-09-08",
"status_as_of": "2026-09-08",
"evidence_label": "Source candidate · not reviewed",
"note": "Full public directory statement. Source lane: backlog; source resolution: open. Source verification claims are attributed only; no local Lean verification has be... | {"created_at":"2026-08-17T19:40:19+00:00","display_standard":{"editorial_review":"submitter_supplied","formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":8912,"public_status":{"evidence":{"hasFormalization":false,"hasInformalProof":false,"hasRecordedWork":false,"solutionRecord":null},"lean":{... | [
"formalization-contains-placeholder",
"formalization-not-compiled-during-export",
"current-status-unassessed",
"source-full-text-not-present"
] | [] | {
"created_at": "2026-09-08T03:32:15.935Z",
"updated_at": "2026-09-08T03:32:15.935Z",
"published_at": "2026-09-08T03:32:58.464Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"d55cc0f9-a077-4901-b974-09774ded643c","format_issues":null,"number":5188} |
1.0.0 | problem:theoremdb-p8914 | problem | theoremdb-p8914 | Strict lower bound for nonzero signed harmonic sums | {
"main": "math",
"sub": "number theory",
"slug": "number-theory",
"name": "Number theory",
"original_discipline": "mathematics"
} | [] | [
{
"code": "11",
"basis": "community-map",
"evidence": "Inferred from community field: number-theory"
}
] | Let $n$ be a positive integer and let $\delta_1, \delta_2, \ldots, \delta_n \in \{-1, 0, 1\}$. Define the signed harmonic sum
$$
S = \left\lvert \sum_{k=1}^{n} \frac{\delta_k}{k} \right\rvert.
$$
Let $[1, 2, \ldots, n]$ denote the least common multiple of the integers $1, 2, \ldots, n$. Is it true that for all suffic... | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P8914",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P8914",
"citation": null,
"doi": null,
"arxiv_id": null,
"license": "Selected packets label their content CC0-1.0; original references retain their own terms",
"notes": "Selected problem... | theorem erdos_317.variants.claim2 : answer(sorry) ↔
∀ᶠ n in atTop, ∀ δ : (Fin n) → ℚ, δ '' Set.univ ⊆ {-1,0,1} →
letI lhs := |∑ k, ((δ k : ℚ) / (k + 1))|
lhs ≠ 0 → lhs > 1 / (Icc 1 n).lcm id | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem erdos_317.variants.claim2 : answer(sorry) ↔\n ∀ᶠ n in atTop, ∀ δ : (Fin n) → ℚ, δ '' Set.univ ⊆ {-1,0,1} →\n letI lhs := |∑ k, ((δ k : ℚ) / (k + 1))|\n lhs ≠ 0 → lhs > 1 / (Icc 1 n).lcm id",
"target": null,
"source_url": "https://theoremdb.org/statem... | [] | TheoremDB P8914 · Candidate awaiting source review | null | Source-reported assessment, retrieved 8 September 2026. Frontier Science has not independently reviewed this status.
Open. No signed Lean verification is attached to this problem or its selected resolution.
This record is in the source backlog; its novelty and unresolved status have not passed source review. | null | published | incremental | hard | [
"source-candidate",
"theoremdb"
] | null | null | null | null | [
{
"kind": "website",
"citation": "TheoremDB P8914",
"url": "https://theoremdb.org/statements/P8914",
"note": null,
"sort_order": 0
},
{
"kind": "formalization",
"citation": "Source formal statement (not locally verified)",
"url": "https://theoremdb.org/statements/P8914",
"not... | {
"source_record": "P8914",
"imported_at": "2026-09-08",
"status_as_of": "2026-09-08",
"evidence_label": "Source candidate · not reviewed",
"note": "Full public directory statement. Source lane: backlog; source resolution: open. Source verification claims are attributed only; no local Lean verification has be... | {"created_at":"2026-08-17T19:40:23+00:00","display_standard":{"editorial_review":"submitter_supplied","formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":8914,"public_status":{"evidence":{"hasFormalization":false,"hasInformalProof":false,"hasRecordedWork":false,"solutionRecord":null},"lean":{... | [
"formalization-contains-placeholder",
"formalization-not-compiled-during-export",
"current-status-unassessed",
"source-full-text-not-present"
] | [] | {
"created_at": "2026-09-08T03:32:15.935Z",
"updated_at": "2026-09-08T03:32:15.935Z",
"published_at": "2026-09-08T03:33:06.083Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"0f489bbf-dae8-4c4a-8cc1-a74e2531d405","format_issues":null,"number":6667} |
1.0.0 | problem:theoremdb-p8916 | problem | theoremdb-p8916 | Uniqueness of Crystal Components | {
"main": "math",
"sub": "number theory",
"slug": "number-theory",
"name": "Number theory",
"original_discipline": "mathematics"
} | [] | [
{
"code": "11",
"basis": "community-map",
"evidence": "Inferred from community field: number-theory"
}
] | Let $n$, $a$, $b$, $c$, and $d$ be natural numbers. Suppose that $n$ is odd, that $a > 1$ and $b > 1$, that $n = ab$, and that $2(a+1)(b+1)$ divides $(a+b)^2 + (ab+1)^2$. Suppose further that $c > 1$ and $d > 1$, that $n = cd$, and that $2(c+1)(d+1)$ divides $(c+d)^2 + (cd+1)^2$. Does it follow that $\{a, b\} = \{c, d\... | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P8916",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P8916",
"citation": null,
"doi": null,
"arxiv_id": null,
"license": "Selected packets label their content CC0-1.0; original references retain their own terms",
"notes": "Selected problem... | theorem crystals_components_unique (n a b c d : ℕ)
(hab : IsCrystalWithComponents n a b) (hcd : IsCrystalWithComponents n c d) :
({a, b} : Finset ℕ) = {c, d} | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem crystals_components_unique (n a b c d : ℕ)\n (hab : IsCrystalWithComponents n a b) (hcd : IsCrystalWithComponents n c d) :\n ({a, b} : Finset ℕ) = {c, d}",
"target": null,
"source_url": "https://theoremdb.org/statements/P8916",
"format": "declaratio... | [] | TheoremDB P8916 · Candidate awaiting source review | null | Source-reported assessment, retrieved 8 September 2026. Frontier Science has not independently reviewed this status.
Open. No signed Lean verification is attached to this problem or its selected resolution.
This record is in the source backlog; its novelty and unresolved status have not passed source review. | null | published | incremental | hard | [
"source-candidate",
"theoremdb"
] | null | null | null | null | [
{
"kind": "website",
"citation": "TheoremDB P8916",
"url": "https://theoremdb.org/statements/P8916",
"note": null,
"sort_order": 0
},
{
"kind": "formalization",
"citation": "Source formal statement (not locally verified)",
"url": "https://theoremdb.org/statements/P8916",
"not... | {
"source_record": "P8916",
"imported_at": "2026-09-08",
"status_as_of": "2026-09-08",
"evidence_label": "Source candidate · not reviewed",
"note": "Full public directory statement. Source lane: backlog; source resolution: open. Source verification claims are attributed only; no local Lean verification has be... | {"created_at":"2026-08-17T19:40:26+00:00","display_standard":{"editorial_review":"submitter_supplied","formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":8916,"public_status":{"evidence":{"hasFormalization":false,"hasInformalProof":false,"hasRecordedWork":false,"solutionRecord":null},"lean":{... | [
"formalization-not-compiled-during-export",
"current-status-unassessed",
"source-full-text-not-present"
] | [] | {
"created_at": "2026-09-08T03:32:15.935Z",
"updated_at": "2026-09-08T03:32:15.935Z",
"published_at": "2026-09-08T03:33:07.247Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"7eac7f40-c1c0-43a5-975f-d083bbe9b5d5","format_issues":null,"number":6918} |
1.0.0 | problem:theoremdb-p8918 | problem | theoremdb-p8918 | The Second Hardy–Littlewood Conjecture | {
"main": "math",
"sub": "number theory",
"slug": "number-theory",
"name": "Number theory",
"original_discipline": "mathematics"
} | [] | [
{
"code": "11",
"basis": "community-map",
"evidence": "Inferred from community field: number-theory"
}
] | For integers $x, y \geq 2$, let $\pi(z)$ denote the prime-counting function, which gives the number of primes less than or equal to $z$. Prove that $\pi(x + y) \leq \pi(x) + \pi(y)$. | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P8918",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P8918",
"citation": null,
"doi": null,
"arxiv_id": null,
"license": "Selected packets label their content CC0-1.0; original references retain their own terms",
"notes": "Selected problem... | theorem second_hardy_littlewood_conjecture {x y : ℕ} (hx : 2 ≤ x) (hy : 2 ≤ y) :
SecondHardyLittlewoodConjectureFor x y | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem second_hardy_littlewood_conjecture {x y : ℕ} (hx : 2 ≤ x) (hy : 2 ≤ y) :\n SecondHardyLittlewoodConjectureFor x y",
"target": null,
"source_url": "https://theoremdb.org/statements/P8918",
"format": "declaration-or-fragment",
"sha256": "c182e47b0aa4... | [] | TheoremDB P8918 · Candidate awaiting source review | null | Source-reported assessment, retrieved 8 September 2026. Frontier Science has not independently reviewed this status.
Open. No signed Lean verification is attached to this problem or its selected resolution.
This record is in the source backlog; its novelty and unresolved status have not passed source review. | null | published | incremental | hard | [
"source-candidate",
"theoremdb"
] | null | null | null | null | [
{
"kind": "website",
"citation": "TheoremDB P8918",
"url": "https://theoremdb.org/statements/P8918",
"note": null,
"sort_order": 0
},
{
"kind": "formalization",
"citation": "Source formal statement (not locally verified)",
"url": "https://theoremdb.org/statements/P8918",
"not... | {
"source_record": "P8918",
"imported_at": "2026-09-08",
"status_as_of": "2026-09-08",
"evidence_label": "Source candidate · not reviewed",
"note": "Full public directory statement. Source lane: backlog; source resolution: open. Source verification claims are attributed only; no local Lean verification has be... | {"created_at":"2026-08-17T19:40:29+00:00","display_standard":{"editorial_review":"submitter_supplied","formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":8918,"public_status":{"evidence":{"hasFormalization":false,"hasInformalProof":false,"hasRecordedWork":false,"solutionRecord":null},"lean":{... | [
"formalization-not-compiled-during-export",
"current-status-unassessed",
"source-full-text-not-present"
] | [] | {
"created_at": "2026-09-08T03:32:15.935Z",
"updated_at": "2026-09-08T03:32:15.935Z",
"published_at": "2026-09-08T03:33:06.754Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"51a19192-f9f6-48d8-af36-31bed1871022","format_issues":null,"number":6810} |
1.0.0 | problem:theoremdb-p8920 | problem | theoremdb-p8920 | Linear lower bound for the arithmetic theta sum $S(k)$ | {
"main": "math",
"sub": "number theory",
"slug": "number-theory",
"name": "Number theory",
"original_discipline": "mathematics"
} | [] | [
{
"code": "11",
"basis": "community-map",
"evidence": "Inferred from community field: number-theory"
}
] | For an odd prime $k$, define
$$
S'(h,k)=\sum_{j=1}^{k-1}(-1)^{j+1+\lfloor hj/k\rfloor}
$$
for $h\in\{1,\dots,k-1\}$, and define
$$
S(k)=\sum_{h=1}^{k-1}S'(h,k).
$$
Prove that for every odd prime $k>3119$, the inequality $S(k)>3k$ holds. | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P8920",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P8920",
"citation": null,
"doi": null,
"arxiv_id": null,
"license": "Selected packets label their content CC0-1.0; original references retain their own terms",
"notes": "Selected problem... | theorem conjecture_4_3 (k : ℕ) (hprim : k.Prime) (hodd : Odd k) (hgt : k > 3119) : 3 * k < S k | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem conjecture_4_3 (k : ℕ) (hprim : k.Prime) (hodd : Odd k) (hgt : k > 3119) : 3 * k < S k",
"target": null,
"source_url": "https://theoremdb.org/statements/P8920",
"format": "declaration-or-fragment",
"sha256": "d4ef670636dc5381df812a6d386c05f1f5c57a192b... | [] | TheoremDB P8920 · Candidate awaiting source review | null | Source-reported assessment, retrieved 8 September 2026. Frontier Science has not independently reviewed this status.
Open. No signed Lean verification is attached to this problem or its selected resolution.
This record is in the source backlog; its novelty and unresolved status have not passed source review. | null | published | incremental | hard | [
"source-candidate",
"theoremdb"
] | null | null | null | null | [
{
"kind": "website",
"citation": "TheoremDB P8920",
"url": "https://theoremdb.org/statements/P8920",
"note": null,
"sort_order": 0
},
{
"kind": "formalization",
"citation": "Source formal statement (not locally verified)",
"url": "https://theoremdb.org/statements/P8920",
"not... | {
"source_record": "P8920",
"imported_at": "2026-09-08",
"status_as_of": "2026-09-08",
"evidence_label": "Source candidate · not reviewed",
"note": "Full public directory statement. Source lane: backlog; source resolution: open. Source verification claims are attributed only; no local Lean verification has be... | {"created_at":"2026-08-17T19:40:33+00:00","display_standard":{"editorial_review":"submitter_supplied","formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":8920,"public_status":{"evidence":{"hasFormalization":false,"hasInformalProof":false,"hasRecordedWork":false,"solutionRecord":null},"lean":{... | [
"formalization-not-compiled-during-export",
"current-status-unassessed",
"source-full-text-not-present"
] | [] | {
"created_at": "2026-09-08T03:32:15.935Z",
"updated_at": "2026-09-08T03:32:15.935Z",
"published_at": "2026-09-08T03:33:03.242Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"ec635993-8c87-4575-a121-5a70abe34c35","format_issues":null,"number":6053} |
1.0.0 | problem:theoremdb-p8922 | problem | theoremdb-p8922 | Zhi-Wei Sun's 1680-Conjecture on Refinements of Lagrange's Four-Square Theorem | {
"main": "math",
"sub": "number theory",
"slug": "number-theory",
"name": "Number theory",
"original_discipline": "mathematics"
} | [] | [
{
"code": "11",
"basis": "community-map",
"evidence": "Inferred from community field: number-theory"
}
] | For every nonnegative integer $n$, does there exist nonnegative integers $x$, $y$, $z$, and $w$ such that $n = x^2 + y^2 + z^2 + w^2$ and $x^4 + 1680 y^3 z$ is a perfect square? | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P8922",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P8922",
"citation": null,
"doi": null,
"arxiv_id": null,
"license": "Selected packets label their content CC0-1.0; original references retain their own terms",
"notes": "Selected problem... | theorem conjecture (n : ℕ) : A n | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem conjecture (n : ℕ) : A n",
"target": null,
"source_url": "https://theoremdb.org/statements/P8922",
"format": "declaration-or-fragment",
"sha256": "79122c62c8b920d59aac24d0de97415fd020e2a20cec64f4a2ce6f643313560f",
"license": "Selected packets labe... | [] | TheoremDB P8922 · Candidate awaiting source review | null | Source-reported assessment, retrieved 8 September 2026. Frontier Science has not independently reviewed this status.
Open. No signed Lean verification is attached to this problem or its selected resolution.
This record is in the source backlog; its novelty and unresolved status have not passed source review. | null | published | incremental | hard | [
"source-candidate",
"theoremdb"
] | null | null | null | null | [
{
"kind": "website",
"citation": "TheoremDB P8922",
"url": "https://theoremdb.org/statements/P8922",
"note": null,
"sort_order": 0
},
{
"kind": "formalization",
"citation": "Source formal statement (not locally verified)",
"url": "https://theoremdb.org/statements/P8922",
"not... | {
"source_record": "P8922",
"imported_at": "2026-09-08",
"status_as_of": "2026-09-08",
"evidence_label": "Source candidate · not reviewed",
"note": "Full public directory statement. Source lane: backlog; source resolution: open. Source verification claims are attributed only; no local Lean verification has be... | {"created_at":"2026-08-17T19:40:36+00:00","display_standard":{"formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":8922,"public_status":{"evidence":{"hasFormalization":false,"hasInformalProof":false,"hasRecordedWork":false,"solutionRecord":null},"lean":{"kernelAccepted":false,"label":"","reaso... | [
"formalization-not-compiled-during-export",
"current-status-unassessed",
"source-full-text-not-present"
] | [] | {
"created_at": "2026-09-08T03:32:15.935Z",
"updated_at": "2026-09-08T03:32:15.935Z",
"published_at": "2026-09-08T03:33:07.641Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"2ae128a6-7531-48ee-a0f3-81ebe42ca3ef","format_issues":null,"number":7003} |
1.0.0 | problem:theoremdb-p8924 | problem | theoremdb-p8924 | Linear-sized independent sets in dense octahedron-free graphs | {
"main": "math",
"sub": "combinatorics",
"slug": "combinatorics",
"name": "Combinatorics",
"original_discipline": "mathematics"
} | [] | [
{
"code": "05",
"basis": "community-map",
"evidence": "Inferred from community field: combinatorics"
}
] | Let the octahedron denote the complete tripartite graph $K_{2,2,2}$, which has three parts of size $2$ with every vertex joined to every vertex in the other parts and no edges within any part. A graph is said to be octahedron-free if it contains no subgraph isomorphic to the octahedron. The independence number of a gra... | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P8924",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P8924",
"citation": null,
"doi": null,
"arxiv_id": null,
"license": "Selected packets label their content CC0-1.0; original references retain their own terms",
"notes": "Selected problem... | theorem erdos_579 : answer(sorry) ↔
∀ δ : ℝ, 0 < δ → ∃ c : ℝ, 0 < c ∧ ∀ᶠ n : ℕ in atTop,
∀ G : SimpleGraph (Fin n), octahedron.Free G →
δ * (n : ℝ) ^ 2 ≤ G.edgeFinset.card →
c * n ≤ (G.indepNum : ℝ) | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem erdos_579 : answer(sorry) ↔\n ∀ δ : ℝ, 0 < δ → ∃ c : ℝ, 0 < c ∧ ∀ᶠ n : ℕ in atTop,\n ∀ G : SimpleGraph (Fin n), octahedron.Free G →\n δ * (n : ℝ) ^ 2 ≤ G.edgeFinset.card →\n c * n ≤ (G.indepNum : ℝ)",
"target": null,
"source_url": "htt... | [] | TheoremDB P8924 · Candidate awaiting source review | null | Source-reported assessment, retrieved 8 September 2026. Frontier Science has not independently reviewed this status.
Open. No signed Lean verification is attached to this problem or its selected resolution.
This record is in the source backlog; its novelty and unresolved status have not passed source review. | null | published | incremental | hard | [
"source-candidate",
"theoremdb"
] | null | null | null | null | [
{
"kind": "website",
"citation": "TheoremDB P8924",
"url": "https://theoremdb.org/statements/P8924",
"note": null,
"sort_order": 0
},
{
"kind": "formalization",
"citation": "Source formal statement (not locally verified)",
"url": "https://theoremdb.org/statements/P8924",
"not... | {
"source_record": "P8924",
"imported_at": "2026-09-08",
"status_as_of": "2026-09-08",
"evidence_label": "Source candidate · not reviewed",
"note": "Full public directory statement. Source lane: backlog; source resolution: open. Source verification claims are attributed only; no local Lean verification has be... | {"created_at":"2026-08-17T19:40:39+00:00","display_standard":{"editorial_review":"submitter_supplied","formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":8924,"public_status":{"evidence":{"hasFormalization":false,"hasInformalProof":false,"hasRecordedWork":false,"solutionRecord":null},"lean":{... | [
"formalization-contains-placeholder",
"formalization-not-compiled-during-export",
"current-status-unassessed",
"source-full-text-not-present"
] | [] | {
"created_at": "2026-09-08T03:32:15.935Z",
"updated_at": "2026-09-08T03:32:15.935Z",
"published_at": "2026-09-08T03:33:03.265Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"cd5d40c1-fc45-4b7d-9f72-3428ca5e22e2","format_issues":null,"number":6058} |
1.0.0 | problem:theoremdb-p8926 | problem | theoremdb-p8926 | Boundedness of Divisors Near the Square Root | {
"main": "math",
"sub": "number theory",
"slug": "number-theory",
"name": "Number theory",
"original_discipline": "mathematics"
} | [] | [
{
"code": "11",
"basis": "community-map",
"evidence": "Inferred from community field: number-theory"
}
] | For a positive integer $n$ and real numbers $\varepsilon > 0$ and $C > 0$, define the set of near-square-root divisors of $n$ with parameters $\varepsilon$ and $C$ to be the set of divisors $d$ of $n$ satisfying
$$
n^{1/2} < d < n^{1/2} + C \cdot n^{1/2-\varepsilon}.
$$
Is it true that for every $\varepsilon > 0$,... | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P8926",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P8926",
"citation": null,
"doi": null,
"arxiv_id": null,
"license": "Selected packets label their content CC0-1.0; original references retain their own terms",
"notes": "Selected problem... | theorem erdos_886 :
answer(sorry) ↔ ∀ ε > 0, ∃ K : ℕ, ∀ᶠ n in atTop, (Erdos886Divisors n ε 1).card ≤ K | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem erdos_886 :\n answer(sorry) ↔ ∀ ε > 0, ∃ K : ℕ, ∀ᶠ n in atTop, (Erdos886Divisors n ε 1).card ≤ K",
"target": null,
"source_url": "https://theoremdb.org/statements/P8926",
"format": "declaration-or-fragment",
"sha256": "cda66348851830f97830f026d05a9... | [] | TheoremDB P8926 · Candidate awaiting source review | null | Source-reported assessment, retrieved 8 September 2026. Frontier Science has not independently reviewed this status.
Open. No signed Lean verification is attached to this problem or its selected resolution.
This record is in the source backlog; its novelty and unresolved status have not passed source review. | null | published | incremental | hard | [
"source-candidate",
"theoremdb"
] | null | null | null | null | [
{
"kind": "website",
"citation": "TheoremDB P8926",
"url": "https://theoremdb.org/statements/P8926",
"note": null,
"sort_order": 0
},
{
"kind": "formalization",
"citation": "Source formal statement (not locally verified)",
"url": "https://theoremdb.org/statements/P8926",
"not... | {
"source_record": "P8926",
"imported_at": "2026-09-08",
"status_as_of": "2026-09-08",
"evidence_label": "Source candidate · not reviewed",
"note": "Full public directory statement. Source lane: backlog; source resolution: open. Source verification claims are attributed only; no local Lean verification has be... | {"created_at":"2026-08-17T19:40:41+00:00","display_standard":{"editorial_review":"submitter_supplied","formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":8926,"public_status":{"evidence":{"hasFormalization":false,"hasInformalProof":false,"hasRecordedWork":false,"solutionRecord":null},"lean":{... | [
"formalization-contains-placeholder",
"formalization-not-compiled-during-export",
"current-status-unassessed",
"source-full-text-not-present"
] | [] | {
"created_at": "2026-09-08T03:32:15.935Z",
"updated_at": "2026-09-08T03:32:15.935Z",
"published_at": "2026-09-08T03:32:50.725Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"60c589ba-13c5-47ad-9dab-f512ec634ba4","format_issues":null,"number":3479} |
1.0.0 | problem:theoremdb-p8928 | problem | theoremdb-p8928 | The Kummer–Vandiver Conjecture | {
"main": "math",
"sub": "number theory",
"slug": "number-theory",
"name": "Number theory",
"original_discipline": "mathematics"
} | [] | [
{
"code": "11",
"basis": "community-map",
"evidence": "Inferred from community field: number-theory"
}
] | Let $p$ be a prime number. Let $\zeta_p$ denote a primitive $p$-th root of unity, and let $\mathbb{Q}(\zeta_p)$ be the $p$-th cyclotomic field. The maximal real subfield of $\mathbb{Q}(\zeta_p)$ is the subfield fixed by complex conjugation, denoted $\mathbb{Q}(\zeta_p)^+$. The class number of $\mathbb{Q}(\zeta_p)^+$ is... | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P8928",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P8928",
"citation": null,
"doi": null,
"arxiv_id": null,
"license": "Selected packets label their content CC0-1.0; original references retain their own terms",
"notes": "Selected problem... | theorem kummer_vandiver (p : ℕ+) (hp : p.Prime) :
¬ ↑p ∣ (classNumber (maximalRealSubfield (CyclotomicField p ℚ))) | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem kummer_vandiver (p : ℕ+) (hp : p.Prime) :\n ¬ ↑p ∣ (classNumber (maximalRealSubfield (CyclotomicField p ℚ)))",
"target": null,
"source_url": "https://theoremdb.org/statements/P8928",
"format": "declaration-or-fragment",
"sha256": "45cd93c6b9b93f370... | [] | TheoremDB P8928 · Candidate awaiting source review | null | Source-reported assessment, retrieved 8 September 2026. Frontier Science has not independently reviewed this status.
Open. No signed Lean verification is attached to this problem or its selected resolution.
This record is in the source backlog; its novelty and unresolved status have not passed source review. | null | published | incremental | hard | [
"source-candidate",
"theoremdb"
] | null | null | null | null | [
{
"kind": "website",
"citation": "TheoremDB P8928",
"url": "https://theoremdb.org/statements/P8928",
"note": null,
"sort_order": 0
},
{
"kind": "formalization",
"citation": "Source formal statement (not locally verified)",
"url": "https://theoremdb.org/statements/P8928",
"not... | {
"source_record": "P8928",
"imported_at": "2026-09-08",
"status_as_of": "2026-09-08",
"evidence_label": "Source candidate · not reviewed",
"note": "Full public directory statement. Source lane: backlog; source resolution: open. Source verification claims are attributed only; no local Lean verification has be... | {"created_at":"2026-08-17T19:40:44+00:00","display_standard":{"formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":8928,"public_status":{"evidence":{"hasFormalization":false,"hasInformalProof":false,"hasRecordedWork":false,"solutionRecord":null},"lean":{"kernelAccepted":false,"label":"","reaso... | [
"formalization-not-compiled-during-export",
"current-status-unassessed",
"source-full-text-not-present"
] | [] | {
"created_at": "2026-09-08T03:32:15.935Z",
"updated_at": "2026-09-08T03:32:15.935Z",
"published_at": "2026-09-08T03:33:06.585Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"09d2de8a-b448-490e-a7c5-36540098963d","format_issues":null,"number":6774} |
1.0.0 | problem:theoremdb-p8930 | problem | theoremdb-p8930 | Eventual validity of the no-three-in-line conjecture | {
"main": "math",
"sub": "combinatorics",
"slug": "combinatorics",
"name": "Combinatorics",
"original_discipline": "mathematics"
} | [] | [
{
"code": "05",
"basis": "community-map",
"evidence": "Inferred from community field: combinatorics"
}
] | For a positive integer $N$, consider the $N \times N$ grid $[N]^2 = \{0, 1, \ldots, N-1\} \times \{0, 1, \ldots, N-1\}$. A subset $S \subseteq [N]^2$ is called allowed if no three points of $S$ lie on a common straight line in $\mathbb{R}^2$. Let $A(N)$ denote the maximum possible cardinality of an allowed subset of $[... | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P8930",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P8930",
"citation": null,
"doi": null,
"arxiv_id": null,
"license": "Selected packets label their content CC0-1.0; original references retain their own terms",
"notes": "Selected problem... | theorem green_72.variants.eventually : answer(sorry) ↔ ∀ᶠ N in Filter.atTop, NoKInLineFor 3 N | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem green_72.variants.eventually : answer(sorry) ↔ ∀ᶠ N in Filter.atTop, NoKInLineFor 3 N",
"target": null,
"source_url": "https://theoremdb.org/statements/P8930",
"format": "declaration-or-fragment",
"sha256": "39ba8e1dae2a528cce3f328bb8ad820a02fadc0acf1... | [] | TheoremDB P8930 · Candidate awaiting source review | null | Source-reported assessment, retrieved 8 September 2026. Frontier Science has not independently reviewed this status.
Open. No signed Lean verification is attached to this problem or its selected resolution.
This record is in the source backlog; its novelty and unresolved status have not passed source review. | null | published | incremental | hard | [
"source-candidate",
"theoremdb"
] | null | null | null | null | [
{
"kind": "website",
"citation": "TheoremDB P8930",
"url": "https://theoremdb.org/statements/P8930",
"note": null,
"sort_order": 0
},
{
"kind": "formalization",
"citation": "Source formal statement (not locally verified)",
"url": "https://theoremdb.org/statements/P8930",
"not... | {
"source_record": "P8930",
"imported_at": "2026-09-08",
"status_as_of": "2026-09-08",
"evidence_label": "Source candidate · not reviewed",
"note": "Full public directory statement. Source lane: backlog; source resolution: open. Source verification claims are attributed only; no local Lean verification has be... | {"created_at":"2026-08-17T19:40:47+00:00","display_standard":{"formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":8930,"public_status":{"evidence":{"hasFormalization":false,"hasInformalProof":false,"hasRecordedWork":false,"solutionRecord":null},"lean":{"kernelAccepted":false,"label":"","reaso... | [
"formalization-contains-placeholder",
"formalization-not-compiled-during-export",
"current-status-unassessed",
"source-full-text-not-present"
] | [] | {
"created_at": "2026-09-08T03:32:15.935Z",
"updated_at": "2026-09-08T03:32:15.935Z",
"published_at": "2026-09-08T03:32:56.814Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"036399d8-e819-4b52-9d9f-204d0d3c2140","format_issues":null,"number":4821} |
1.0.0 | problem:theoremdb-p8932 | problem | theoremdb-p8932 | Convergent sequences in infinite homogeneous compact Hausdorff spaces | {
"main": "math",
"sub": "geometry and topology",
"slug": "geometry-and-topology",
"name": "Geometry and topology",
"original_discipline": "mathematics"
} | [] | [] | A topological space $X$ is called *homogeneous* if for all points $x, y \in X$ there exists a homeomorphism $f : X \to X$ such that $f(x) = y$. Determine whether every infinite homogeneous compact Hausdorff space contains a nontrivial convergent sequence. That is, does every infinite homogeneous compact Hausdorff space... | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P8932",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P8932",
"citation": null,
"doi": null,
"arxiv_id": null,
"license": "Selected packets label their content CC0-1.0; original references retain their own terms",
"notes": "Selected problem... | theorem homogeneousSpace_exists_inj_tendsto :
answer(sorry) ↔ ∀ (X : Type) (_ : TopologicalSpace X), ¬ Finite X → T2Space X → CompactSpace X →
HomogeneousSpace X → ∃ s : ℕ → X, s.Injective ∧ ∃ a : X, Tendsto s atTop (nhds a) | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem homogeneousSpace_exists_inj_tendsto :\n answer(sorry) ↔ ∀ (X : Type) (_ : TopologicalSpace X), ¬ Finite X → T2Space X → CompactSpace X →\n HomogeneousSpace X → ∃ s : ℕ → X, s.Injective ∧ ∃ a : X, Tendsto s atTop (nhds a)",
"target": null,
"source_url"... | [] | TheoremDB P8932 · Candidate awaiting source review | null | Source-reported assessment, retrieved 8 September 2026. Frontier Science has not independently reviewed this status.
Open. No signed Lean verification is attached to this problem or its selected resolution.
This record is in the source backlog; its novelty and unresolved status have not passed source review. | null | published | incremental | hard | [
"source-candidate",
"theoremdb"
] | null | null | null | null | [
{
"kind": "website",
"citation": "TheoremDB P8932",
"url": "https://theoremdb.org/statements/P8932",
"note": null,
"sort_order": 0
},
{
"kind": "formalization",
"citation": "Source formal statement (not locally verified)",
"url": "https://theoremdb.org/statements/P8932",
"not... | {
"source_record": "P8932",
"imported_at": "2026-09-08",
"status_as_of": "2026-09-08",
"evidence_label": "Source candidate · not reviewed",
"note": "Full public directory statement. Source lane: backlog; source resolution: open. Source verification claims are attributed only; no local Lean verification has be... | {"created_at":"2026-08-17T19:40:50+00:00","display_standard":{"editorial_review":"submitter_supplied","formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":8932,"public_status":{"evidence":{"hasFormalization":false,"hasInformalProof":false,"hasRecordedWork":false,"solutionRecord":null},"lean":{... | [
"formalization-contains-placeholder",
"formalization-not-compiled-during-export",
"current-status-unassessed",
"source-full-text-not-present"
] | [] | {
"created_at": "2026-09-08T03:32:15.935Z",
"updated_at": "2026-09-08T03:32:15.935Z",
"published_at": "2026-09-08T03:32:52.693Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"ddaaf78e-9c47-40ba-bbd9-0d3ba0d4e321","format_issues":null,"number":3915} |
1.0.0 | problem:theoremdb-p8934 | problem | theoremdb-p8934 | Weak form of Hall's conjecture | {
"main": "math",
"sub": "number theory",
"slug": "number-theory",
"name": "Number theory",
"original_discipline": "mathematics"
} | [] | [
{
"code": "11",
"basis": "community-map",
"evidence": "Inferred from community field: number-theory"
}
] | Let $\varepsilon > 0$ be a real number. A pair of real numbers $(C, e)$ with $C > 0$ is said to satisfy the Hall inequality if for all integers $x$ and $y$ with $y^2 \neq x^3$,
$$
|y^2 - x^3| > C \cdot |x|^e.
$$
The Hall conjecture with exponent $e$ holds if there exists $C > 0$ such that $(C, e)$ satisfies the Hall ... | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P8934",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P8934",
"citation": null,
"doi": null,
"arxiv_id": null,
"license": "Selected packets label their content CC0-1.0; original references retain their own terms",
"notes": "Selected problem... | theorem weak_hall_conjecture (ε : ℝ) (hε : ε > 0) : HallConjectureExp (2⁻¹ - ε) | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem weak_hall_conjecture (ε : ℝ) (hε : ε > 0) : HallConjectureExp (2⁻¹ - ε)",
"target": null,
"source_url": "https://theoremdb.org/statements/P8934",
"format": "declaration-or-fragment",
"sha256": "28888089832642c6ea5d8760bd2ca86306baff59e047d614980ffad17... | [] | TheoremDB P8934 · Candidate awaiting source review | null | Source-reported assessment, retrieved 8 September 2026. Frontier Science has not independently reviewed this status.
Open. No signed Lean verification is attached to this problem or its selected resolution.
This record is in the source backlog; its novelty and unresolved status have not passed source review. | null | published | incremental | hard | [
"source-candidate",
"theoremdb"
] | null | null | null | null | [
{
"kind": "website",
"citation": "TheoremDB P8934",
"url": "https://theoremdb.org/statements/P8934",
"note": null,
"sort_order": 0
},
{
"kind": "formalization",
"citation": "Source formal statement (not locally verified)",
"url": "https://theoremdb.org/statements/P8934",
"not... | {
"source_record": "P8934",
"imported_at": "2026-09-08",
"status_as_of": "2026-09-08",
"evidence_label": "Source candidate · not reviewed",
"note": "Full public directory statement. Source lane: backlog; source resolution: open. Source verification claims are attributed only; no local Lean verification has be... | {"created_at":"2026-08-17T19:40:53+00:00","display_standard":{"editorial_review":"submitter_supplied","formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":8934,"public_status":{"evidence":{"hasFormalization":false,"hasInformalProof":false,"hasRecordedWork":false,"solutionRecord":null},"lean":{... | [
"formalization-not-compiled-during-export",
"current-status-unassessed",
"source-full-text-not-present"
] | [] | {
"created_at": "2026-09-08T03:32:15.935Z",
"updated_at": "2026-09-08T03:32:15.935Z",
"published_at": "2026-09-08T03:33:07.450Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"66c77957-3b9f-4b88-920e-3ea2a0523f05","format_issues":null,"number":6962} |
1.0.0 | problem:theoremdb-p8936 | problem | theoremdb-p8936 | Equidistribution of Scalar Multiples of Powers of Three Halves Modulo One | {
"main": "math",
"sub": "number theory",
"slug": "number-theory",
"name": "Number theory",
"original_discipline": "mathematics"
} | [] | [
{
"code": "11",
"basis": "community-map",
"evidence": "Inferred from community field: number-theory"
}
] | Let $x$ be a real number that is transcendental over the rational numbers $\mathbb{Q}$, meaning that $x$ is not a root of any non-zero polynomial with rational coefficients. Consider the sequence whose $n$-th term is $x \cdot (3/2)^n$ for $n = 1, 2, \ldots$. A sequence $(a_n)$ of real numbers is said to be equidistribu... | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P8936",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P8936",
"citation": null,
"doi": null,
"arxiv_id": null,
"license": "Selected packets label their content CC0-1.0; original references retain their own terms",
"notes": "Selected problem... | theorem isEquidistributedModuloOne_transcendental_three_halves_pow (x : ℝ)
(hx : Transcendental ℚ x) :
IsEquidistributedModuloOne (fun n ↦ x * (3 / 2 : ℝ) ^ n) | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem isEquidistributedModuloOne_transcendental_three_halves_pow (x : ℝ)\n (hx : Transcendental ℚ x) :\n IsEquidistributedModuloOne (fun n ↦ x * (3 / 2 : ℝ) ^ n)",
"target": null,
"source_url": "https://theoremdb.org/statements/P8936",
"format": "declarat... | [] | TheoremDB P8936 · Candidate awaiting source review | null | Source-reported assessment, retrieved 8 September 2026. Frontier Science has not independently reviewed this status.
Open. No signed Lean verification is attached to this problem or its selected resolution.
This record is in the source backlog; its novelty and unresolved status have not passed source review. | null | published | incremental | hard | [
"source-candidate",
"theoremdb"
] | null | null | null | null | [
{
"kind": "website",
"citation": "TheoremDB P8936",
"url": "https://theoremdb.org/statements/P8936",
"note": null,
"sort_order": 0
},
{
"kind": "formalization",
"citation": "Source formal statement (not locally verified)",
"url": "https://theoremdb.org/statements/P8936",
"not... | {
"source_record": "P8936",
"imported_at": "2026-09-08",
"status_as_of": "2026-09-08",
"evidence_label": "Source candidate · not reviewed",
"note": "Full public directory statement. Source lane: backlog; source resolution: open. Source verification claims are attributed only; no local Lean verification has be... | {"created_at":"2026-08-17T19:40:55+00:00","display_standard":{"formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":8936,"public_status":{"evidence":{"hasFormalization":false,"hasInformalProof":false,"hasRecordedWork":false,"solutionRecord":null},"lean":{"kernelAccepted":false,"label":"","reaso... | [
"formalization-not-compiled-during-export",
"current-status-unassessed",
"source-full-text-not-present"
] | [] | {
"created_at": "2026-09-08T03:32:15.935Z",
"updated_at": "2026-09-08T03:32:15.935Z",
"published_at": "2026-09-08T03:32:54.777Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"6b45b9a9-f850-4a2a-b0ef-a5a63bfafb91","format_issues":null,"number":4376} |
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