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1.0.0 | problem:theoremdb-p8326 | problem | theoremdb-p8326 | De Giorgi's conjecture for dimensions at most 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 $n$ be a positive integer with $n \leq 8$. Let $u : \mathbb{R}^n \to \mathbb{R}$ be a function. We say that $u$ is a bounded classical solution to $\Delta u + u - u^3 = 0$ if $u$ is twice continuously differentiable, there exists a constant $C$ such that $|u(x)| \leq C$ for all $x \in \mathbb{R}^n$, and $\Delta u(x... | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P8326",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P8326",
"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_le_eight (hn : n ≤ 8) : (DeGiorgi_conclusion n) | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem DeGiorgi_le_eight (hn : n ≤ 8) : (DeGiorgi_conclusion n)",
"target": null,
"source_url": "https://theoremdb.org/statements/P8326",
"format": "declaration-or-fragment",
"sha256": "08dda57de52b7196eb0c56aa070fb73ef75efb179b2e4d3c6edb2554966f864b",
"... | [] | TheoremDB P8326 · 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 P8326",
"url": "https://theoremdb.org/statements/P8326",
"note": null,
"sort_order": 0
},
{
"kind": "formalization",
"citation": "Source formal statement (not locally verified)",
"url": "https://theoremdb.org/statements/P8326",
"not... | {
"source_record": "P8326",
"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":"2026-08-17T18:39:42+00:00","display_standard":{"editorial_review":"submitter_supplied","formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":8326,"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": "2026-09-08T03:32:52.891Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"1b436df2-31f1-4d11-8c86-c600c2afedc7","format_issues":null,"number":3958} |
1.0.0 | problem:theoremdb-p8328 | problem | theoremdb-p8328 | Property B for Almost-Disjoint Families with Forbidden Intersection Size | {
"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 $\alpha$ be a set and let $(A_i)_{i \in I}$ be a family of subsets of $\alpha$. A family is called countably infinite if each set $A_i$ is both countable and infinite. The family is called almost-disjoint if for all distinct $i, j \in I$, the intersection $A_i \cap A_j$ is finite. A 2-colouring of $\alpha$ is a fun... | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P8328",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P8328",
"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_602 : answer(sorry) ↔
∀ {α : Type*} {I : Type*} (A : I → Set α),
(∀ i, (A i).Countable ∧ (A i).Infinite) →
(∀ i j, i ≠ j → (A i ∩ A j).Finite) →
(∀ i j, i ≠ j → Set.ncard (A i ∩ A j) ≠ 1) →
HasPropertyB I A | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem erdos_602 : answer(sorry) ↔\n ∀ {α : Type*} {I : Type*} (A : I → Set α),\n (∀ i, (A i).Countable ∧ (A i).Infinite) →\n (∀ i j, i ≠ j → (A i ∩ A j).Finite) →\n (∀ i j, i ≠ j → Set.ncard (A i ∩ A j) ≠ 1) →\n HasPropertyB I A",
"target": null,... | [] | TheoremDB P8328 · 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 P8328",
"url": "https://theoremdb.org/statements/P8328",
"note": null,
"sort_order": 0
},
{
"kind": "formalization",
"citation": "Source formal statement (not locally verified)",
"url": "https://theoremdb.org/statements/P8328",
"not... | {
"source_record": "P8328",
"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":"2026-08-17T18:39:46+00:00","display_standard":{"editorial_review":"submitter_supplied","formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":8328,"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": "2026-09-08T03:33:05.322Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"64f9d321-d127-4bf9-8079-368655368373","format_issues":null,"number":6503} |
1.0.0 | problem:theoremdb-p8330 | problem | theoremdb-p8330 | Asymptotic growth of a prime-reciprocal sum 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 each positive integer $n$, define
$$
f(n)=\sum_{\substack{p\text{ prime}\\ p<n}}\frac{1}{n-p}.
$$
Is it true that $f(n)=o(\log\log n)$ as $n\to\infty$? | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P8330",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P8330",
"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_950.parts.iii :
answer(sorry) ↔ f =o[atTop] (fun n : ℕ ↦ Real.log (Real.log n)) | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem erdos_950.parts.iii :\n answer(sorry) ↔ f =o[atTop] (fun n : ℕ ↦ Real.log (Real.log n))",
"target": null,
"source_url": "https://theoremdb.org/statements/P8330",
"format": "declaration-or-fragment",
"sha256": "20b974429fe5d196138f1f9b51ae8f85c72c80... | [] | TheoremDB P8330 · 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 P8330",
"url": "https://theoremdb.org/statements/P8330",
"note": null,
"sort_order": 0
},
{
"kind": "formalization",
"citation": "Source formal statement (not locally verified)",
"url": "https://theoremdb.org/statements/P8330",
"not... | {
"source_record": "P8330",
"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":"2026-08-17T18:39:49+00:00","display_standard":{"editorial_review":"submitter_supplied","formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":8330,"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": "2026-09-08T03:32:50.065Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"e35fff66-9295-4bb0-9a9b-f644f683065b","format_issues":null,"number":3343} |
1.0.0 | problem:theoremdb-p8332 | problem | theoremdb-p8332 | Positivity of the arithmetic theta sum $S(k)$ for odd 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 integers $h$ and $k$ with $k \geq 1$, define
$$
S'(h, k) = \sum_{j=1}^{k-1} (-1)^{j + 1 + \lfloor hj/k \rfloor},
$$
where $\lfloor \cdot \rfloor$ denotes the floor function. For a positive integer $k$, define
$$
S(k) = \sum_{h=1}^{k-1} S'(h, k).
$$
Let $k$ be an odd prime. Prove that $S(k) > 0$. | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P8332",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P8332",
"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_1_1 (k : ℕ) (hprim : k.Prime) (hodd : Odd k) : 0 < S k | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem conjecture_1_1 (k : ℕ) (hprim : k.Prime) (hodd : Odd k) : 0 < S k",
"target": null,
"source_url": "https://theoremdb.org/statements/P8332",
"format": "declaration-or-fragment",
"sha256": "69568979954df2885a5512f2dc243de0aa4e5271b8fbed435af9113835bfd90... | [] | TheoremDB P8332 · 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 P8332",
"url": "https://theoremdb.org/statements/P8332",
"note": null,
"sort_order": 0
},
{
"kind": "formalization",
"citation": "Source formal statement (not locally verified)",
"url": "https://theoremdb.org/statements/P8332",
"not... | {
"source_record": "P8332",
"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":"2026-08-17T18:39:58+00:00","display_standard":{"formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":8332,"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": "2026-09-08T03:33:04.979Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"4da4fe65-c6d0-4bdd-8c41-d37b00843f7d","format_issues":null,"number":6427} |
1.0.0 | problem:theoremdb-p8334 | problem | theoremdb-p8334 | Limit inferior of the minimal coprime base 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 each positive integer $n$, define $h(n)$ to be the smallest integer $m > 2$ such that the integers $2^n - 1, 3^n - 1, \dots, m^n - 1$ are collectively coprime; that is, their greatest common divisor equals $1$. If no such $m$ exists, set $h(n) = \infty$. Here $\mathbb{N}^\infty$ denotes the extended natural numbers... | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P8334",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P8334",
"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_770.parts.ii : answer(sorry) ↔ liminf h atTop = ⊤ | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem erdos_770.parts.ii : answer(sorry) ↔ liminf h atTop = ⊤",
"target": null,
"source_url": "https://theoremdb.org/statements/P8334",
"format": "declaration-or-fragment",
"sha256": "b1c587c57c73769c6922bd2832c63308dbe0641589d694faae0ceaa2ae54c6b5",
"l... | [] | TheoremDB P8334 · 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 P8334",
"url": "https://theoremdb.org/statements/P8334",
"note": null,
"sort_order": 0
},
{
"kind": "formalization",
"citation": "Source formal statement (not locally verified)",
"url": "https://theoremdb.org/statements/P8334",
"not... | {
"source_record": "P8334",
"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":"2026-08-17T18:40:02+00:00","display_standard":{"editorial_review":"submitter_supplied","formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":8334,"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": "2026-09-08T03:33:03.187Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"dc0179fb-9421-41e9-bf73-70cba450adeb","format_issues":null,"number":6042} |
1.0.0 | problem:theoremdb-p8336 | problem | theoremdb-p8336 | Existence of an absolutely maximally entangled state for 12 parties with local dimension 6 | {
"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 a positive integer and let $d \geq 2$ be a positive integer. Consider an $n$-partite quantum system where each party has a $d$-dimensional Hilbert space $\mathbb{C}^d$. The total Hilbert space is $(\mathbb{C}^d)^{\otimes n}$. A pure state $|\psi\rangle \in (\mathbb{C}^d)^{\otimes n}$ is called absolut... | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P8336",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P8336",
"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_12_6_open :
answer(sorry) ↔ ExistsAME 12 6 | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem ame_12_6_open :\n answer(sorry) ↔ ExistsAME 12 6",
"target": null,
"source_url": "https://theoremdb.org/statements/P8336",
"format": "declaration-or-fragment",
"sha256": "3facf3086afe4d175bd81dfcb53f229b7fc4aec7c7334dd64b1c0283d985244f",
"licen... | [] | TheoremDB P8336 · 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 P8336",
"url": "https://theoremdb.org/statements/P8336",
"note": null,
"sort_order": 0
},
{
"kind": "formalization",
"citation": "Source formal statement (not locally verified)",
"url": "https://theoremdb.org/statements/P8336",
"not... | {
"source_record": "P8336",
"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":"2026-08-17T18:40:03+00:00","display_standard":{"editorial_review":"submitter_supplied","formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":8336,"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": "2026-09-08T03:32:58.617Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"d173d5b9-605a-4516-afeb-d1f67bb3c0d9","format_issues":null,"number":5220} |
1.0.0 | problem:theoremdb-p8338 | problem | theoremdb-p8338 | Asymptotic behavior 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 $1 \leq a_1 \leq a_2 \leq \dots \leq a_k \leq n$ such that all sums of the form $\sum_{i=u}^{v} a_i$ with $1 \leq u \leq v \leq k$ are distinct. Is it true that $h(n) = o(n)$? Equivalently, does $\displaystyle\lim_{n... | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P8338",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P8338",
"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.i : (fun n ↦ (h n : ℝ)) =o[atTop] (fun n ↦ (n : ℝ)) | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem erdos_357.variants.monotone.parts.i : (fun n ↦ (h n : ℝ)) =o[atTop] (fun n ↦ (n : ℝ))",
"target": null,
"source_url": "https://theoremdb.org/statements/P8338",
"format": "declaration-or-fragment",
"sha256": "154c1180d5e9e50b8647f551b2ff98b93dd5118ff3d... | [] | TheoremDB P8338 · 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 P8338",
"url": "https://theoremdb.org/statements/P8338",
"note": null,
"sort_order": 0
},
{
"kind": "formalization",
"citation": "Source formal statement (not locally verified)",
"url": "https://theoremdb.org/statements/P8338",
"not... | {
"source_record": "P8338",
"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":"2026-08-17T18:40:03+00:00","display_standard":{"editorial_review":"submitter_supplied","formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":8338,"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": "2026-09-08T03:32:49.908Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"26d10661-3fa0-44d3-98aa-4d9f661a34ec","format_issues":null,"number":3311} |
1.0.0 | problem:theoremdb-p8340 | problem | theoremdb-p8340 | Asymptotic count of elliptic curves with bounded rank and height | {
"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 $E$ be an elliptic curve over $\mathbb{Q}$. Every such curve is isomorphic to a unique curve with Weierstrass equation $y^2 = x^3 + Ax + B$ where $A, B \in \mathbb{Z}$ satisfy the reduced condition: for every prime $p$, it is not the case that both $p^4 \mid A$ and $p^6 \mid B$, and the discriminant condition $4A^3... | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P8340",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P8340",
"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 rank_height_count_asymptotic (r : ℕ) (h₁ : 1 ≤ r) (h₂ : r ≤ 20) :
∃ f : ℕ → ℝ, atTop.Tendsto f (𝓝 0) ∧
∀ H : ℕ, 1 < H → {E ∈ heightLE H | r ≤ E.rank}.ncard = (H : ℝ) ^ ((21 - r) / 24 + f H) | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem rank_height_count_asymptotic (r : ℕ) (h₁ : 1 ≤ r) (h₂ : r ≤ 20) :\n ∃ f : ℕ → ℝ, atTop.Tendsto f (𝓝 0) ∧\n ∀ H : ℕ, 1 < H → {E ∈ heightLE H | r ≤ E.rank}.ncard = (H : ℝ) ^ ((21 - r) / 24 + f H)",
"target": null,
"source_url": "https://theoremdb.org/s... | [] | TheoremDB P8340 · 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 P8340",
"url": "https://theoremdb.org/statements/P8340",
"note": null,
"sort_order": 0
},
{
"kind": "formalization",
"citation": "Source formal statement (not locally verified)",
"url": "https://theoremdb.org/statements/P8340",
"not... | {
"source_record": "P8340",
"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":"2026-08-17T18:40:05+00:00","display_standard":{"editorial_review":"submitter_supplied","formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":8340,"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": "2026-09-08T03:32:49.981Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"ed9204ff-78a9-4a01-afdd-150c37f3da81","format_issues":null,"number":3326} |
1.0.0 | problem:theoremdb-p8342 | problem | theoremdb-p8342 | Erdős's distinct distances diameter problem | {
"main": "math",
"sub": "geometry and topology",
"slug": "geometry-and-topology",
"name": "Geometry and topology",
"original_discipline": "mathematics"
} | [] | [] | Let a finite set of points in the plane be called distance-separated if whenever two distances determined by pairs of its points are unequal, they differ by at least 1. For each positive integer $n$, let $A$ be a distance-separated set of $n$ points in $\mathbb{R}^2$, and let $\operatorname{diam}(A)$ denote the diamete... | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P8342",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P8342",
"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_100 :
answer(sorry) ↔ ∃ C > (0 : ℝ), ∀ᶠ n in atTop, ∀ A : Finset ℝ²,
A.card = n →
DistancesSeparated A →
diam (A : Set ℝ²) > C * n | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem erdos_100 :\n answer(sorry) ↔ ∃ C > (0 : ℝ), ∀ᶠ n in atTop, ∀ A : Finset ℝ²,\n A.card = n →\n DistancesSeparated A →\n diam (A : Set ℝ²) > C * n",
"target": null,
"source_url": "https://theoremdb.org/statements/P8342",
"format": "declara... | [] | TheoremDB P8342 · 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 P8342",
"url": "https://theoremdb.org/statements/P8342",
"note": null,
"sort_order": 0
},
{
"kind": "formalization",
"citation": "Source formal statement (not locally verified)",
"url": "https://theoremdb.org/statements/P8342",
"not... | {
"source_record": "P8342",
"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":"2026-08-17T18:40:06+00:00","display_standard":{"editorial_review":"submitter_supplied","formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":8342,"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": "2026-09-08T03:32:56.365Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"94030bf2-2486-4aed-acf9-26d17ad279f4","format_issues":null,"number":4725} |
1.0.0 | problem:theoremdb-p8344 | problem | theoremdb-p8344 | Irrationality of the product of Euler's number and pi | {
"main": "math",
"sub": "analysis",
"slug": "analysis",
"name": "Analysis",
"original_discipline": "mathematics"
} | [] | [] | Let $e = \exp(1)$ denote Euler's number, and let $\pi$ denote the ratio of the circumference of a circle to its diameter. Is the product $e \pi$ irrational? | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P8344",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P8344",
"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_times_pi :
answer(sorry) ↔ Irrational (e * π) | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem irrational_e_times_pi :\n answer(sorry) ↔ Irrational (e * π)",
"target": null,
"source_url": "https://theoremdb.org/statements/P8344",
"format": "declaration-or-fragment",
"sha256": "104500f6840a0c55b9543c883d82d668f7c3518ffeef69fc3ee94e6b0dfed2b8"... | [] | TheoremDB P8344 · 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 P8344",
"url": "https://theoremdb.org/statements/P8344",
"note": null,
"sort_order": 0
},
{
"kind": "formalization",
"citation": "Source formal statement (not locally verified)",
"url": "https://theoremdb.org/statements/P8344",
"not... | {
"source_record": "P8344",
"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":"2026-08-17T18:40:08+00:00","display_standard":{"editorial_review":"submitter_supplied","formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":8344,"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": "2026-09-08T03:33:01.807Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"a128828e-a0ca-432e-b17f-df941b7f8955","format_issues":null,"number":5739} |
1.0.0 | problem:theoremdb-p8346 | problem | theoremdb-p8346 | Smooth Poincaré conjecture in remaining higher dimensions | {
"main": "math",
"sub": "geometry and topology",
"slug": "geometry-and-topology",
"name": "Geometry and topology",
"original_discipline": "mathematics"
} | [] | [] | Let $n$ be a natural number with $n > 4$ and $n \notin \{1, 2, 3, 5, 6, 12, 56, 61\}$. The smooth Poincaré conjecture in dimension $n$ asserts that every smooth $n$-manifold homotopy equivalent to the $n$-sphere is diffeomorphic to the $n$-sphere. Does this conjecture fail for every such $n$? | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P8346",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P8346",
"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 poincare_conjecture.variants.smooth_other_cases (n : ℕ) (hn : n > 4)
(hn' : n ∉ SmoothTrueValues) : ¬ SmoothConjectureFor n | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem poincare_conjecture.variants.smooth_other_cases (n : ℕ) (hn : n > 4)\n (hn' : n ∉ SmoothTrueValues) : ¬ SmoothConjectureFor n",
"target": null,
"source_url": "https://theoremdb.org/statements/P8346",
"format": "declaration-or-fragment",
"sha256": "... | [] | TheoremDB P8346 · 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 P8346",
"url": "https://theoremdb.org/statements/P8346",
"note": null,
"sort_order": 0
},
{
"kind": "formalization",
"citation": "Source formal statement (not locally verified)",
"url": "https://theoremdb.org/statements/P8346",
"not... | {
"source_record": "P8346",
"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":"2026-08-17T18:40:09+00:00","display_standard":{"editorial_review":"submitter_supplied","formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":8346,"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": "2026-09-08T03:33:05.964Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"2a4fa395-2aa8-4bd0-b7e3-56dec731a579","format_issues":null,"number":6641} |
1.0.0 | problem:theoremdb-p8348 | problem | theoremdb-p8348 | Covering Number of Graphs by Cycles and Edges | {
"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 $n \geq 1$ vertices. A subgraph $H$ of $G$ is called a cycle or an edge if either $H$ is connected and every vertex of $H$ has degree $2$, or $H$ has exactly one edge. Determine whether the edges of $G$ can always be covered by at most $n-1$ subgraphs, each of which is a cycle or an ... | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P8348",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P8348",
"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_184.variants.covering :
answer(sorry) ↔
∀ {V : Type} [Fintype V] [DecidableEq V] [Nonempty V] (G : SimpleGraph V),
∃ (D : Finset G.Subgraph),
(∀ H ∈ D, IsCycleOrEdge H.coe) ∧
(⋃ H ∈ D, H.edgeSet) = G.edgeSet ∧
(D.card : ℝ) ≤ (Fintype.card V : ℝ) - 1 | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem erdos_184.variants.covering :\n answer(sorry) ↔\n ∀ {V : Type} [Fintype V] [DecidableEq V] [Nonempty V] (G : SimpleGraph V),\n ∃ (D : Finset G.Subgraph),\n (∀ H ∈ D, IsCycleOrEdge H.coe) ∧\n (⋃ H ∈ D, H.edgeSet) = G.edgeSet ∧\n (D.ca... | [] | TheoremDB P8348 · 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 P8348",
"url": "https://theoremdb.org/statements/P8348",
"note": null,
"sort_order": 0
},
{
"kind": "formalization",
"citation": "Source formal statement (not locally verified)",
"url": "https://theoremdb.org/statements/P8348",
"not... | {
"source_record": "P8348",
"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":"2026-08-17T18:40:13+00:00","display_standard":{"editorial_review":"submitter_supplied","formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":8348,"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": "2026-09-08T03:32:52.841Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"72e61c5d-eee5-47fd-ac7f-0883062c16d0","format_issues":null,"number":3947} |
1.0.0 | problem:theoremdb-p8350 | problem | theoremdb-p8350 | Existence of Integer-Distance Point Sets in the Plane with No Three Collinear and No Four Concyclic | {
"main": "math",
"sub": "geometry and topology",
"slug": "geometry-and-topology",
"name": "Geometry and topology",
"original_discipline": "mathematics"
} | [] | [] | For a natural number $n$, let $E(n)$ denote the property that there exists a finite set $S$ of $n$ points in $\mathbb{R}^2$ such that: no three points of $S$ lie on a common line; no four points of $S$ lie on a common circle; and the Euclidean distance between every pair of distinct points in $S$ is an integer. Does $E... | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P8350",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P8350",
"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_213 : answer(sorry) ↔ ∀ n : ℕ, n ≥ 4 → Erdos213For n | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem erdos_213 : answer(sorry) ↔ ∀ n : ℕ, n ≥ 4 → Erdos213For n",
"target": null,
"source_url": "https://theoremdb.org/statements/P8350",
"format": "declaration-or-fragment",
"sha256": "f0756925c65d131037c8a891933b41fae7eb0333a9508fb2c96a3f8e4d309fa2",
... | [] | TheoremDB P8350 · 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 P8350",
"url": "https://theoremdb.org/statements/P8350",
"note": null,
"sort_order": 0
},
{
"kind": "formalization",
"citation": "Source formal statement (not locally verified)",
"url": "https://theoremdb.org/statements/P8350",
"not... | {
"source_record": "P8350",
"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":"2026-08-17T18:40:13+00:00","display_standard":{"editorial_review":"submitter_supplied","formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":8350,"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": "2026-09-08T03:32:58.752Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"192fd711-bb23-47fb-bcf1-3c1b94fc0918","format_issues":null,"number":5248} |
1.0.0 | problem:theoremdb-p8352 | problem | theoremdb-p8352 | Additive Complements to the Primes with Sub-Log-Squared Density | {
"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 $A \subseteq \mathbb{N}$ is called an additive complement to the primes if every sufficiently large natural number can be written as $p + a$ for some prime $p$ and some element $a \in A$. Erdős proved that there exists an additive complement to the primes whose counting function satisfies $|A \cap \{1, \ldots, N\... | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P8352",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P8352",
"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_32 : answer(sorry) ↔ ∃ A : Set ℕ,
IsAdditiveComplementToPrimes A ∧
(fun N => (((Finset.Icc 1 N).filter (· ∈ A)).card : ℝ)) =o[atTop]
fun N => (Real.log N) ^ 2 | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem erdos_32 : answer(sorry) ↔ ∃ A : Set ℕ,\n IsAdditiveComplementToPrimes A ∧\n (fun N => (((Finset.Icc 1 N).filter (· ∈ A)).card : ℝ)) =o[atTop]\n fun N => (Real.log N) ^ 2",
"target": null,
"source_url": "https://theoremdb.org/statements/P8352",
... | [] | TheoremDB P8352 · 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 P8352",
"url": "https://theoremdb.org/statements/P8352",
"note": null,
"sort_order": 0
},
{
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"citation": "Source formal statement (not locally verified)",
"url": "https://theoremdb.org/statements/P8352",
"not... | {
"source_record": "P8352",
"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":"2026-08-17T18:40:15+00:00","display_standard":{"editorial_review":"submitter_supplied","formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":8352,"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": "2026-09-08T03:32:49.468Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"d9e59c9e-776d-4db5-a2ee-f8b82d7531e3","format_issues":null,"number":3214} |
1.0.0 | problem:theoremdb-p8354 | problem | theoremdb-p8354 | Erdős's Problem on Non-Unique Representability as 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 a set $A \subseteq \mathbb{N}$, let $B$ denote the set of positive integers that can be represented in exactly one way as the sum of two (not necessarily distinct) elements of $A$. For each positive integer $N$, define $f_A(N)$ to be the number of integers in $\{1, 2, \ldots, N\}$ that do not belong to $B$; that is... | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P8354",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P8354",
"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_14.parts.ii :
answer(sorry) ↔ ∃ (A : Set ℕ), IsLittleO atTop (nonUniqueSumCount A) squareRoot | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem erdos_14.parts.ii :\n answer(sorry) ↔ ∃ (A : Set ℕ), IsLittleO atTop (nonUniqueSumCount A) squareRoot",
"target": null,
"source_url": "https://theoremdb.org/statements/P8354",
"format": "declaration-or-fragment",
"sha256": "a21145c13e468ca51bc3f960... | [] | TheoremDB P8354 · 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 P8354",
"url": "https://theoremdb.org/statements/P8354",
"note": null,
"sort_order": 0
},
{
"kind": "formalization",
"citation": "Source formal statement (not locally verified)",
"url": "https://theoremdb.org/statements/P8354",
"not... | {
"source_record": "P8354",
"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":"2026-08-17T18:40:16+00:00","display_standard":{"editorial_review":"submitter_supplied","formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":8354,"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": "2026-09-08T03:32:56.519Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"45a6fc73-070c-4d70-9a9d-c1a62e326267","format_issues":null,"number":4758} |
1.0.0 | problem:theoremdb-p8356 | problem | theoremdb-p8356 | Asymptotic constant for the number of distinct exponents in prime factorizations of 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 each positive integer $n$, write the prime factorization of $n!$ as $n! = \prod_i p_i^{k_i}$ with distinct primes $p_i$. Define $h(n)$ to be the number of distinct values among the exponents $k_i$ in this factorization. Erdős and Selfridge proved that $h(n) \asymp \left(\frac{n}{\log n}\right)^{1/2}$ as $n \to \inf... | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P8356",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P8356",
"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_912 : ∃ c > 0,
(fun n => (h n : ℝ)) ~[atTop] (fun n => c * (n / Real.log n) ^ (1 / 2 : ℝ)) | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem erdos_912 : ∃ c > 0,\n (fun n => (h n : ℝ)) ~[atTop] (fun n => c * (n / Real.log n) ^ (1 / 2 : ℝ))",
"target": null,
"source_url": "https://theoremdb.org/statements/P8356",
"format": "declaration-or-fragment",
"sha256": "8eedac2f0ca9ac45465307c2318... | [] | TheoremDB P8356 · 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 P8356",
"url": "https://theoremdb.org/statements/P8356",
"note": null,
"sort_order": 0
},
{
"kind": "formalization",
"citation": "Source formal statement (not locally verified)",
"url": "https://theoremdb.org/statements/P8356",
"not... | {
"source_record": "P8356",
"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":"2026-08-17T18:40:18+00:00","display_standard":{"editorial_review":"submitter_supplied","formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":8356,"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": "2026-09-08T03:32:49.967Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"bc488c8e-e4a8-486f-b2c0-9172c4a76d61","format_issues":null,"number":3323} |
1.0.0 | problem:theoremdb-p8358 | problem | theoremdb-p8358 | The 3-uniform partition relation for the continuum | {
"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 $\mathfrak{c}$ denote the cardinality of the continuum. For ordinals $\alpha$, $\beta$ and a cardinal $c$, the 3-uniform ordinal Ramsey property $\alpha \to (\beta, c)^3_2$ is defined as follows: for every 2-coloring of the 3-element subsets of $\alpha$ (identified with its ordinal type), either there exists a subs... | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P8358",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P8358",
"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_70 :
answer(sorry) ↔
∀ᵉ (β : Ordinal.{0}) (n : ℕ) (_ : β.card ≤ ℵ₀) (_ : 2 ≤ n),
OrdinalCardinalRamsey3 (𝔠).ord β n | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem erdos_70 :\n answer(sorry) ↔\n ∀ᵉ (β : Ordinal.{0}) (n : ℕ) (_ : β.card ≤ ℵ₀) (_ : 2 ≤ n),\n OrdinalCardinalRamsey3 (𝔠).ord β n",
"target": null,
"source_url": "https://theoremdb.org/statements/P8358",
"format": "declaration-or-fragment",
... | [] | TheoremDB P8358 · 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 P8358",
"url": "https://theoremdb.org/statements/P8358",
"note": null,
"sort_order": 0
},
{
"kind": "formalization",
"citation": "Source formal statement (not locally verified)",
"url": "https://theoremdb.org/statements/P8358",
"not... | {
"source_record": "P8358",
"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":"2026-08-17T18:40:18+00:00","display_standard":{"formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":8358,"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": "2026-09-08T03:33:06.351Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"029f8e7e-1d23-4219-b9ec-d4ae1a24148c","format_issues":null,"number":6724} |
1.0.0 | problem:theoremdb-p8360 | problem | theoremdb-p8360 | The Hilbert–Smith Conjecture | {
"main": "math",
"sub": "geometry and topology",
"slug": "geometry-and-topology",
"name": "Geometry and topology",
"original_discipline": "mathematics"
} | [] | [] | Let $G$ be a topological group that is locally compact. Let $n$ be a natural number and let $X$ be a Hausdorff, connected topological space equipped with a charted space structure modeled on the $n$-dimensional Euclidean space $\mathbb{R}^n$. Suppose $G$ acts on $X$ via a multiplicative action that is continuous and fa... | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P8360",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P8360",
"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 hilbert_smith_conjecture
[IsTopologicalGroup G] [LocallyCompactSpace G]
[MulAction G X] [ContinuousSMul G X] [FaithfulSMul G X] :
AdmitsLieGroupStructure G | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem hilbert_smith_conjecture\n [IsTopologicalGroup G] [LocallyCompactSpace G]\n [MulAction G X] [ContinuousSMul G X] [FaithfulSMul G X] :\n AdmitsLieGroupStructure G",
"target": null,
"source_url": "https://theoremdb.org/statements/P8360",
"format": ... | [] | TheoremDB P8360 · 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 P8360",
"url": "https://theoremdb.org/statements/P8360",
"note": null,
"sort_order": 0
},
{
"kind": "formalization",
"citation": "Source formal statement (not locally verified)",
"url": "https://theoremdb.org/statements/P8360",
"not... | {
"source_record": "P8360",
"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":"2026-08-17T18:40:21+00:00","display_standard":{"formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":8360,"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": "2026-09-08T03:33:06.551Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"c250c314-5d93-4d4f-b845-5f4aac90d5a9","format_issues":null,"number":6767} |
1.0.0 | problem:theoremdb-p8362 | problem | theoremdb-p8362 | Irrationality of the Riemann zeta value at 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"
}
] | The Riemann zeta function is defined for a complex number $s$ with real part greater than $1$ by
$$
\zeta(s) = \sum_{n=1}^{\infty} \frac{1}{n^s}.
$$
A real number is said to be irrational if it cannot be expressed as a ratio of two integers. Prove that $\zeta(5)$ is irrational. Equivalently, prove that there exists... | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P8362",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P8362",
"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_five : ∃ x, Irrational x ∧ riemannZeta 5 = x | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem irrational_five : ∃ x, Irrational x ∧ riemannZeta 5 = x",
"target": null,
"source_url": "https://theoremdb.org/statements/P8362",
"format": "declaration-or-fragment",
"sha256": "1a852befe322ed1199f5edac2884e619920e3dc5184a243e507fa73cabf7aa1d",
"l... | [] | TheoremDB P8362 · 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 P8362",
"url": "https://theoremdb.org/statements/P8362",
"note": null,
"sort_order": 0
},
{
"kind": "formalization",
"citation": "Source formal statement (not locally verified)",
"url": "https://theoremdb.org/statements/P8362",
"not... | {
"source_record": "P8362",
"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":"2026-08-17T18:40:22+00:00","display_standard":{"editorial_review":"submitter_supplied","formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":8362,"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": "2026-09-08T03:33:01.815Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"5be0b5c6-9c83-4a55-8bfc-5ca65d8542b6","format_issues":null,"number":5741} |
1.0.0 | problem:theoremdb-p8364 | problem | theoremdb-p8364 | Maximal number of mutually unbiased bases in dimension 10 | {
"main": "math",
"sub": "combinatorics",
"slug": "combinatorics",
"name": "Combinatorics",
"original_discipline": "mathematics"
} | [] | [
{
"code": "05",
"basis": "community-map",
"evidence": "Inferred from community field: combinatorics"
}
] | Let $\mu(d)$ denote the maximum number $k$ for which there exist orthonormal bases $\mathcal{B}_1, \dots, \mathcal{B}_k$ of the complex Hilbert space $\mathbb{C}^d$ such that any two distinct bases are mutually unbiased. Two orthonormal bases $\mathcal{B}_r = \{ e_0^{(r)}, \dots, e_{d-1}^{(r)} \}$ and $\mathcal{B}_s = ... | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P8364",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P8364",
"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 mutuallyUnbiasedBases_dim10 :
IsMaxMUBCount 10 (answer(sorry)) | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem mutuallyUnbiasedBases_dim10 :\n IsMaxMUBCount 10 (answer(sorry))",
"target": null,
"source_url": "https://theoremdb.org/statements/P8364",
"format": "declaration-or-fragment",
"sha256": "6c4f5c35398ab8904b4d23149e39e59b5a4712446ddc2e59a0ebc62cc6d6d... | [] | TheoremDB P8364 · 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 P8364",
"url": "https://theoremdb.org/statements/P8364",
"note": null,
"sort_order": 0
},
{
"kind": "formalization",
"citation": "Source formal statement (not locally verified)",
"url": "https://theoremdb.org/statements/P8364",
"not... | {
"source_record": "P8364",
"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":"2026-08-17T18:40:24+00:00","display_standard":{"formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":8364,"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": "2026-09-08T03:33:03.650Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"316bc911-71d4-4c36-a9d6-77a1ca378d47","format_issues":null,"number":6140} |
1.0.0 | problem:theoremdb-p8366 | problem | theoremdb-p8366 | Finiteness of the set of natural numbers with $v_1(n)=1$ | {
"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 positive integer $n$ and a non-negative integer $k$, let $v(n,k)$ denote the number of prime factors of $n+k$ that do not divide $n+i$ for any $i$ with $0 \leq i < k$. For a non-negative integer $l$, define $v_l(n) = \sup_{k \geq l} v(n,k)$, where the supremum is taken in the extended natural numbers $\mathbb{N} ... | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P8366",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P8366",
"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.variants.v1_eq_1_finite :
answer(sorry) ↔ {n | v_l 1 n = 1}.Finite | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem erdos_889.variants.v1_eq_1_finite :\n answer(sorry) ↔ {n | v_l 1 n = 1}.Finite",
"target": null,
"source_url": "https://theoremdb.org/statements/P8366",
"format": "declaration-or-fragment",
"sha256": "536c5516e081a05b6f2c50ddbcc809941abd4cdcae596e4... | [] | TheoremDB P8366 · 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",
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"url": "https://theoremdb.org/statements/P8366",
"note": null,
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},
{
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"citation": "Source formal statement (not locally verified)",
"url": "https://theoremdb.org/statements/P8366",
"not... | {
"source_record": "P8366",
"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":"2026-08-17T18:40:27+00:00","display_standard":{"editorial_review":"submitter_supplied","formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":8366,"public_status":{"evidence":{"hasFormalization":false,"hasInformalProof":false,"hasRecordedWork":false,"solutionRecord":null},"lean":{... | [
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"created_at": "2026-09-08T03:32:15.935Z",
"updated_at": "2026-09-08T03:32:15.935Z",
"published_at": "2026-09-08T03:32:59.112Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"f8901d77-9578-4384-856f-9f58eccf89f2","format_issues":null,"number":5325} |
1.0.0 | problem:theoremdb-p8368 | problem | theoremdb-p8368 | Near-Transversal Existence in 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 $n$ different symbols, each occurring exactly once in each row and exactly once in each column. A near-transversal of a Latin square is a selection of $n - 1$ cells, no two in the same row, no two in the same column, and no two containing the same symbol.... | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P8368",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P8368",
"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 latinSquareNearTransversal : answer(sorry) ↔
∀ (L : LatinSquare n), ∃ ρ σ, IsNearTransversal L ρ σ | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem latinSquareNearTransversal : answer(sorry) ↔\n ∀ (L : LatinSquare n), ∃ ρ σ, IsNearTransversal L ρ σ",
"target": null,
"source_url": "https://theoremdb.org/statements/P8368",
"format": "declaration-or-fragment",
"sha256": "9b2eec565337f162f54d01db6... | [] | TheoremDB P8368 · 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 P8368",
"url": "https://theoremdb.org/statements/P8368",
"note": null,
"sort_order": 0
},
{
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"citation": "Source formal statement (not locally verified)",
"url": "https://theoremdb.org/statements/P8368",
"not... | {
"source_record": "P8368",
"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":"2026-08-17T18:40:44+00:00","display_standard":{"editorial_review":"submitter_supplied","formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":8368,"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": "2026-09-08T03:33:04.217Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"e730c780-6e0d-478f-84db-a6b42eb9d295","format_issues":null,"number":6264} |
1.0.0 | problem:theoremdb-p8370 | problem | theoremdb-p8370 | The minimal upper density of a square-additive basis for the 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"
}
] | A set $A \subseteq \mathbb{N}$ is called a square-additive basis if every natural number $k$ can be written as $k = a + n^2$ for some $a \in A$ and some $n \geq 0$. For a square-additive basis $A$, define its upper square-root density as
$$
\limsup_{N \to \infty} \frac{|A \cap \{1, 2, \ldots, N\}|}{\sqrt{N}}.
$$
De... | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P8370",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P8370",
"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_33 : ⨅ A : {A : Set ℕ | AdditiveBasisCondition A}, Filter.atTop.limsup (fun N =>
(A.1 ∩ Icc 1 N).ncard / (√N : EReal)) = answer(sorry) | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem erdos_33 : ⨅ A : {A : Set ℕ | AdditiveBasisCondition A}, Filter.atTop.limsup (fun N =>\n (A.1 ∩ Icc 1 N).ncard / (√N : EReal)) = answer(sorry)",
"target": null,
"source_url": "https://theoremdb.org/statements/P8370",
"format": "declaration-or-fragment"... | [] | TheoremDB P8370 · 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 P8370",
"url": "https://theoremdb.org/statements/P8370",
"note": null,
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{
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"citation": "Source formal statement (not locally verified)",
"url": "https://theoremdb.org/statements/P8370",
"not... | {
"source_record": "P8370",
"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":"2026-08-17T18:40:50+00:00","display_standard":{"formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":8370,"public_status":{"evidence":{"hasFormalization":false,"hasInformalProof":false,"hasRecordedWork":false,"solutionRecord":null},"lean":{"kernelAccepted":false,"label":"","reaso... | [
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"created_at": "2026-09-08T03:32:15.935Z",
"updated_at": "2026-09-08T03:32:15.935Z",
"published_at": "2026-09-08T03:33:06.649Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"40420381-a720-4377-8deb-81f5b8a0986c","format_issues":null,"number":6788} |
1.0.0 | problem:theoremdb-p8372 | problem | theoremdb-p8372 | Existence of Almost Perfect Numbers Not 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"
}
] | A natural number $n$ is called almost perfect if the sum of its positive divisors equals $2n - 1$, that is, if $\sigma(n) = 2n - 1$ where $\sigma(n)$ denotes the sum of all positive divisors of $n$. It is known that every power of two $n = 2^k$ with $k \geq 0$ is almost perfect. Determine whether there exists an almost... | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P8372",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P8372",
"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 exists_almost_perfect_not_power_of_two :
answer(sorry) ↔ ∃ n : ℕ, AlmostPerfect n ∧ ¬ ∃ k : ℕ, n = 2^k | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem exists_almost_perfect_not_power_of_two :\n answer(sorry) ↔ ∃ n : ℕ, AlmostPerfect n ∧ ¬ ∃ k : ℕ, n = 2^k",
"target": null,
"source_url": "https://theoremdb.org/statements/P8372",
"format": "declaration-or-fragment",
"sha256": "40135ff1029f91ac8d563... | [] | TheoremDB P8372 · 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 P8372",
"url": "https://theoremdb.org/statements/P8372",
"note": null,
"sort_order": 0
},
{
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"citation": "Source formal statement (not locally verified)",
"url": "https://theoremdb.org/statements/P8372",
"not... | {
"source_record": "P8372",
"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":"2026-08-17T18:40:59+00:00","display_standard":{"editorial_review":"submitter_supplied","formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":8372,"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": "2026-09-08T03:32:58.594Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"714664fb-4c97-45f4-9c15-a67231d41daf","format_issues":null,"number":5215} |
1.0.0 | problem:theoremdb-p8376 | problem | theoremdb-p8376 | Existence of a Large Weakly First-Countable Compact Hausdorff Space | {
"main": "math",
"sub": "geometry and topology",
"slug": "geometry-and-topology",
"name": "Geometry and topology",
"original_discipline": "mathematics"
} | [] | [] | A topological space $X$ is called weakly first countable if there exists a function $V : X \times \mathbb{N} \to \mathcal{P}(X)$ such that:
\begin{itemize}
\item for every $x \in X$ and $n \in \mathbb{N}$, we have $x \in V(x,n)$;
\item for every $x \in X$ and $n \in \mathbb{N}$, we have $V(x,n+1) \subseteq V(x,n)$;
\it... | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P8376",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P8376",
"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 existsWeaklyFirstCountableCompactBig : answer(sorry) ↔
∃ (X : Type) (_ : TopologicalSpace X),
WeaklyFirstCountableTopology X ∧ CompactSpace X ∧ T2Space X ∧
𝔠 < #X | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem existsWeaklyFirstCountableCompactBig : answer(sorry) ↔\n ∃ (X : Type) (_ : TopologicalSpace X),\n WeaklyFirstCountableTopology X ∧ CompactSpace X ∧ T2Space X ∧\n 𝔠 < #X",
"target": null,
"source_url": "https://theoremdb.org/statements/P8376",
... | [] | TheoremDB P8376 · 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 P8376",
"url": "https://theoremdb.org/statements/P8376",
"note": null,
"sort_order": 0
},
{
"kind": "formalization",
"citation": "Source formal statement (not locally verified)",
"url": "https://theoremdb.org/statements/P8376",
"not... | {
"source_record": "P8376",
"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":"2026-08-17T18:48:41+00:00","display_standard":{"editorial_review":"submitter_supplied","formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":8376,"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": "2026-09-08T03:32:58.454Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"a8541a0c-fd2d-4b20-91d5-9955964322a2","format_issues":null,"number":5186} |
1.0.0 | problem:theoremdb-p8378 | problem | theoremdb-p8378 | Asymptotic Growth of Longest Prime Arithmetic Progressions | {
"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 $n$, let $L(n)$ denote the maximum length $k$ of an arithmetic progression of prime numbers contained in the interval $\{1, 2, \ldots, n\}$. That is, $L(n)$ is the supremum of all $k$ for which there exists a subset $s \subseteq \{1, 2, \ldots, n\}$ such that $s$ forms an arithmetic progressio... | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P8378",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P8378",
"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_200 : answer(sorry) ↔
(fun n => (longestPrimeArithmeticProgressions n : ℝ)) =o[atTop] (fun n => log n) | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem erdos_200 : answer(sorry) ↔\n (fun n => (longestPrimeArithmeticProgressions n : ℝ)) =o[atTop] (fun n => log n)",
"target": null,
"source_url": "https://theoremdb.org/statements/P8378",
"format": "declaration-or-fragment",
"sha256": "ef2668265cdaa02... | [] | TheoremDB P8378 · 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 P8378",
"url": "https://theoremdb.org/statements/P8378",
"note": null,
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},
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"citation": "Source formal statement (not locally verified)",
"url": "https://theoremdb.org/statements/P8378",
"not... | {
"source_record": "P8378",
"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":"2026-08-17T18:48:45+00:00","display_standard":{"editorial_review":"submitter_supplied","formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":8378,"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": "2026-09-08T03:32:50.094Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"dfc389a0-f299-40aa-9df6-934cd7a898b4","format_issues":null,"number":3349} |
1.0.0 | problem:theoremdb-p8380 | problem | theoremdb-p8380 | The inscribed square problem | {
"main": "math",
"sub": "geometry and topology",
"slug": "geometry-and-topology",
"name": "Geometry and topology",
"original_discipline": "mathematics"
} | [] | [] | A Jordan curve is a simple closed curve in the plane, that is, the image of an injective continuous map from the unit circle into $\mathbb{R}^2$. Four points $a, b, c, d$ in the plane form a rectangle with $a$ opposite to $c$ if the line segments from $a$ to $c$ and from $b$ to $d$ have the same length and the same mid... | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P8380",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P8380",
"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 inscribed_square_problem :
answer(sorry) ↔ ∀ (γ : Circle → ℝ²) (hγ : IsEmbedding γ),
∃ t₁ t₂ t₃ t₄, IsRectangle (γ t₁) (γ t₂) (γ t₃) (γ t₄) 1 | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem inscribed_square_problem :\n answer(sorry) ↔ ∀ (γ : Circle → ℝ²) (hγ : IsEmbedding γ),\n ∃ t₁ t₂ t₃ t₄, IsRectangle (γ t₁) (γ t₂) (γ t₃) (γ t₄) 1",
"target": null,
"source_url": "https://theoremdb.org/statements/P8380",
"format": "declaration-or-f... | [] | TheoremDB P8380 · 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 P8380",
"url": "https://theoremdb.org/statements/P8380",
"note": null,
"sort_order": 0
},
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"citation": "Source formal statement (not locally verified)",
"url": "https://theoremdb.org/statements/P8380",
"not... | {
"source_record": "P8380",
"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":"2026-08-17T18:48:49+00:00","display_standard":{"formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":8380,"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": "2026-09-08T03:33:06.556Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"45826be5-f6fc-466d-a5ef-c2d62d85e0f2","format_issues":null,"number":6768} |
1.0.0 | problem:theoremdb-p8382 | problem | theoremdb-p8382 | Existence and smoothness of periodic solutions to the three-dimensional Navier–Stokes equations | {
"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 $\nu > 0$ be a positive real number. Let $u_0 : \mathbb{R}^3 \to \mathbb{R}^3$ be a smooth vector field that is divergence-free, meaning $\nabla \cdot u_0 = 0$ everywhere, and is $1$-periodic in each coordinate, meaning $u_0(x + e_i) = u_0(x)$ for all $x \in \mathbb{R}^3$ and each standard basis vector $e_i$. Prove... | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P8382",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P8382",
"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 navier_stokes_existence_and_smoothness_periodic (nu : ℝ) (hnu : nu > 0)
(u₀ : ℝ³ → ℝ³) (hu₀ : InitialVelocityConditionPeriodic u₀) :
∃ v p, NavierStokesExistenceAndSmoothnessPeriodic nu u₀ (f := 0) v p | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem navier_stokes_existence_and_smoothness_periodic (nu : ℝ) (hnu : nu > 0)\n (u₀ : ℝ³ → ℝ³) (hu₀ : InitialVelocityConditionPeriodic u₀) :\n ∃ v p, NavierStokesExistenceAndSmoothnessPeriodic nu u₀ (f := 0) v p",
"target": null,
"source_url": "https://theore... | [] | TheoremDB P8382 · 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 | [
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"url": "https://theoremdb.org/statements/P8382",
"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":"2026-08-17T18:48:51+00:00","display_standard":{"editorial_review":"submitter_supplied","formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":8382,"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": "2026-09-08T03:32:58.374Z",
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} | 2026-10-10T16:07:49.246Z | {"database_id":"08be4e1e-0c42-4e8f-a9c5-c2fc4ea22aca","format_issues":null,"number":5170} |
1.0.0 | problem:theoremdb-p8384 | problem | theoremdb-p8384 | Iteration count to reach a prime via Euler's totient | {
"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 the function $f(n) = \varphi(n) + 1$, where $\varphi$ denotes Euler's totient function, which counts the number of integers from $1$ to $n$ that are coprime to $n$. Let $f^{[i]}(n)$ denote the $i$-fold iteration of $f$ applied to $n$, with $f^{[0]}(n) = n$. It is known that for eve... | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P8384",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P8384",
"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.i (n : ℕ) (hn : 0 < n) :
IsLeast { i | (φ · + 1)^[i] n |>.Prime } answer(sorry) | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem erdos_409.parts.i (n : ℕ) (hn : 0 < n) :\n IsLeast { i | (φ · + 1)^[i] n |>.Prime } answer(sorry)",
"target": null,
"source_url": "https://theoremdb.org/statements/P8384",
"format": "declaration-or-fragment",
"sha256": "465afa7e59dead0e0ffd6e775380... | [] | TheoremDB P8384 · 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",
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"url": "https://theoremdb.org/statements/P8384",
"not... | {
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"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":"2026-08-17T18:48:56+00:00","display_standard":{"editorial_review":"submitter_supplied","formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":8384,"public_status":{"evidence":{"hasFormalization":false,"hasInformalProof":false,"hasRecordedWork":false,"solutionRecord":null},"lean":{... | [
"formalization-contains-placeholder",
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"updated_at": "2026-09-08T03:32:15.935Z",
"published_at": "2026-09-08T03:33:02.148Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"556b4b5d-5430-4045-8606-54e9fa0c3e65","format_issues":null,"number":5814} |
1.0.0 | problem:theoremdb-p8386 | problem | theoremdb-p8386 | Kaplansky's idempotent 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 has infinite order. Let $K[G]$ denote the monoid algebra of $G$ over $K$, which consists of formal finite $K$-linear combinations of elements of $G$ with multiplication induced by the group operation. An ... | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P8386",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P8386",
"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 idempotent_conjecture (a : MonoidAlgebra K G) (h : IsIdempotentElem a) :
a = 0 ∨ a = 1 | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem idempotent_conjecture (a : MonoidAlgebra K G) (h : IsIdempotentElem a) :\n a = 0 ∨ a = 1",
"target": null,
"source_url": "https://theoremdb.org/statements/P8386",
"format": "declaration-or-fragment",
"sha256": "76c77d4f966976f47a2159be2bc43efb689a0... | [] | TheoremDB P8386 · 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 P8386",
"url": "https://theoremdb.org/statements/P8386",
"note": null,
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},
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"citation": "Source formal statement (not locally verified)",
"url": "https://theoremdb.org/statements/P8386",
"not... | {
"source_record": "P8386",
"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":"2026-08-17T18:48:59+00:00","display_standard":{"editorial_review":"submitter_supplied","formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":8386,"public_status":{"evidence":{"hasFormalization":false,"hasInformalProof":false,"hasRecordedWork":false,"solutionRecord":null},"lean":{... | [
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"created_at": "2026-09-08T03:32:15.935Z",
"updated_at": "2026-09-08T03:32:15.935Z",
"published_at": "2026-09-08T03:33:02.170Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"d731803d-75fa-41d9-ad8b-cdbda65322d2","format_issues":null,"number":5819} |
1.0.0 | problem:theoremdb-p8388 | problem | theoremdb-p8388 | Mahler's lower bound for the distance from $e^n$ to the nearest integer | {
"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 real number $x$, let $\|x\|$ denote the distance from $x$ to the nearest integer, defined as $\|x\| = \min_{m \in \mathbb{Z}} |x - m|$. Determine whether there exists a positive real number $c$ such that $\|e^n\| > e^{-cn}$ for every integer $n \geq 1$. | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P8388",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P8388",
"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 problem_10_3 :
∃ c : ℝ, 0 < c ∧ ∀ n : ℕ, 1 ≤ n → Real.exp (-c * n) < distToNearestInt (Real.exp n) | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem problem_10_3 :\n ∃ c : ℝ, 0 < c ∧ ∀ n : ℕ, 1 ≤ n → Real.exp (-c * n) < distToNearestInt (Real.exp n)",
"target": null,
"source_url": "https://theoremdb.org/statements/P8388",
"format": "declaration-or-fragment",
"sha256": "85a335df22a0d1013fc08ba27... | [] | TheoremDB P8388 · 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 P8388",
"url": "https://theoremdb.org/statements/P8388",
"note": null,
"sort_order": 0
},
{
"kind": "formalization",
"citation": "Source formal statement (not locally verified)",
"url": "https://theoremdb.org/statements/P8388",
"not... | {
"source_record": "P8388",
"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":"2026-08-17T18:49:03+00:00","display_standard":{"formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":8388,"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": "2026-09-08T03:33:03.604Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"932ca716-ebee-4b5e-b4f9-bc734f81f87b","format_issues":null,"number":6130} |
1.0.0 | problem:theoremdb-p8390 | problem | theoremdb-p8390 | Asymptotic Growth of Minimal Egyptian Fraction Lengths | {
"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 $N$, define $k(N)$ to be the smallest positive integer $k$ for which there exist integers $n_1, n_2, \ldots, n_k$ satisfying $N \leq n_1 < n_2 < \cdots < n_k$ and $\sum_{i=1}^{k} \frac{1}{n_i} = 1$. Determine whether $\lim_{N \to \infty} \bigl(k(N) - (e-1)N\bigr) = \infty$, where $e$ denotes t... | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P8390",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P8390",
"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_295 :
answer(sorry) ↔ Filter.atTop.Tendsto (fun N => k N - (rexp 1 - 1)*N) Filter.atTop | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem erdos_295 :\n answer(sorry) ↔ Filter.atTop.Tendsto (fun N => k N - (rexp 1 - 1)*N) Filter.atTop",
"target": null,
"source_url": "https://theoremdb.org/statements/P8390",
"format": "declaration-or-fragment",
"sha256": "f619ffa5512d93abb9d9984b328c51... | [] | TheoremDB P8390 · 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 | [
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"kind": "website",
"citation": "TheoremDB P8390",
"url": "https://theoremdb.org/statements/P8390",
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"kind": "formalization",
"citation": "Source formal statement (not locally verified)",
"url": "https://theoremdb.org/statements/P8390",
"not... | {
"source_record": "P8390",
"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":"2026-08-17T18:49:07+00:00","display_standard":{"editorial_review":"submitter_supplied","formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":8390,"public_status":{"evidence":{"hasFormalization":false,"hasInformalProof":false,"hasRecordedWork":false,"solutionRecord":null},"lean":{... | [
"formalization-contains-placeholder",
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"source-full-text-not-present"
] | [] | {
"created_at": "2026-09-08T03:32:15.935Z",
"updated_at": "2026-09-08T03:32:15.935Z",
"published_at": "2026-09-08T03:32:50.135Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"30ba8ed6-c1ab-48cc-a624-5d2344bb1125","format_issues":null,"number":3358} |
1.0.0 | problem:theoremdb-p8392 | problem | theoremdb-p8392 | Growth of the van der Waerden root 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 $k$, 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 the sequence $(W(k))^{1/k}$ tends to infinity as $k \to \infty$.... | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P8392",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P8392",
"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 : answer(sorry) ↔ atTop.Tendsto (fun k => (W k : ℝ)^(1/(k : ℝ))) atTop | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem erdos_138 : answer(sorry) ↔ atTop.Tendsto (fun k => (W k : ℝ)^(1/(k : ℝ))) atTop",
"target": null,
"source_url": "https://theoremdb.org/statements/P8392",
"format": "declaration-or-fragment",
"sha256": "26b7e0c5e4c94e64d6e74c0521c2e949de03857877d79e99... | [] | TheoremDB P8392 · 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 | [
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"citation": "TheoremDB P8392",
"url": "https://theoremdb.org/statements/P8392",
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"citation": "Source formal statement (not locally verified)",
"url": "https://theoremdb.org/statements/P8392",
"not... | {
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"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":"2026-08-17T18:49:10+00:00","display_standard":{"editorial_review":"submitter_supplied","formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":8392,"public_status":{"evidence":{"hasFormalization":false,"hasInformalProof":false,"hasRecordedWork":false,"solutionRecord":null},"lean":{... | [
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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": "2026-09-08T03:32:59.497Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"cd865c6f-a6ec-4295-87b0-db04146877c2","format_issues":null,"number":5408} |
1.0.0 | problem:theoremdb-p8394 | problem | theoremdb-p8394 | Mahler-Popken complexity of 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 a positive integer $n$, define the (Mahler-Popken) complexity of $n$, denoted $\|n\|$, to be the minimum number of $1$s needed to express $n$ using only addition, multiplication, and parentheses. Equivalently, $\|n\|$ is the smallest $k$ such that there exists a sequence of $k$ applications of addition and multipli... | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P8394",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P8394",
"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 complexity_two_pow : answer(sorry) ↔ ∀ n : ℕ, 0 < n → complexity (2 ^ n) = 2 * n | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem complexity_two_pow : answer(sorry) ↔ ∀ n : ℕ, 0 < n → complexity (2 ^ n) = 2 * n",
"target": null,
"source_url": "https://theoremdb.org/statements/P8394",
"format": "declaration-or-fragment",
"sha256": "e9e5383214ccd279388b181e48e59f4841d42b1ed299cec1... | [] | TheoremDB P8394 · 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 | [
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"kind": "website",
"citation": "TheoremDB P8394",
"url": "https://theoremdb.org/statements/P8394",
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"citation": "Source formal statement (not locally verified)",
"url": "https://theoremdb.org/statements/P8394",
"not... | {
"source_record": "P8394",
"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":"2026-08-17T18:49:14+00:00","display_standard":{"formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":8394,"public_status":{"evidence":{"hasFormalization":false,"hasInformalProof":false,"hasRecordedWork":false,"solutionRecord":null},"lean":{"kernelAccepted":false,"label":"","reaso... | [
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] | [] | {
"created_at": "2026-09-08T03:32:15.935Z",
"updated_at": "2026-09-08T03:32:15.935Z",
"published_at": "2026-09-08T03:33:03.609Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"3de76ad5-7aff-42a5-bf4e-0644e121848e","format_issues":null,"number":6131} |
1.0.0 | problem:theoremdb-p8396 | problem | theoremdb-p8396 | Covering Sublevel Sets of Monic Polynomials by Discs of Bounded Total Radius | {
"main": "math",
"sub": "analysis",
"slug": "analysis",
"name": "Analysis",
"original_discipline": "mathematics"
} | [] | [] | Let $f \in \mathbb{C}[z]$ be a monic polynomial of positive degree. Consider the sublevel set
$$
S_f = \{z \in \mathbb{C} : |f(z)| \leq 1\}.
$$
An $r$-bounded disc cover of a subset $S \subseteq \mathbb{C}$ is an indexed family of closed discs with centers $C_i \in \mathbb{C}$ and radii $R_i > 0$ such that $S \subset... | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P8396",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P8396",
"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_509 : answer(sorry) ↔ ∀ (f : ℂ[X]), f.Monic → f.natDegree ≠ 0 →
∃ (ι : Type), Nonempty (BoundedDiscCover {z | ‖f.eval z‖ ≤ 1} 2 ι) | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem erdos_509 : answer(sorry) ↔ ∀ (f : ℂ[X]), f.Monic → f.natDegree ≠ 0 →\n ∃ (ι : Type), Nonempty (BoundedDiscCover {z | ‖f.eval z‖ ≤ 1} 2 ι)",
"target": null,
"source_url": "https://theoremdb.org/statements/P8396",
"format": "declaration-or-fragment",
... | [] | TheoremDB P8396 · 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 P8396",
"url": "https://theoremdb.org/statements/P8396",
"note": null,
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},
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"url": "https://theoremdb.org/statements/P8396",
"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":"2026-08-17T18:49:17+00:00","display_standard":{"editorial_review":"submitter_supplied","formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":8396,"public_status":{"evidence":{"hasFormalization":false,"hasInformalProof":false,"hasRecordedWork":false,"solutionRecord":null},"lean":{... | [
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"created_at": "2026-09-08T03:32:15.935Z",
"updated_at": "2026-09-08T03:32:15.935Z",
"published_at": "2026-09-08T03:32:52.850Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"dab0be32-698c-4bf3-8ae9-cd02b91a6b51","format_issues":null,"number":3949} |
1.0.0 | problem:theoremdb-p8398 | problem | theoremdb-p8398 | Non-existence of monochromatic quantum graphs with six vertices and three colours over the reals | {
"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 even integer and let $D \geq 1$ be an integer. A weight function on the complete graph $K_N$ with $D$ colours and coefficients in a ring $\alpha$ is a function $W$ that assigns to every unordered pair of distinct vertices $\{i,j\}$ with $i,j \in \{0,1,\dots,N-1\}$ and every colour $c \in \{0,1,\dot... | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P8398",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P8398",
"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 eqSystem6_no_solution_d3_real :
answer(sorry) ↔
¬ ∃ W : WeightsN 6 3 ℝ, EqSystemN 6 3 W | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem eqSystem6_no_solution_d3_real :\n answer(sorry) ↔\n ¬ ∃ W : WeightsN 6 3 ℝ, EqSystemN 6 3 W",
"target": null,
"source_url": "https://theoremdb.org/statements/P8398",
"format": "declaration-or-fragment",
"sha256": "b2908329744f2f4272966d2777fac... | [] | TheoremDB P8398 · 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 | [
{
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"citation": "Source formal statement (not locally verified)",
"url": "https://theoremdb.org/statements/P8398",
"not... | {
"source_record": "P8398",
"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":"2026-08-17T18:49:21+00:00","display_standard":{"editorial_review":"submitter_supplied","formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":8398,"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": "2026-09-08T03:33:04.347Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"c10b1403-0596-45ef-8e89-35c81c566ba7","format_issues":null,"number":6291} |
1.0.0 | problem:theoremdb-p8400 | problem | theoremdb-p8400 | Irrationality of π raised to the power π | {
"main": "math",
"sub": "analysis",
"slug": "analysis",
"name": "Analysis",
"original_discipline": "mathematics"
} | [] | [] | Let $\pi$ denote the ratio of the circumference of a circle to its diameter. Determine whether $\pi^\pi$ is irrational. | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P8400",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P8400",
"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_pi_to_pi :
answer(sorry) ↔ Irrational (π ^ π) | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem irrational_pi_to_pi :\n answer(sorry) ↔ Irrational (π ^ π)",
"target": null,
"source_url": "https://theoremdb.org/statements/P8400",
"format": "declaration-or-fragment",
"sha256": "0c88a97cf98fe5e1ede760cf3fbf4505b5edda9c2567739281e67d8142904b67",
... | [] | TheoremDB P8400 · 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 | [
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"citation": "TheoremDB P8400",
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"citation": "Source formal statement (not locally verified)",
"url": "https://theoremdb.org/statements/P8400",
"not... | {
"source_record": "P8400",
"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":"2026-08-17T18:49:25+00:00","display_standard":{"editorial_review":"submitter_supplied","formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":8400,"public_status":{"evidence":{"hasFormalization":false,"hasInformalProof":false,"hasRecordedWork":false,"solutionRecord":null},"lean":{... | [
"formalization-contains-placeholder",
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"source-full-text-not-present"
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"created_at": "2026-09-08T03:32:15.935Z",
"updated_at": "2026-09-08T03:32:15.935Z",
"published_at": "2026-09-08T03:33:01.845Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"8620f6a4-55fc-483f-9755-c29db4a777b8","format_issues":null,"number":5747} |
1.0.0 | problem:theoremdb-p8402 | problem | theoremdb-p8402 | Infinitude of Fermat 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 $n$, the $n$th Fermat number is defined as $F_n = 2^{2^n} + 1$. A Fermat prime is a Fermat number that is prime. Determine whether the set $\{n \in \mathbb{N} : F_n \text{ is prime}\}$ is infinite. | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P8402",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P8402",
"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_fermat_primes : answer(sorry) ↔ Infinite {n : ℕ | Prime n.fermatNumber} | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem infinite_fermat_primes : answer(sorry) ↔ Infinite {n : ℕ | Prime n.fermatNumber}",
"target": null,
"source_url": "https://theoremdb.org/statements/P8402",
"format": "declaration-or-fragment",
"sha256": "2ea42bbe3a7544aaeeece8729a50ac7f316c30c5e6c657f3... | [] | TheoremDB P8402 · 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 | [
{
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"citation": "TheoremDB P8402",
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"citation": "Source formal statement (not locally verified)",
"url": "https://theoremdb.org/statements/P8402",
"not... | {
"source_record": "P8402",
"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":"2026-08-17T18:49:29+00:00","display_standard":{"editorial_review":"submitter_supplied","formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":8402,"public_status":{"evidence":{"hasFormalization":false,"hasInformalProof":false,"hasRecordedWork":false,"solutionRecord":null},"lean":{... | [
"formalization-contains-placeholder",
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"created_at": "2026-09-08T03:32:15.935Z",
"updated_at": "2026-09-08T03:32:15.935Z",
"published_at": "2026-09-08T03:33:01.150Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"3b8f61a6-3f95-481e-a2a4-170cf6563d0d","format_issues":null,"number":5668} |
1.0.0 | problem:theoremdb-p8404 | problem | theoremdb-p8404 | Non-existence of monochromatic quantum graphs with integer weights | {
"main": "math",
"sub": "combinatorics",
"slug": "combinatorics",
"name": "Combinatorics",
"original_discipline": "mathematics"
} | [] | [
{
"code": "05",
"basis": "community-map",
"evidence": "Inferred from community field: combinatorics"
}
] | Let $N \geq 6$ be an even integer and let $D \geq 3$ be an integer. A quantum graph with $N$ vertices and $D$ colours consists of a complete graph $K_N$ with each edge assigned a colour from $\{1, \ldots, D\}$ and a weight from a coefficient domain. For each vertex colouring $\iota: \{1, \ldots, N\} \to \{1, \ldots, D\... | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P8404",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P8404",
"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 eqSystem_no_solution_ge6_ge3_int :
answer(sorry) ↔
∀ N D : Nat, N ≥ 6 → Even N → D ≥ 3 →
¬ ∃ W : WeightsN N D ℤ, EqSystemN N D W | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem eqSystem_no_solution_ge6_ge3_int :\n answer(sorry) ↔\n ∀ N D : Nat, N ≥ 6 → Even N → D ≥ 3 →\n ¬ ∃ W : WeightsN N D ℤ, EqSystemN N D W",
"target": null,
"source_url": "https://theoremdb.org/statements/P8404",
"format": "declaration-or-fragm... | [] | TheoremDB P8404 · 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 P8404",
"url": "https://theoremdb.org/statements/P8404",
"note": null,
"sort_order": 0
},
{
"kind": "formalization",
"citation": "Source formal statement (not locally verified)",
"url": "https://theoremdb.org/statements/P8404",
"not... | {
"source_record": "P8404",
"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":"2026-08-17T18:49:33+00:00","display_standard":{"editorial_review":"submitter_supplied","formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":8404,"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": "2026-09-08T03:33:04.319Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"fda8b801-b385-4a61-9765-7a2367d2e6b2","format_issues":null,"number":6286} |
1.0.0 | problem:theoremdb-p8406 | problem | theoremdb-p8406 | Primitive Prime Roots Modulo a Prime | {
"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 odd prime number $p$, does there exist a prime number $q$ with $q < p$ such that the multiplicative order of $q$ modulo $p$ equals $p-1$? In other words, is there always a smaller prime that serves as a primitive root modulo $p$? | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P8406",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P8406",
"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_985 : answer(sorry) ↔ ∀ᵉ (p : ℕ) (hp_prime : p.Prime) (hp_nontrivial : p ≠ 2),
∃ q, q.Prime ∧ q < p ∧ orderOf (q : ZMod p) = p - 1 | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem erdos_985 : answer(sorry) ↔ ∀ᵉ (p : ℕ) (hp_prime : p.Prime) (hp_nontrivial : p ≠ 2),\n ∃ q, q.Prime ∧ q < p ∧ orderOf (q : ZMod p) = p - 1",
"target": null,
"source_url": "https://theoremdb.org/statements/P8406",
"format": "declaration-or-fragment",
... | [] | TheoremDB P8406 · 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 P8406",
"url": "https://theoremdb.org/statements/P8406",
"note": null,
"sort_order": 0
},
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"citation": "Source formal statement (not locally verified)",
"url": "https://theoremdb.org/statements/P8406",
"not... | {
"source_record": "P8406",
"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":"2026-08-17T18:49:40+00:00","display_standard":{"formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":8406,"public_status":{"evidence":{"hasFormalization":false,"hasInformalProof":false,"hasRecordedWork":false,"solutionRecord":null},"lean":{"kernelAccepted":false,"label":"","reaso... | [
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"created_at": "2026-09-08T03:32:15.935Z",
"updated_at": "2026-09-08T03:32:15.935Z",
"published_at": "2026-09-08T03:33:05.264Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"777fa491-1755-4555-9016-9d86aba7b220","format_issues":null,"number":6490} |
1.0.0 | problem:theoremdb-p8408 | problem | theoremdb-p8408 | Firoozbakht's conjecture on the monotonicity of prime root means | {
"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 $p_n$ denote the $n$-th prime number, and define the sequence $a_n = \sqrt[n]{p_n} = p_n^{1/n}$. Does the inequality $a_{n+1} < a_n$ hold for every positive integer $n$? Equivalently, does $\sqrt[n+1]{p_{n+1}} < \sqrt[n]{p_n}$ hold for all $n \geq 1$? | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P8408",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P8408",
"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 firoozbakht_conjecture (n : ℕ) :
firoozbakhtSeq (n+1) < firoozbakhtSeq n | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem firoozbakht_conjecture (n : ℕ) :\n firoozbakhtSeq (n+1) < firoozbakhtSeq n",
"target": null,
"source_url": "https://theoremdb.org/statements/P8408",
"format": "declaration-or-fragment",
"sha256": "7c5a19d2cb404f0eae6e4cbcfce445dbc6d1c8b4dfd9262ee03... | [] | TheoremDB P8408 · 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 P8408",
"url": "https://theoremdb.org/statements/P8408",
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"citation": "Source formal statement (not locally verified)",
"url": "https://theoremdb.org/statements/P8408",
"not... | {
"source_record": "P8408",
"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":"2026-08-17T18:49:50+00:00","display_standard":{"editorial_review":"submitter_supplied","formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":8408,"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": "2026-09-08T03:32:59.139Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"9480fcc7-d046-43ae-b5a8-cf1ef2626f4e","format_issues":null,"number":5331} |
1.0.0 | problem:theoremdb-p8410 | problem | theoremdb-p8410 | Infinitude of Regular 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 $p$ is called regular if $p$ is coprime to the cardinality of the class group of the $p$-th cyclotomic field $\mathbb{Q}(\zeta_p)$, where $\zeta_p$ denotes a primitive $p$-th root of unity. Equivalently, a prime $p$ is regular if and only if $p$ does not divide the numerator of any of the Bernoulli numbe... | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P8410",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P8410",
"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 regularprime_conjecture : RegularPrimeConjecture | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem regularprime_conjecture : RegularPrimeConjecture",
"target": null,
"source_url": "https://theoremdb.org/statements/P8410",
"format": "declaration-or-fragment",
"sha256": "5c7c433576794471729becdf77d5cccdfa5112da699fa22963866134b6d082b3",
"license"... | [] | TheoremDB P8410 · 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 P8410",
"url": "https://theoremdb.org/statements/P8410",
"note": null,
"sort_order": 0
},
{
"kind": "formalization",
"citation": "Source formal statement (not locally verified)",
"url": "https://theoremdb.org/statements/P8410",
"not... | {
"source_record": "P8410",
"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":"2026-08-17T18:50:00+00:00","display_standard":{"editorial_review":"submitter_supplied","formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":8410,"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": "2026-09-08T03:33:01.253Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"f46d761b-9c6b-4cb9-b641-2bc9dd3ad532","format_issues":null,"number":5690} |
1.0.0 | problem:theoremdb-p8412 | problem | theoremdb-p8412 | Zhi-Wei Sun's Conjecture on Sums of Four Squares with Square Conditions | {
"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 nonnegative integer. Does there always exist nonnegative integers $x$, $y$, $z$, and $w$ with $z \leq w$ such that $n = x^2 + y^2 + z^2 + w^2$, both $x$ and $x + 24y$ are perfect squares? | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P8412",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P8412",
"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/P8412",
"format": "declaration-or-fragment",
"sha256": "79122c62c8b920d59aac24d0de97415fd020e2a20cec64f4a2ce6f643313560f",
"license": "Selected packets labe... | [] | TheoremDB P8412 · 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 P8412",
"url": "https://theoremdb.org/statements/P8412",
"note": null,
"sort_order": 0
},
{
"kind": "formalization",
"citation": "Source formal statement (not locally verified)",
"url": "https://theoremdb.org/statements/P8412",
"not... | {
"source_record": "P8412",
"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":"2026-08-17T18:50:14+00:00","display_standard":{"formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":8412,"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": "2026-09-08T03:33:07.653Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"164da357-79cb-463d-a053-871a90128ee6","format_issues":null,"number":7006} |
1.0.0 | problem:theoremdb-p8414 | problem | theoremdb-p8414 | Asymptotic lower bound for sums of three $k$th 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 a positive integer $k$, an integer $n$ is called a sum of three $k$th powers if there exist natural numbers $a$, $b$, and $c$ such that $a^k + b^k + c^k = n$. For a natural number $x$, let $f_{k,3}(x)$ denote the number of integers $n$ with $1 \leq n \leq x$ that are sums of three $k$th powers. Determine whether it... | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P8414",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P8414",
"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_325 :
answer(sorry) ↔ ∀ k : ℕ, 3 ≤ k → (fun x : ℕ => (x : ℝ) ^ (3 / k : ℝ)) =O[atTop]
(fun x : ℕ => (cardIsSumThreePowerBelow k x : ℝ)) | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem erdos_325 :\n answer(sorry) ↔ ∀ k : ℕ, 3 ≤ k → (fun x : ℕ => (x : ℝ) ^ (3 / k : ℝ)) =O[atTop]\n (fun x : ℕ => (cardIsSumThreePowerBelow k x : ℝ))",
"target": null,
"source_url": "https://theoremdb.org/statements/P8414",
"format": "declaration-or-... | [] | TheoremDB P8414 · 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 P8414",
"url": "https://theoremdb.org/statements/P8414",
"note": null,
"sort_order": 0
},
{
"kind": "formalization",
"citation": "Source formal statement (not locally verified)",
"url": "https://theoremdb.org/statements/P8414",
"not... | {
"source_record": "P8414",
"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":"2026-08-17T18:50:26+00:00","display_standard":{"editorial_review":"submitter_supplied","formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":8414,"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": "2026-09-08T03:32:50.311Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"10169f56-2961-455b-bb70-d2352b9de548","format_issues":null,"number":3394} |
1.0.0 | problem:theoremdb-p8416 | problem | theoremdb-p8416 | The determinantal conjecture for normal matrices | {
"main": "math",
"sub": "algebra",
"slug": "algebra",
"name": "Algebra",
"original_discipline": "mathematics"
} | [] | [] | Let $n$ be a finite index type. Let $d_1, d_2 : n \to \mathbb{C}$ be two functions, and let $U_1, U_2$ be unitary $n \times n$ complex matrices. Consider the two normal matrices $A = U_1 \cdot \mathrm{diag}(d_1) \cdot U_1^*$ and $B = U_2 \cdot \mathrm{diag}(d_2) \cdot U_2^*$, where $\mathrm{diag}(d)$ denotes the diagon... | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P8416",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P8416",
"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 determinantal_conjecture
(n : Type) [Fintype n] [DecidableEq n]
(d1 d2 : n → ℂ) (U1 U2 : unitary (Matrix n n ℂ)) :
(U1 * Matrix.diagonal d1 * star U1 + U2 * Matrix.diagonal d2 * star U2).det
∈ convexHull ℝ { ∏ i, (d1 i + d2 (σ i)) | σ : Equiv.Perm n } | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem determinantal_conjecture\n (n : Type) [Fintype n] [DecidableEq n]\n (d1 d2 : n → ℂ) (U1 U2 : unitary (Matrix n n ℂ)) :\n (U1 * Matrix.diagonal d1 * star U1 + U2 * Matrix.diagonal d2 * star U2).det\n ∈ convexHull ℝ { ∏ i, (d1 i + d2 (σ i)) | σ : Equiv.Pe... | [] | TheoremDB P8416 · 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 P8416",
"url": "https://theoremdb.org/statements/P8416",
"note": null,
"sort_order": 0
},
{
"kind": "formalization",
"citation": "Source formal statement (not locally verified)",
"url": "https://theoremdb.org/statements/P8416",
"not... | {
"source_record": "P8416",
"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":"2026-08-17T18:50:32+00:00","display_standard":{"formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":8416,"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": "2026-09-08T03:33:06.444Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"9661cc71-bbe2-4e88-a912-4b8f93b20dd9","format_issues":null,"number":6744} |
1.0.0 | problem:theoremdb-p8418 | problem | theoremdb-p8418 | Erdős Problem 203 on Prime-Avoiding Integers Coprime to 6 | {
"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 an integer $m$ that is coprime to $6$ such that for all non-negative integers $k$ and $\ell$, the number $2^k \cdot 3^\ell \cdot m + 1$ is not prime? | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P8418",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P8418",
"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_203 : answer(sorry) ↔ ∃ m, m.Coprime 6 ∧ ∀ k l, ¬ (2^k * 3^l * m + 1).Prime | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem erdos_203 : answer(sorry) ↔ ∃ m, m.Coprime 6 ∧ ∀ k l, ¬ (2^k * 3^l * m + 1).Prime",
"target": null,
"source_url": "https://theoremdb.org/statements/P8418",
"format": "declaration-or-fragment",
"sha256": "4f6f2934b94782d5a4011405d38765839a7d00f711c8e6e... | [] | TheoremDB P8418 · 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 P8418",
"url": "https://theoremdb.org/statements/P8418",
"note": null,
"sort_order": 0
},
{
"kind": "formalization",
"citation": "Source formal statement (not locally verified)",
"url": "https://theoremdb.org/statements/P8418",
"not... | {
"source_record": "P8418",
"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":"2026-08-17T18:50:39+00:00","display_standard":{"formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":8418,"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": "2026-09-08T03:32:55.235Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"bf6c80b0-4765-440b-9a21-3976dd1d15a3","format_issues":null,"number":4476} |
1.0.0 | problem:theoremdb-p8420 | problem | theoremdb-p8420 | Asymptotic formula for the minimum sunflower-forcing family size | {
"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 $k$, let $m(n,k)$ denote the minimum integer $m$ such that every family $\mathcal{F}$ of subsets of $\{1, 2, \ldots, n\}$ with $|\mathcal{F}| \geq m$ contains a $k$-sunflower. A family $\mathcal{S}$ of $k$ distinct sets is called a $k$-sunflower if there exists a set $C$ (the core) such th... | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P8420",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P8420",
"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_857 :
let f : ℕ → ℕ → ℝ := answer(sorry)
∀ k : ℕ, 3 ≤ k → Tendsto (fun n : ℕ => (m n k : ℝ) / f k n) atTop (nhds 1) | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem erdos_857 :\n let f : ℕ → ℕ → ℝ := answer(sorry)\n ∀ k : ℕ, 3 ≤ k → Tendsto (fun n : ℕ => (m n k : ℝ) / f k n) atTop (nhds 1)",
"target": null,
"source_url": "https://theoremdb.org/statements/P8420",
"format": "declaration-or-fragment",
"sha256"... | [] | TheoremDB P8420 · 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 P8420",
"url": "https://theoremdb.org/statements/P8420",
"note": null,
"sort_order": 0
},
{
"kind": "formalization",
"citation": "Source formal statement (not locally verified)",
"url": "https://theoremdb.org/statements/P8420",
"not... | {
"source_record": "P8420",
"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":"2026-08-17T18:50:49+00:00","display_standard":{"editorial_review":"submitter_supplied","formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":8420,"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": "2026-09-08T03:32:50.048Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"8e356c6a-e44c-4815-8b1a-7e28fbfd0eee","format_issues":null,"number":3340} |
1.0.0 | problem:theoremdb-p8422 | problem | theoremdb-p8422 | Existence of vertex-critical graphs with large critical edge 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"
}
] | Let $k$ and $r$ be positive integers with $k \ge 4$ and $r \ge 1$. A finite simple graph $G$ with vertex set $V$ is called $k$-critical if its chromatic number equals $k$, but the chromatic number of every proper subgraph is strictly less than $k$. A subset of edges $E' \subseteq E(G)$ is called critical if the chromat... | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P8422",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P8422",
"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_944 :
answer(sorry) ↔ ∀ k ≥ 4, ∀ r ≥ 1, ∃ (V : Type u) (G : SimpleGraph V), G.IsErdos944 k r | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem erdos_944 :\n answer(sorry) ↔ ∀ k ≥ 4, ∀ r ≥ 1, ∃ (V : Type u) (G : SimpleGraph V), G.IsErdos944 k r",
"target": null,
"source_url": "https://theoremdb.org/statements/P8422",
"format": "declaration-or-fragment",
"sha256": "b4aab7e07e8b9c109f84c51aa... | [] | TheoremDB P8422 · 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 P8422",
"url": "https://theoremdb.org/statements/P8422",
"note": null,
"sort_order": 0
},
{
"kind": "formalization",
"citation": "Source formal statement (not locally verified)",
"url": "https://theoremdb.org/statements/P8422",
"not... | {
"source_record": "P8422",
"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":"2026-08-17T18:52:00+00:00","display_standard":{"editorial_review":"submitter_supplied","formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":8422,"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": "2026-09-08T03:32:58.875Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"62893f54-8534-4f4a-bf0f-ac7d82c2b33d","format_issues":null,"number":5274} |
1.0.0 | problem:theoremdb-p8424 | problem | theoremdb-p8424 | Normality of π in base 10 | {
"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 real number is said to be normal in base $b$ if, in its base-$b$ expansion, every finite string of digits of length $k$ occurs with limiting frequency $b^{-k}$. Equivalently, the sequence of digits is uniformly distributed in the most equidistributed sense possible. Determine whether the mathematical constant $\pi$ (... | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P8424",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P8424",
"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 pi_normal_base_ten : IsNormalInBase 10 π | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem pi_normal_base_ten : IsNormalInBase 10 π",
"target": null,
"source_url": "https://theoremdb.org/statements/P8424",
"format": "declaration-or-fragment",
"sha256": "1c28b3b36fe161b0d3c38af9676c114ecd05bcd73ac799fd9c259ec75c030489",
"license": "Selec... | [] | TheoremDB P8424 · 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 P8424",
"url": "https://theoremdb.org/statements/P8424",
"note": null,
"sort_order": 0
},
{
"kind": "formalization",
"citation": "Source formal statement (not locally verified)",
"url": "https://theoremdb.org/statements/P8424",
"not... | {
"source_record": "P8424",
"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":"2026-08-17T18:52:02+00:00","display_standard":{"editorial_review":"submitter_supplied","formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":8424,"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": "2026-09-08T03:33:04.530Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"33f13787-f24c-4a9f-9fa4-371dd9a2e35a","format_issues":null,"number":6330} |
1.0.0 | problem:theoremdb-p8426 | problem | theoremdb-p8426 | Beaver Math Olympiad Problem 5 | {
"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 $(a_0, b_0) = (0, 5)$ and for all non-negative integers $n$,
$$
(a_{n+1}, b_{n+1}) = \begin{cases}
(a_n + 1, b_n - f(a_n)) & \text{if } f(a_n) \le b_n, \\
(a_n, 3b_n + a_n + 5) & \text{if } f(a_n) > b_n,
\end{cases}
$$
where $f... | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P8426",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P8426",
"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_5 : answer(sorry) ↔
∀ (a b f : ℕ → ℕ), ∀ᵉ (hf : f = fun x ↦ 10 * 2 ^ x - 1)
(a_ini : a 0 = 0) (b_ini : b 0 = 5)
(a_rec : ∀ n, a (n + 1) = if f (a n) ≤ b n then a n + 1 else a n)
(b_rec : ∀ n, b (n+1) = if f (a n) ≤ b n then b n - f (a n) else 3 * b n + a n + 5),
... | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem beaver_math_olympiad_problem_5 : answer(sorry) ↔\n ∀ (a b f : ℕ → ℕ), ∀ᵉ (hf : f = fun x ↦ 10 * 2 ^ x - 1)\n (a_ini : a 0 = 0) (b_ini : b 0 = 5)\n (a_rec : ∀ n, a (n + 1) = if f (a n) ≤ b n then a n + 1 else a n)\n (b_rec : ∀ n, b (n+1) = if f (a n) ≤ b n... | [] | TheoremDB P8426 · 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 P8426",
"url": "https://theoremdb.org/statements/P8426",
"note": null,
"sort_order": 0
},
{
"kind": "formalization",
"citation": "Source formal statement (not locally verified)",
"url": "https://theoremdb.org/statements/P8426",
"not... | {
"source_record": "P8426",
"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":"2026-08-17T18:52:02+00:00","display_standard":{"editorial_review":"submitter_supplied","formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":8426,"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": "2026-09-08T03:32:50.627Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"9181fd6d-a266-4ced-be4b-7505362a2af0","format_issues":null,"number":3457} |
1.0.0 | problem:theoremdb-p8428 | problem | theoremdb-p8428 | Positive Density of Increasing Steps in the Normalized Prime 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 $p_n$ denote the $n$th prime number, and define the normalized prime sequence by $u_n = p_n / n$. Determine whether the set $\{n \in \mathbb{N} : u_n < u_{n+1}\}$ has positive natural density. | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P8428",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P8428",
"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 : answer(sorry) ↔ 0 < {n : ℕ | u n < u (n + 1)}.lowerDensity | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem erdos_968 : answer(sorry) ↔ 0 < {n : ℕ | u n < u (n + 1)}.lowerDensity",
"target": null,
"source_url": "https://theoremdb.org/statements/P8428",
"format": "declaration-or-fragment",
"sha256": "d7c448bd589c8b4bb54a9fc277c48f712f6a2b0fe0bd8d2f044cf80801... | [] | TheoremDB P8428 · 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 P8428",
"url": "https://theoremdb.org/statements/P8428",
"note": null,
"sort_order": 0
},
{
"kind": "formalization",
"citation": "Source formal statement (not locally verified)",
"url": "https://theoremdb.org/statements/P8428",
"not... | {
"source_record": "P8428",
"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":"2026-08-17T18:52:03+00:00","display_standard":{"formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":8428,"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": "2026-09-08T03:33:04.940Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"e77f995e-7470-4d48-905e-04893b4ccd4c","format_issues":null,"number":6419} |
1.0.0 | problem:theoremdb-p8430 | problem | theoremdb-p8430 | Sun's Conjecture on Prime-Generating Partitions | {
"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 integer $n > 1$, determine whether there exist positive integers $x$ and $y$ such that $n = x + y$ and $2^x + y$ is prime. | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P8430",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P8430",
"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/P8430",
"format": "declaration-or-fragment",
"sha256": "9c3552ea3b574a89c5f6b8cd5a92ea437c055998af76a271622edaaf8ee3d6e7",
"license": "Selected... | [] | TheoremDB P8430 · 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 P8430",
"url": "https://theoremdb.org/statements/P8430",
"note": null,
"sort_order": 0
},
{
"kind": "formalization",
"citation": "Source formal statement (not locally verified)",
"url": "https://theoremdb.org/statements/P8430",
"not... | {
"source_record": "P8430",
"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":"2026-08-17T18:52:03+00:00","display_standard":{"editorial_review":"submitter_supplied","formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":8430,"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": "2026-09-08T03:33:06.177Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"9695f18b-53f9-4619-a905-dc992bb6d793","format_issues":null,"number":6687} |
1.0.0 | problem:theoremdb-p8432 | problem | theoremdb-p8432 | Non-existence of monochromatic quantum graphs with six vertices, five colours, and integer weights | {
"main": "math",
"sub": "combinatorics",
"slug": "combinatorics",
"name": "Combinatorics",
"original_discipline": "mathematics"
} | [] | [
{
"code": "05",
"basis": "community-map",
"evidence": "Inferred from community field: combinatorics"
}
] | Let $N = 6$ and $D = 5$. Consider a complete graph on $N$ vertices whose edges are coloured with $D$ colours and assigned integer weights. For each vertex colouring $\iota$ (an assignment of colours to vertices), define the perfect-matching sum to be the sum over all perfect matchings of the complete graph on $N$ verti... | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P8432",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P8432",
"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 eqSystem6_no_solution_d5_int :
answer(sorry) ↔
¬ ∃ W : WeightsN 6 5 ℤ, EqSystemN 6 5 W | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem eqSystem6_no_solution_d5_int :\n answer(sorry) ↔\n ¬ ∃ W : WeightsN 6 5 ℤ, EqSystemN 6 5 W",
"target": null,
"source_url": "https://theoremdb.org/statements/P8432",
"format": "declaration-or-fragment",
"sha256": "9913cd56657eb9c820ea1b79859930... | [] | TheoremDB P8432 · 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 P8432",
"url": "https://theoremdb.org/statements/P8432",
"note": null,
"sort_order": 0
},
{
"kind": "formalization",
"citation": "Source formal statement (not locally verified)",
"url": "https://theoremdb.org/statements/P8432",
"not... | {
"source_record": "P8432",
"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":"2026-08-17T18:52:04+00:00","display_standard":{"editorial_review":"submitter_supplied","formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":8432,"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": "2026-09-08T03:33:04.357Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"7bf569ba-2781-4185-99ad-af8c7e32c439","format_issues":null,"number":6293} |
1.0.0 | problem:theoremdb-p8434 | problem | theoremdb-p8434 | Erdős's strong conjecture on gaps between 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. Let $n_k$ denote the $k$th powerful number in increasing order. Does there exist a real constant $c > 0$ such that for every natural number $k$, the inequality $n_{k+2} - n_k > n_k^c$ holds? | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P8434",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P8434",
"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.variants.strong :
∃ (c : ℝ) (h : c > 0), ∀ (k : ℕ),
Nat.nth Powerful (k + 2) - Nat.nth Powerful k > (Nat.nth Powerful k : ℝ) ^ c | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem erdos_364.variants.strong :\n ∃ (c : ℝ) (h : c > 0), ∀ (k : ℕ),\n Nat.nth Powerful (k + 2) - Nat.nth Powerful k > (Nat.nth Powerful k : ℝ) ^ c",
"target": null,
"source_url": "https://theoremdb.org/statements/P8434",
"format": "declaration-or-fragme... | [] | TheoremDB P8434 · 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 P8434",
"url": "https://theoremdb.org/statements/P8434",
"note": null,
"sort_order": 0
},
{
"kind": "formalization",
"citation": "Source formal statement (not locally verified)",
"url": "https://theoremdb.org/statements/P8434",
"not... | {
"source_record": "P8434",
"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":"2026-08-17T18:52:05+00:00","display_standard":{"editorial_review":"submitter_supplied","formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":8434,"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": "2026-09-08T03:32:56.589Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"f3177428-9118-462d-942d-ab39130b91ae","format_issues":null,"number":4773} |
1.0.0 | problem:theoremdb-p8436 | problem | theoremdb-p8436 | Opposite-parity amicable pair existence | {
"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"
}
] | Two distinct positive integers $a$ and $b$ are said to form an amicable pair if each equals the sum of the proper divisors of the other; equivalently, the sum of all positive divisors of $a$ equals $a + b$, and the sum of all positive divisors of $b$ equals $a + b$. Does there exist an amicable pair $(a, b)$ such that ... | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P8436",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P8436",
"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 opposite_parity_amicable :
answer(sorry) ↔ ∃ a b : ℕ, IsAmicable a b ∧ (Even a ↔ Odd b) | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem opposite_parity_amicable :\n answer(sorry) ↔ ∃ a b : ℕ, IsAmicable a b ∧ (Even a ↔ Odd b)",
"target": null,
"source_url": "https://theoremdb.org/statements/P8436",
"format": "declaration-or-fragment",
"sha256": "c3984a91bc1fccafffaef908a5363d1dbe6b... | [] | TheoremDB P8436 · 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 P8436",
"url": "https://theoremdb.org/statements/P8436",
"note": null,
"sort_order": 0
},
{
"kind": "formalization",
"citation": "Source formal statement (not locally verified)",
"url": "https://theoremdb.org/statements/P8436",
"not... | {
"source_record": "P8436",
"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":"2026-08-17T18:52:05+00:00","display_standard":{"editorial_review":"submitter_supplied","formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":8436,"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": "2026-09-08T03:33:04.663Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"9a908ad7-e472-42bf-bb0e-851f8e09d8fe","format_issues":null,"number":6359} |
1.0.0 | problem:theoremdb-p8438 | problem | theoremdb-p8438 | Equal Least Common Multiples in Dense Subsets of Initial 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 every real number $\varepsilon > 0$, does there exist a positive integer $N_0$ such that for all integers $N \geq N_0$, the following holds? Every subset $A$ of $\{1, 2, \ldots, N\}$ with $|A| \geq \varepsilon N$ contains three distinct elements $a, b, c \in A$ such that $\operatorname{lcm}(a,b) = \operatorname{lcm... | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P8438",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P8438",
"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_536 :
answer(sorry) ↔ ∀ᵉ (ε > (0: ℝ)), ∀ᶠ N in atTop,
∀ (A : Finset ℕ), A ⊆ Icc 1 N → (ε * (N : ℝ)) ≤ (A.card : ℝ) →
∃ᵉ (a ∈ A) (b ∈ A) (c ∈ A),
# {a, b, c} = 3 ∧ a.lcm b = b.lcm c ∧ b.lcm c = a.lcm c | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem erdos_536 :\n answer(sorry) ↔ ∀ᵉ (ε > (0: ℝ)), ∀ᶠ N in atTop,\n ∀ (A : Finset ℕ), A ⊆ Icc 1 N → (ε * (N : ℝ)) ≤ (A.card : ℝ) →\n ∃ᵉ (a ∈ A) (b ∈ A) (c ∈ A),\n # {a, b, c} = 3 ∧ a.lcm b = b.lcm c ∧ b.lcm c = a.lcm c",
"target": null,
"source_url":... | [] | TheoremDB P8438 · 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 P8438",
"url": "https://theoremdb.org/statements/P8438",
"note": null,
"sort_order": 0
},
{
"kind": "formalization",
"citation": "Source formal statement (not locally verified)",
"url": "https://theoremdb.org/statements/P8438",
"not... | {
"source_record": "P8438",
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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":"2026-08-17T18:52:06+00:00","display_standard":{"formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":8438,"public_status":{"evidence":{"hasFormalization":false,"hasInformalProof":false,"hasRecordedWork":false,"solutionRecord":null},"lean":{"kernelAccepted":false,"label":"","reaso... | [
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"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"e82cbc5d-0040-480a-adaf-88e99b6621db","format_issues":null,"number":4369} |
1.0.0 | problem:theoremdb-p8440 | problem | theoremdb-p8440 | Infinitely many consecutive primes in arithmetic progression of length k | {
"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 $k \geq 3$, let $\mathcal{P}_k$ denote the set of all subsets $s \subseteq \mathbb{N}$ such that $s$ is both an arithmetic progression of length $k$ and a set of $k$ consecutive prime numbers. Here, a set $s$ is an arithmetic progression of length $k$ if there exist integers $a$ and $d$ with $d > 0$ su... | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P8440",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P8440",
"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_general_case : answer(sorry) ↔
∀ k ≥ 3, (consecutivePrimeArithmeticProgressions k).Infinite | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem erdos_141.variants.infinite_general_case : answer(sorry) ↔\n ∀ k ≥ 3, (consecutivePrimeArithmeticProgressions k).Infinite",
"target": null,
"source_url": "https://theoremdb.org/statements/P8440",
"format": "declaration-or-fragment",
"sha256": "393b... | [] | TheoremDB P8440 · 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 P8440",
"url": "https://theoremdb.org/statements/P8440",
"note": null,
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},
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"kind": "formalization",
"citation": "Source formal statement (not locally verified)",
"url": "https://theoremdb.org/statements/P8440",
"not... | {
"source_record": "P8440",
"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":"2026-08-17T18:52:06+00:00","display_standard":{"editorial_review":"submitter_supplied","formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":8440,"public_status":{"evidence":{"hasFormalization":false,"hasInformalProof":false,"hasRecordedWork":false,"solutionRecord":null},"lean":{... | [
"formalization-contains-placeholder",
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"source-full-text-not-present"
] | [] | {
"created_at": "2026-09-08T03:32:15.935Z",
"updated_at": "2026-09-08T03:32:15.935Z",
"published_at": "2026-09-08T03:33:00.848Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"850d3f11-c56e-49e3-a653-a2bc140fd212","format_issues":null,"number":5603} |
1.0.0 | problem:theoremdb-p8442 | problem | theoremdb-p8442 | Goldbach'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"
}
] | A natural number is called even if it is divisible by $2$, and prime if it is greater than $1$ and has no positive divisors other than $1$ and itself. Determine whether every even natural number greater than $2$ can be expressed as the sum of two prime numbers. | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P8442",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P8442",
"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 goldbach :
answer(sorry) ↔ ∀ n : ℕ, 2 < n → Even n → ∃ p q, Prime p ∧ Prime q ∧ n = p + q | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem goldbach :\n answer(sorry) ↔ ∀ n : ℕ, 2 < n → Even n → ∃ p q, Prime p ∧ Prime q ∧ n = p + q",
"target": null,
"source_url": "https://theoremdb.org/statements/P8442",
"format": "declaration-or-fragment",
"sha256": "1ca6186d2bdcd9f048ce357cd761599af7... | [] | TheoremDB P8442 · 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 P8442",
"url": "https://theoremdb.org/statements/P8442",
"note": null,
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},
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"kind": "formalization",
"citation": "Source formal statement (not locally verified)",
"url": "https://theoremdb.org/statements/P8442",
"not... | {
"source_record": "P8442",
"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":"2026-08-17T18:52:07+00:00","display_standard":{"editorial_review":"submitter_supplied","formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":8442,"public_status":{"evidence":{"hasFormalization":false,"hasInformalProof":false,"hasRecordedWork":false,"solutionRecord":null},"lean":{... | [
"formalization-contains-placeholder",
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] | [] | {
"created_at": "2026-09-08T03:32:15.935Z",
"updated_at": "2026-09-08T03:32:15.935Z",
"published_at": "2026-09-08T03:32:59.371Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"c57719f5-7786-477b-a2ac-bb26604419e3","format_issues":null,"number":5381} |
1.0.0 | problem:theoremdb-p8444 | problem | theoremdb-p8444 | Scholz conjecture on addition chains | {
"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"
}
] | An *addition chain* is a strictly increasing sequence $1 = a_0 < a_1 < \cdots < a_r$ of natural numbers in which every entry after the first is the sum of two (not necessarily distinct) earlier entries. The *length* of a natural number $n$, denoted $\ell(n)$, is the minimal number of addition steps (that is, the number... | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P8444",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P8444",
"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 scholz_conjecture :
answer(sorry) ↔ ∀ (n : ℕ), 0 < n → ℓ(2 ^ n - 1) ≤ n - 1 + ℓ(n) | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem scholz_conjecture :\n answer(sorry) ↔ ∀ (n : ℕ), 0 < n → ℓ(2 ^ n - 1) ≤ n - 1 + ℓ(n)",
"target": null,
"source_url": "https://theoremdb.org/statements/P8444",
"format": "declaration-or-fragment",
"sha256": "aeae49d43f119924a3bce6a2e4977098bacd8476c... | [] | TheoremDB P8444 · 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 P8444",
"url": "https://theoremdb.org/statements/P8444",
"note": null,
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},
{
"kind": "formalization",
"citation": "Source formal statement (not locally verified)",
"url": "https://theoremdb.org/statements/P8444",
"not... | {
"source_record": "P8444",
"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":"2026-08-17T18:52:08+00:00","display_standard":{"editorial_review":"submitter_supplied","formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":8444,"public_status":{"evidence":{"hasFormalization":false,"hasInformalProof":false,"hasRecordedWork":false,"solutionRecord":null},"lean":{... | [
"formalization-contains-placeholder",
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"created_at": "2026-09-08T03:32:15.935Z",
"updated_at": "2026-09-08T03:32:15.935Z",
"published_at": "2026-09-08T03:33:05.738Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"a3c4f9fb-f901-470d-ac50-2ccee7beab26","format_issues":null,"number":6592} |
1.0.0 | problem:theoremdb-p8446 | problem | theoremdb-p8446 | Irrationality of the Natural Logarithm of Pi | {
"main": "math",
"sub": "analysis",
"slug": "analysis",
"name": "Analysis",
"original_discipline": "mathematics"
} | [] | [] | Let $\pi$ denote the ratio of the circumference of a circle to its diameter, and let $\ln(\pi)$ denote the natural logarithm of $\pi$. Is $\ln(\pi)$ irrational? In other words, does there exist no pair of integers $p$ and $q$ with $q \neq 0$ such that $\ln(\pi) = p/q$? | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P8446",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P8446",
"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_ln_pi :
answer(sorry) ↔ Irrational (log π) | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem irrational_ln_pi :\n answer(sorry) ↔ Irrational (log π)",
"target": null,
"source_url": "https://theoremdb.org/statements/P8446",
"format": "declaration-or-fragment",
"sha256": "84aec8131d911ba46ed723fde3f114b79f90881054b1ba4daa5bd98d5dafb9a4",
... | [] | TheoremDB P8446 · 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 P8446",
"url": "https://theoremdb.org/statements/P8446",
"note": null,
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},
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"citation": "Source formal statement (not locally verified)",
"url": "https://theoremdb.org/statements/P8446",
"not... | {
"source_record": "P8446",
"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":"2026-08-17T18:52:08+00:00","display_standard":{"editorial_review":"submitter_supplied","formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":8446,"public_status":{"evidence":{"hasFormalization":false,"hasInformalProof":false,"hasRecordedWork":false,"solutionRecord":null},"lean":{... | [
"formalization-contains-placeholder",
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"source-full-text-not-present"
] | [] | {
"created_at": "2026-09-08T03:32:15.935Z",
"updated_at": "2026-09-08T03:32:15.935Z",
"published_at": "2026-09-08T03:33:01.793Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"5c27500a-cb44-4890-96e2-b155bc6e3f50","format_issues":null,"number":5736} |
1.0.0 | problem:theoremdb-p8448 | problem | theoremdb-p8448 | Explicit coloring for the quadratic van der Waerden lower bound | {
"main": "math",
"sub": "combinatorics",
"slug": "combinatorics",
"name": "Combinatorics",
"original_discipline": "mathematics"
} | [] | [
{
"code": "05",
"basis": "community-map",
"evidence": "Inferred from community field: combinatorics"
}
] | For integers $k, r \geq 1$, the mixed van der Waerden number $W(k, r)$ is defined as the least positive integer $N$ such that every 2-coloring of $\{1, 2, \ldots, N\}$ contains either a monochromatic arithmetic progression of length $k$ in color 0 or a monochromatic arithmetic progression of length $r$ in color 1. Dete... | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P8448",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P8448",
"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_14_variant_2r2 :
-- Provide a pair (r, associated coloring) that avoids the monochromatic APs
-- To show $W(3, r) > 2r^2 - 1$, we need a coloring of $\{1, \ldots, 2r^2 - 1\}$
-- that avoids monochromatic APs of length 3 and $r$.
let ans : Σ r : ℕ, Icc 1 (2 * r^2 - 1) → Fin 2 := answer(sorr... | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem green_14_variant_2r2 :\n -- Provide a pair (r, associated coloring) that avoids the monochromatic APs\n -- To show $W(3, r) > 2r^2 - 1$, we need a coloring of $\\{1, \\ldots, 2r^2 - 1\\}$\n -- that avoids monochromatic APs of length 3 and $r$.\n let ans :... | [] | TheoremDB P8448 · 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 P8448",
"url": "https://theoremdb.org/statements/P8448",
"note": null,
"sort_order": 0
},
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"kind": "formalization",
"citation": "Source formal statement (not locally verified)",
"url": "https://theoremdb.org/statements/P8448",
"not... | {
"source_record": "P8448",
"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":"2026-08-17T18:52:09+00:00","display_standard":{"editorial_review":"submitter_supplied","formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":8448,"public_status":{"evidence":{"hasFormalization":false,"hasInformalProof":false,"hasRecordedWork":false,"solutionRecord":null},"lean":{... | [
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"source-full-text-not-present"
] | [] | {
"created_at": "2026-09-08T03:32:15.935Z",
"updated_at": "2026-09-08T03:32:15.935Z",
"published_at": "2026-09-08T03:32:58.883Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"81c6a1dc-9326-44e3-b7eb-d25a0a4ccd0c","format_issues":null,"number":5276} |
1.0.0 | problem:theoremdb-p8450 | problem | theoremdb-p8450 | Erdős–Mirsky constant for repeated divisor counts 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 a positive integer $m$, let $\tau(m)$ denote the number of positive divisors of $m$. For a real number $x \geq 1$, let $F(x)$ be the maximum integer $k$ such that there exist consecutive integers $n+1, n+2, \dots, n+k \leq x$ with $\tau(n+1), \tau(n+2), \dots, \tau(n+k)$ all distinct. Determine whether there exists... | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P8450",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P8450",
"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_945.variants.constant : answer(sorry) ↔ Erdos945Constant | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem erdos_945.variants.constant : answer(sorry) ↔ Erdos945Constant",
"target": null,
"source_url": "https://theoremdb.org/statements/P8450",
"format": "declaration-or-fragment",
"sha256": "fd4450df72b979d58baef94d8de2efcdcdcf8753a356325ef06e87ad03c610fa",... | [] | TheoremDB P8450 · 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 P8450",
"url": "https://theoremdb.org/statements/P8450",
"note": null,
"sort_order": 0
},
{
"kind": "formalization",
"citation": "Source formal statement (not locally verified)",
"url": "https://theoremdb.org/statements/P8450",
"not... | {
"source_record": "P8450",
"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":"2026-08-17T18:52:09+00:00","display_standard":{"editorial_review":"submitter_supplied","formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":8450,"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": "2026-09-08T03:32:56.687Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"49bc58ed-6467-44a9-ba28-921d5bf9af48","format_issues":null,"number":4793} |
1.0.0 | problem:theoremdb-p8452 | problem | theoremdb-p8452 | Optimality of the Cohn–Elkies Upper Bound for Circle Packings in Two Dimensions | {
"main": "math",
"sub": "geometry and topology",
"slug": "geometry-and-topology",
"name": "Geometry and topology",
"original_discipline": "mathematics"
} | [] | [] | Let a function $f : \mathbb{R}^d \to \mathbb{R}$ be called Cohn–Elkies admissible if there exist constants $C > 0$ and $\delta > 0$ such that for all $x, t \in \mathbb{R}^d$,
$$
|f(x)| \leq \frac{C}{(1 + \|x\|)^{d + \delta}} \quad \text{and} \quad |\hat{f}(t)| \leq \frac{C}{(1 + \|t\|)^{d + \delta}},
$$
where $\hat{f... | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P8452",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P8452",
"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_42 :
answer(sorry) ↔ CohnElkiesOptimal 2 (Real.sqrt 3 / 6) | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem green_42 :\n answer(sorry) ↔ CohnElkiesOptimal 2 (Real.sqrt 3 / 6)",
"target": null,
"source_url": "https://theoremdb.org/statements/P8452",
"format": "declaration-or-fragment",
"sha256": "f2f857ed8b27ff2378c0859874673e983ff6572d4898f3485caa1803c8d... | [] | TheoremDB P8452 · 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 | [
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"url": "https://theoremdb.org/statements/P8452",
"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":"2026-08-17T18:52:09+00:00","display_standard":{"editorial_review":"submitter_supplied","formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":8452,"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": "2026-09-08T03:33:04.685Z",
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} | 2026-10-10T16:07:49.246Z | {"database_id":"c32d838d-d624-4e14-9b83-250a9a16c24c","format_issues":null,"number":6364} |
1.0.0 | problem:theoremdb-p8454 | problem | theoremdb-p8454 | The Köthe Conjecture on Nil Ideals | {
"main": "math",
"sub": "algebra",
"slug": "algebra",
"name": "Algebra",
"original_discipline": "mathematics"
} | [] | [] | Let $R$ be a ring. A subset $I$ of $R$ is called nil if every element of $I$ is nilpotent, that is, for each $i \in I$ there exists a positive integer $n$ such that $i^n = 0$. Let $I$ and $J$ be left ideals of $R$. Suppose that $I$ is nil and $J$ is nil. Is the sum $I + J = \{i + j : i \in I, j \in J\}$ a nil left idea... | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P8454",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P8454",
"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 KotheConjecture (I J : Ideal R) (hI : IsNil I) (hJ : IsNil J) : IsNil (I + J) | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem KotheConjecture (I J : Ideal R) (hI : IsNil I) (hJ : IsNil J) : IsNil (I + J)",
"target": null,
"source_url": "https://theoremdb.org/statements/P8454",
"format": "declaration-or-fragment",
"sha256": "0c7600a4373820db3d160e2e7898bd6c9bcdd78445a0af7bb22... | [] | TheoremDB P8454 · 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 | [
{
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"url": "https://theoremdb.org/statements/P8454",
"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":"2026-08-17T18:52:10+00:00","display_standard":{"formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":8454,"public_status":{"evidence":{"hasFormalization":false,"hasInformalProof":false,"hasRecordedWork":false,"solutionRecord":null},"lean":{"kernelAccepted":false,"label":"","reaso... | [
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"published_at": "2026-09-08T03:33:06.589Z",
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} | 2026-10-10T16:07:49.246Z | {"database_id":"642df195-691e-4fce-9809-885cf3dcca42","format_issues":null,"number":6775} |
1.0.0 | problem:theoremdb-p8456 | problem | theoremdb-p8456 | Improved upper bound for the largest admissible cardinality in Green's Problem 38 | {
"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 $n$, let $\mathbb{F}_7^n$ denote the $n$-dimensional vector space over the field $\mathbb{Z}/7\mathbb{Z}$ of integers modulo $7$. A finite subset $A \subseteq \mathbb{F}_7^n$ is called admissible if whenever $a - b \in \{-1, 0, 1\}^n$ for some $a, b \in A$, we have $a = b$; equivalently, the d... | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P8456",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P8456",
"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_38.upper :
let ans := (answer(sorry) : ℕ → ℝ)
LargestAdmissibleCardinality ≤ᶠ[atTop] ans ∧
∃ c < C₂, ans =O[atTop] (fun n ↦ c ^ n) | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem green_38.upper :\n let ans := (answer(sorry) : ℕ → ℝ)\n LargestAdmissibleCardinality ≤ᶠ[atTop] ans ∧\n ∃ c < C₂, ans =O[atTop] (fun n ↦ c ^ n)",
"target": null,
"source_url": "https://theoremdb.org/statements/P8456",
"format": "declaration-or-fra... | [] | TheoremDB P8456 · 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 | [
{
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"citation": "Source formal statement (not locally verified)",
"url": "https://theoremdb.org/statements/P8456",
"not... | {
"source_record": "P8456",
"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":"2026-08-17T18:52:10+00:00","display_standard":{"editorial_review":"submitter_supplied","formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":8456,"public_status":{"evidence":{"hasFormalization":false,"hasInformalProof":false,"hasRecordedWork":false,"solutionRecord":null},"lean":{... | [
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"updated_at": "2026-09-08T03:32:15.935Z",
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} | 2026-10-10T16:07:49.246Z | {"database_id":"9a571c93-ebb1-4269-a0a6-80a579a0924e","format_issues":null,"number":5445} |
1.0.0 | problem:theoremdb-p8458 | problem | theoremdb-p8458 | Erdős Problem 40 on additive bases with slow growth | {
"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 set $A \subseteq \mathbb{N}$ and a positive integer $N$, let $|A \cap \{1, \ldots, N\}|$ denote the cardinality of the intersection. For $n \in \mathbb{N}$, define the representation function $(1_A * 1_A)(n)$ as the number of ordered pairs $(a, b) \in A \times A$ such that $a + b = n$. We say that a property hold... | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P8458",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P8458",
"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_40 : Erdos40ForSet answer(sorry) | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem erdos_40 : Erdos40ForSet answer(sorry)",
"target": null,
"source_url": "https://theoremdb.org/statements/P8458",
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"sha256": "07e6531efa86802a1e6ac5481803c8fdafa38ca36ad5e9a0e252ddf9a753ac75",
"license": "Selecte... | [] | TheoremDB P8458 · 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 | [
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"url": "https://theoremdb.org/statements/P8458",
"not... | {
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"created_at": "2026-09-08T03:32:15.935Z",
"updated_at": "2026-09-08T03:32:15.935Z",
"published_at": "2026-09-08T03:32:55.425Z",
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} | 2026-10-10T16:07:49.246Z | {"database_id":"54119a11-a4ba-4da6-9d71-4cc68c056f0a","format_issues":null,"number":4516} |
1.0.0 | problem:theoremdb-p8460 | problem | theoremdb-p8460 | Smallest forbidden arithmetic progression length in a two-coloring of the plane avoiding unit-distance red pairs | {
"main": "math",
"sub": "combinatorics",
"slug": "combinatorics",
"name": "Combinatorics",
"original_discipline": "mathematics"
} | [] | [
{
"code": "05",
"basis": "community-map",
"evidence": "Inferred from community field: combinatorics"
}
] | Let $\mathbb{R}^2$ be the Euclidean plane. A coloring of $\mathbb{R}^2$ with red and blue is called admissible if no two red points are at distance 1, and there is no $k$-term arithmetic progression of blue points with common difference of norm 1. Let $s$ be the set of natural numbers $k$ for which there exists an admi... | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P8460",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P8460",
"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_188 : IsLeast s answer(sorry) | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem erdos_188 : IsLeast s answer(sorry)",
"target": null,
"source_url": "https://theoremdb.org/statements/P8460",
"format": "declaration-or-fragment",
"sha256": "9c14e6c2d1756a3b0064f76696dc5884ba07cca2baf9b0da26688631812635d4",
"license": "Selected p... | [] | TheoremDB P8460 · 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 P8460",
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"url": "https://theoremdb.org/statements/P8460",
"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":"2026-08-17T18:52:11+00:00","display_standard":{"editorial_review":"submitter_supplied","formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":8460,"public_status":{"evidence":{"hasFormalization":false,"hasInformalProof":false,"hasRecordedWork":false,"solutionRecord":null},"lean":{... | [
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"created_at": "2026-09-08T03:32:15.935Z",
"updated_at": "2026-09-08T03:32:15.935Z",
"published_at": "2026-09-08T03:33:05.905Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"1f1e162c-2806-489d-bb7c-72a62244ac3e","format_issues":null,"number":6628} |
1.0.0 | problem:theoremdb-p8462 | problem | theoremdb-p8462 | Divergence of the iterated sum-of-divisors root mean | {
"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 $\sigma(n)$ denote the sum of all positive divisors of a positive integer $n$. For each integer $k \geq 1$, define the $k$-fold iterate $\sigma_k(n)$ recursively by $\sigma_1(n) = \sigma(n)$ and $\sigma_k(n) = \sigma(\sigma_{k-1}(n))$ for $k \geq 2$. For a fixed integer $n > 1$, consider the sequence $\sigma_k(n)^{... | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P8462",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P8462",
"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_410 : answer(sorry) ↔ ∀ n > 1,
Tendsto (fun k : ℕ ↦ ((sigma 1)^[k] n : ℝ) ^ (1 / (k : ℝ))) atTop atTop | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem erdos_410 : answer(sorry) ↔ ∀ n > 1,\n Tendsto (fun k : ℕ ↦ ((sigma 1)^[k] n : ℝ) ^ (1 / (k : ℝ))) atTop atTop",
"target": null,
"source_url": "https://theoremdb.org/statements/P8462",
"format": "declaration-or-fragment",
"sha256": "75828a12cd691ab... | [] | TheoremDB P8462 · 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 | [
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"url": "https://theoremdb.org/statements/P8462",
"not... | {
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"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":"2026-08-17T18:52:12+00:00","display_standard":{"editorial_review":"submitter_supplied","formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":8462,"public_status":{"evidence":{"hasFormalization":false,"hasInformalProof":false,"hasRecordedWork":false,"solutionRecord":null},"lean":{... | [
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"created_at": "2026-09-08T03:32:15.935Z",
"updated_at": "2026-09-08T03:32:15.935Z",
"published_at": "2026-09-08T03:32:53.276Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"93ded1eb-257a-4cb5-a5f3-b7c680476f64","format_issues":null,"number":4041} |
1.0.0 | problem:theoremdb-p8464 | problem | theoremdb-p8464 | Irrationality of the Lambert series at rational bases | {
"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 rational number $t > 1$, consider the series
$$
S(t) = \sum_{n=1}^{\infty} \frac{1}{t^n - 1}.
$$
By the classical Lambert series identity, this equals $\sum_{n=1}^{\infty} \frac{\tau(n)}{t^n}$, where $\tau(n)$ denotes the number of positive divisors of $n$. Determine whether $S(t)$ is irrational for every rat... | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P8464",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P8464",
"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_1049 :
answer(sorry) ↔ ∀ t : ℚ, t > 1 → Irrational (∑' n : ℕ+, 1 / ((t : ℝ) ^ (n : ℕ) - 1)) | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem erdos_1049 :\n answer(sorry) ↔ ∀ t : ℚ, t > 1 → Irrational (∑' n : ℕ+, 1 / ((t : ℝ) ^ (n : ℕ) - 1))",
"target": null,
"source_url": "https://theoremdb.org/statements/P8464",
"format": "declaration-or-fragment",
"sha256": "9d4cc938a354d9e20569fd131a... | [] | TheoremDB P8464 · 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 | [
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"not... | {
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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":"2026-08-17T18:52:12+00:00","display_standard":{"editorial_review":"submitter_supplied","formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":8464,"public_status":{"evidence":{"hasFormalization":false,"hasInformalProof":false,"hasRecordedWork":false,"solutionRecord":null},"lean":{... | [
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"created_at": "2026-09-08T03:32:15.935Z",
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"published_at": "2026-09-08T03:33:01.784Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"39704c7d-a0e4-4203-af9d-ee7867a8823e","format_issues":null,"number":5734} |
1.0.0 | problem:theoremdb-p8466 | problem | theoremdb-p8466 | Distinctness of consecutive products in a sequence of natural density 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"
}
] | A sequence of positive integers has natural density one if the proportion of integers up to $n$ that appear in the sequence tends to $1$ as $n \to \infty$. Does there exist a strictly increasing sequence $d = (d_0, d_1, d_2, \dots)$ of positive integers with natural density one such that all products of consecutive ter... | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P8466",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P8466",
"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_421 : answer(sorry) ↔
∃ (d : ℕ → ℕ), StrictMono d ∧ 1 ≤ d 0 ∧ HasDensity (Set.range d) 1 ∧
{(u, v) : ℕ × ℕ | u ≤ v}.InjOn fun (u, v) => ∏ i ∈ Finset.Icc u v, d i | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem erdos_421 : answer(sorry) ↔\n ∃ (d : ℕ → ℕ), StrictMono d ∧ 1 ≤ d 0 ∧ HasDensity (Set.range d) 1 ∧\n {(u, v) : ℕ × ℕ | u ≤ v}.InjOn fun (u, v) => ∏ i ∈ Finset.Icc u v, d i",
"target": null,
"source_url": "https://theoremdb.org/statements/P8466",
"fo... | [] | TheoremDB P8466 · 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 P8466",
"url": "https://theoremdb.org/statements/P8466",
"note": null,
"sort_order": 0
},
{
"kind": "formalization",
"citation": "Source formal statement (not locally verified)",
"url": "https://theoremdb.org/statements/P8466",
"not... | {
"source_record": "P8466",
"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":"2026-08-17T18:52:13+00:00","display_standard":{"editorial_review":"submitter_supplied","formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":8466,"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": "2026-09-08T03:32:53.254Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"9afd8843-c388-41ca-8a0b-458314f60f5c","format_issues":null,"number":4036} |
1.0.0 | problem:theoremdb-p8468 | problem | theoremdb-p8468 | Erdős's sum-product growth problem for finite sets 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"
}
] | Let $A$ be a finite set of integers. For every real number $\varepsilon$ with $0 < \varepsilon < 1$, does there exist a real constant $C > 0$ such that
$$
\max(|A+A|, |A \cdot A|) \geq C \cdot |A|^{2-\varepsilon}
$$
holds for all such sets $A$? Here $A+A$ denotes the sumset $\{a+b : a,b \in A\}$ and $A \cdot A$ den... | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P8468",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P8468",
"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_52 : answer(sorry) ↔ ∀ (ε : ℝ), 0 < ε → ε < 1 → ∃ (C : ℝ), 0 < C ∧ ∀ (A : Finset ℤ),
(max (A + A).card (A * A).card : ℝ) ≥ C * (A.card : ℝ) ^ (2 - ε) | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem erdos_52 : answer(sorry) ↔ ∀ (ε : ℝ), 0 < ε → ε < 1 → ∃ (C : ℝ), 0 < C ∧ ∀ (A : Finset ℤ),\n (max (A + A).card (A * A).card : ℝ) ≥ C * (A.card : ℝ) ^ (2 - ε)",
"target": null,
"source_url": "https://theoremdb.org/statements/P8468",
"format": "declarati... | [] | TheoremDB P8468 · 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 P8468",
"url": "https://theoremdb.org/statements/P8468",
"note": null,
"sort_order": 0
},
{
"kind": "formalization",
"citation": "Source formal statement (not locally verified)",
"url": "https://theoremdb.org/statements/P8468",
"not... | {
"source_record": "P8468",
"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":"2026-08-17T18:52:13+00:00","display_standard":{"editorial_review":"submitter_supplied","formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":8468,"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": "2026-09-08T03:32:56.608Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"f10634a4-20aa-422b-a1ca-14cbaa0e95ee","format_issues":null,"number":4777} |
1.0.0 | problem:theoremdb-p8470 | problem | theoremdb-p8470 | Existence of cycles of power-of-two length in cubic 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 finite simple graph with vertex set $V$. Suppose that every vertex of $G$ has degree at least $3$. Does there exist an integer $k \geq 2$, a vertex $v \in V$, and a cycle $c$ in $G$ starting and ending at $v$ such that the length of $c$ equals $2^k$? | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P8470",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P8470",
"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_64 :
answer(sorry) ↔ ∀ (V : Type*) (G : SimpleGraph V) [Fintype V] [DecidableRel G.Adj],
G.minDegree ≥ 3 → ∃ (k : ℕ) (v : V) (c : G.Walk v v),
k ≥ 2 ∧ c.IsCycle ∧ c.length = 2^k | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem erdos_64 :\n answer(sorry) ↔ ∀ (V : Type*) (G : SimpleGraph V) [Fintype V] [DecidableRel G.Adj],\n G.minDegree ≥ 3 → ∃ (k : ℕ) (v : V) (c : G.Walk v v),\n k ≥ 2 ∧ c.IsCycle ∧ c.length = 2^k",
"target": null,
"source_url": "https://theorem... | [] | TheoremDB P8470 · 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 P8470",
"url": "https://theoremdb.org/statements/P8470",
"note": null,
"sort_order": 0
},
{
"kind": "formalization",
"citation": "Source formal statement (not locally verified)",
"url": "https://theoremdb.org/statements/P8470",
"not... | {
"source_record": "P8470",
"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":"2026-08-17T18:52:15+00:00","display_standard":{"editorial_review":"submitter_supplied","formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":8470,"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": "2026-09-08T03:32:58.721Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"8108fd1d-bae4-48e0-bf86-190ef3b24c8b","format_issues":null,"number":5241} |
1.0.0 | problem:theoremdb-p8472 | problem | theoremdb-p8472 | Infinitely many binomial coefficients $inom{n}{2}$ as products of consecutive 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 positive integer $m$ is called a product of consecutive primes if there exist natural numbers $p$ and $q$ with $p < q$ such that
$$
m = \prod_{i=p}^{q-1} p_i,
$$
where $p_i$ denotes the $i$-th prime number (with $p_0 = 2$, $p_1 = 3$, $p_2 = 5$, and so on). Determine whether there exist infinitely many natural num... | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P8472",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P8472",
"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_386.variants.two :
answer(sorry) ↔ ∃ᶠ n in .atTop,
2 ≤ n - 2 ∧ ∃ p q : ℕ, n.choose 2 = ∏ i ∈ .Ico p q, nth Nat.Prime i | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem erdos_386.variants.two :\n answer(sorry) ↔ ∃ᶠ n in .atTop,\n 2 ≤ n - 2 ∧ ∃ p q : ℕ, n.choose 2 = ∏ i ∈ .Ico p q, nth Nat.Prime i",
"target": null,
"source_url": "https://theoremdb.org/statements/P8472",
"format": "declaration-or-fragment",
"sh... | [] | TheoremDB P8472 · 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 P8472",
"url": "https://theoremdb.org/statements/P8472",
"note": null,
"sort_order": 0
},
{
"kind": "formalization",
"citation": "Source formal statement (not locally verified)",
"url": "https://theoremdb.org/statements/P8472",
"not... | {
"source_record": "P8472",
"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":"2026-08-17T18:52:16+00:00","display_standard":{"formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":8472,"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": "2026-09-08T03:33:00.829Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"2d120103-b3de-43fd-bef6-9fcf03e2d2a3","format_issues":null,"number":5599} |
1.0.0 | problem:theoremdb-p8474 | problem | theoremdb-p8474 | Existence of a SIC-POVM in dimension 58 | {
"main": "math",
"sub": "algebra",
"slug": "algebra",
"name": "Algebra",
"original_discipline": "mathematics"
} | [] | [] | Let $d$ be a positive integer. A state vector in $\mathbb{C}^d$ is a vector of $d$ complex coordinates, and it is normalized if its Euclidean norm equals $1$. For two state vectors $\varphi$ and $\psi$ in $\mathbb{C}^d$, define their squared overlap to be the squared magnitude of their inner product, that is, $\bigl|\s... | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P8474",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P8474",
"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_58 : answer(sorry) ↔ HasSICPOVM 58 | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem hasSICPOVM_58 : answer(sorry) ↔ HasSICPOVM 58",
"target": null,
"source_url": "https://theoremdb.org/statements/P8474",
"format": "declaration-or-fragment",
"sha256": "7a927544480886936c1630c1df15e7b68cbd7729d31547a0c2cdbcb80bfe15b9",
"license": "... | [] | TheoremDB P8474 · 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 P8474",
"url": "https://theoremdb.org/statements/P8474",
"note": null,
"sort_order": 0
},
{
"kind": "formalization",
"citation": "Source formal statement (not locally verified)",
"url": "https://theoremdb.org/statements/P8474",
"not... | {
"source_record": "P8474",
"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":"2026-08-17T18:52:17+00:00","display_standard":{"editorial_review":"submitter_supplied","formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":8474,"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": "2026-09-08T03:32:58.492Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"1a5f2417-e8ea-4a28-b12e-e4e514b59771","format_issues":null,"number":5194} |
1.0.0 | problem:theoremdb-p8476 | problem | theoremdb-p8476 | Representability of 4 as a ratio of consecutive integer products | {
"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 $k \geq 2$, $n \geq 0$, and $m \geq n + k$, define the ratio of products
$$
R(k,n,m) = \frac{\prod_{i=1}^{k}(m+i)}{\prod_{i=1}^{k}(n+i)}.
$$
Does there exist $k \geq 2$, $n \in \mathbb{N}$, and $m \geq n + k$ such that $R(k,n,m) = 4$? | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P8476",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P8476",
"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_686.variants.four :
answer(sorry) ↔ ∃ᵉ (k ≥ 2) (n : ℕ) (m ≥ n + k),
(4 : ℚ) = (∏ i ∈ Finset.Icc 1 k, (m + i)) / (∏ i ∈ Finset.Icc 1 k, (n + i)) | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem erdos_686.variants.four :\n answer(sorry) ↔ ∃ᵉ (k ≥ 2) (n : ℕ) (m ≥ n + k),\n (4 : ℚ) = (∏ i ∈ Finset.Icc 1 k, (m + i)) / (∏ i ∈ Finset.Icc 1 k, (n + i))",
"target": null,
"source_url": "https://theoremdb.org/statements/P8476",
"format": "declarat... | [] | TheoremDB P8476 · 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 P8476",
"url": "https://theoremdb.org/statements/P8476",
"note": null,
"sort_order": 0
},
{
"kind": "formalization",
"citation": "Source formal statement (not locally verified)",
"url": "https://theoremdb.org/statements/P8476",
"not... | {
"source_record": "P8476",
"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":"2026-08-17T18:52:18+00:00","display_standard":{"editorial_review":"submitter_supplied","formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":8476,"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": "2026-09-08T03:33:05.613Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"bb3adf04-ef61-45cf-895e-cb1e1b17b12c","format_issues":null,"number":6565} |
1.0.0 | problem:theoremdb-p8478 | problem | theoremdb-p8478 | Representability of 25 as a ratio of consecutive integer products | {
"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 $k \geq 2$, $n \geq 0$, and $m \geq n + k$, consider the rational number
$$
\frac{\prod_{i=1}^{k}(m+i)}{\prod_{i=1}^{k}(n+i)}.
$$
Does there exist such $k$, $n$, and $m$ for which this ratio equals $25$? | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P8478",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P8478",
"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_686.variants.twenty_five :
answer(sorry) ↔ ∃ᵉ (k ≥ 2) (n : ℕ) (m ≥ n + k),
(25 : ℚ) = (∏ i ∈ Finset.Icc 1 k, (m + i)) / (∏ i ∈ Finset.Icc 1 k, (n + i)) | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem erdos_686.variants.twenty_five :\n answer(sorry) ↔ ∃ᵉ (k ≥ 2) (n : ℕ) (m ≥ n + k),\n (25 : ℚ) = (∏ i ∈ Finset.Icc 1 k, (m + i)) / (∏ i ∈ Finset.Icc 1 k, (n + i))",
"target": null,
"source_url": "https://theoremdb.org/statements/P8478",
"format": "... | [] | TheoremDB P8478 · 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 P8478",
"url": "https://theoremdb.org/statements/P8478",
"note": null,
"sort_order": 0
},
{
"kind": "formalization",
"citation": "Source formal statement (not locally verified)",
"url": "https://theoremdb.org/statements/P8478",
"not... | {
"source_record": "P8478",
"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":"2026-08-17T18:52:19+00:00","display_standard":{"editorial_review":"submitter_supplied","formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":8478,"public_status":{"evidence":{"hasFormalization":false,"hasInformalProof":false,"hasRecordedWork":false,"solutionRecord":null},"lean":{... | [
"formalization-contains-placeholder",
"formalization-not-compiled-during-export",
"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": "2026-09-08T03:33:05.609Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"4f8c0ba4-0870-4d57-9035-2da1d65d0f42","format_issues":null,"number":6564} |
1.0.0 | problem:theoremdb-p8480 | problem | theoremdb-p8480 | Popovych's variant of Agrawal'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 $n$ and $r$ be natural numbers with $n > 1$, $r > 0$, and $\gcd(n,r) = 1$. Let $\mathbb{Z}/n\mathbb{Z}$ denote the ring of integers modulo $n$, and let $R = (\mathbb{Z}/n\mathbb{Z})[X]$ be the polynomial ring in one indeterminate over $\mathbb{Z}/n\mathbb{Z}$. Let $I$ be the ideal of $R$ generated by $X^r - 1$, and... | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P8480",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P8480",
"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.variants.popovych :
∀ (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) →
Ideal.Quotient.mk I ((X + 2)^n) = Ide... | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem agrawal_conjecture.variants.popovych :\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.Quotient.mk I (X^... | [] | TheoremDB P8480 · 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 P8480",
"url": "https://theoremdb.org/statements/P8480",
"note": null,
"sort_order": 0
},
{
"kind": "formalization",
"citation": "Source formal statement (not locally verified)",
"url": "https://theoremdb.org/statements/P8480",
"not... | {
"source_record": "P8480",
"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":"2026-08-17T18:52:20+00:00","display_standard":{"formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":8480,"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": "2026-09-08T03:33:04.929Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"c5d95b06-b2b2-478a-aeb2-c4e25d312583","format_issues":null,"number":6417} |
1.0.0 | problem:theoremdb-p8482 | problem | theoremdb-p8482 | The liminf of a prime reciprocal sum 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 each positive integer $n$, define
$$
f(n)=\sum_{\substack{p<n\\ p\text{ prime}}}\frac{1}{n-p}.
$$
Determine whether $\liminf_{n\to\infty} f(n) = 1$. | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P8482",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P8482",
"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_950.parts.i :
answer(sorry) ↔ atTop.liminf (fun n : ℕ ↦ (f n : EReal)) = 1 | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem erdos_950.parts.i :\n answer(sorry) ↔ atTop.liminf (fun n : ℕ ↦ (f n : EReal)) = 1",
"target": null,
"source_url": "https://theoremdb.org/statements/P8482",
"format": "declaration-or-fragment",
"sha256": "7ce1d3e324cf06df72f60f7ded8b3d208ffd30c295f... | [] | TheoremDB P8482 · 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 P8482",
"url": "https://theoremdb.org/statements/P8482",
"note": null,
"sort_order": 0
},
{
"kind": "formalization",
"citation": "Source formal statement (not locally verified)",
"url": "https://theoremdb.org/statements/P8482",
"not... | {
"source_record": "P8482",
"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":"2026-08-17T18:52:21+00:00","display_standard":{"formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":8482,"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": "2026-09-08T03:33:06.612Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"3f31e45f-585a-4f10-8f2b-fa669f692833","format_issues":null,"number":6780} |
1.0.0 | problem:theoremdb-p8484 | problem | theoremdb-p8484 | Little-o asymptotic for the sigma-iteration prime-hitting time | {
"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 $n > 1$, let $c(n)$ denote the minimum number of iterations of the map $n \mapsto \sigma(n) - 1$ needed to reach a prime number, where $\sigma(n)$ denotes the sum-of-divisors function $\sigma(n) = \sum_{d \mid n} d$. Assuming that $c(n)$ is well-defined for all $n > 1$, determine the simplest function ... | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P8484",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P8484",
"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.variants.sigma_isLittleO (c : ℕ → ℕ)
(h : ∀ n > 1, IsLeast { i | (σ 1 · - 1)^[i] n |>.Prime } (c n)) :
(fun n => (c n : ℝ)) =o[atTop] (answer(sorry) : ℕ → ℝ) | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem erdos_409.variants.sigma_isLittleO (c : ℕ → ℕ)\n (h : ∀ n > 1, IsLeast { i | (σ 1 · - 1)^[i] n |>.Prime } (c n)) :\n (fun n => (c n : ℝ)) =o[atTop] (answer(sorry) : ℕ → ℝ)",
"target": null,
"source_url": "https://theoremdb.org/statements/P8484",
"fo... | [] | TheoremDB P8484 · 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 P8484",
"url": "https://theoremdb.org/statements/P8484",
"note": null,
"sort_order": 0
},
{
"kind": "formalization",
"citation": "Source formal statement (not locally verified)",
"url": "https://theoremdb.org/statements/P8484",
"not... | {
"source_record": "P8484",
"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":"2026-08-17T18:52:23+00:00","display_standard":{"editorial_review":"submitter_supplied","formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":8484,"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": "2026-09-08T03:33:03.283Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"fe1afe9b-f8b5-4cd7-a016-2018b4355bd5","format_issues":null,"number":6062} |
1.0.0 | problem:theoremdb-p8486 | problem | theoremdb-p8486 | Infinitude of consecutive 3-full and 2-full 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 $n$ and a prime $p$, let $v_p(n)$ denote the exponent of $p$ in the prime factorization of $n$. For an integer $k \geq 2$, a positive integer $n$ is called $k$\textit{-full} if for every prime $p$ dividing $n$, we have $v_p(n) \geq k$; equivalently, $n$ is $k$-full if and only if for every prime ... | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P8486",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P8486",
"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_366.variants.three_two :
answer(sorry) ↔ {n | (3).Full n ∧ (2).Full (n + 1)}.Infinite | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem erdos_366.variants.three_two :\n answer(sorry) ↔ {n | (3).Full n ∧ (2).Full (n + 1)}.Infinite",
"target": null,
"source_url": "https://theoremdb.org/statements/P8486",
"format": "declaration-or-fragment",
"sha256": "b9eabc774c985a6e0cd8c10b6d8a3fdb... | [] | TheoremDB P8486 · 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 P8486",
"url": "https://theoremdb.org/statements/P8486",
"note": null,
"sort_order": 0
},
{
"kind": "formalization",
"citation": "Source formal statement (not locally verified)",
"url": "https://theoremdb.org/statements/P8486",
"not... | {
"source_record": "P8486",
"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":"2026-08-17T18:52:24+00:00","display_standard":{"editorial_review":"submitter_supplied","formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":8486,"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": "2026-09-08T03:33:01.127Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"c671bd84-13c5-413c-8fb6-b6a8ac33d7dc","format_issues":null,"number":5663} |
1.0.0 | problem:theoremdb-p8488 | problem | theoremdb-p8488 | Schinzel's Hypothesis H | {
"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 $\mathbb{Z}[X]$ denote the ring of polynomials with integer coefficients. A polynomial $f \in \mathbb{Z}[X]$ satisfies the Bunyakovsky condition if: (i) the leading coefficient of $f$ is positive; (ii) $f$ is irreducible over $\mathbb{Z}$; and (iii) the values $\{f(n) : n \in \mathbb{Z}\}$ have no common divisor gr... | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P8488",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P8488",
"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 schinzel_conjecture (fs : Finset ℤ[X]) (hfs : ∀ f ∈ fs, BunyakovskyCondition f)
(hfs' : SchinzelCondition fs) : Infinite {n : ℕ | ∀ f ∈ fs, (f.eval (n : ℤ)).natAbs.Prime} | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem schinzel_conjecture (fs : Finset ℤ[X]) (hfs : ∀ f ∈ fs, BunyakovskyCondition f)\n (hfs' : SchinzelCondition fs) : Infinite {n : ℕ | ∀ f ∈ fs, (f.eval (n : ℤ)).natAbs.Prime}",
"target": null,
"source_url": "https://theoremdb.org/statements/P8488",
"form... | [] | TheoremDB P8488 · 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 P8488",
"url": "https://theoremdb.org/statements/P8488",
"note": null,
"sort_order": 0
},
{
"kind": "formalization",
"citation": "Source formal statement (not locally verified)",
"url": "https://theoremdb.org/statements/P8488",
"not... | {
"source_record": "P8488",
"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":"2026-08-17T18:52:26+00:00","display_standard":{"editorial_review":"submitter_supplied","formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":8488,"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": "2026-09-08T03:33:05.733Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"5aba0dde-dcd7-43a7-a1cc-413b03fe3e53","format_issues":null,"number":6591} |
1.0.0 | problem:theoremdb-p8490 | problem | theoremdb-p8490 | Uniform Distribution Preservation Under Independent Convolution | {
"main": "math",
"sub": "probability",
"slug": "probability",
"name": "Probability",
"original_discipline": "mathematics"
} | [] | [
{
"code": "60",
"basis": "community-map",
"evidence": "Inferred from community field: probability"
}
] | Let $X$ and $Y$ be finitely-supported independent random variables taking integer values, and suppose that their sum $X + Y$ is uniformly distributed on its range. Must $X$ and $Y$ each be uniformly distributed on their respective ranges? | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P8490",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P8490",
"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_28 : answer(sorry) ↔
∀ (X Y : PMF ℤ), -- marginals, independence is built into indepSum
X.support.Finite ∧ Y.support.Finite ∧ IsUniformOnSupport (indepSum X Y) →
IsUniformOnSupport X ∧ IsUniformOnSupport Y | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem green_28 : answer(sorry) ↔\n ∀ (X Y : PMF ℤ), -- marginals, independence is built into indepSum\n X.support.Finite ∧ Y.support.Finite ∧ IsUniformOnSupport (indepSum X Y) →\n IsUniformOnSupport X ∧ IsUniformOnSupport Y",
"target": null,
"source_url": ... | [] | TheoremDB P8490 · 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 P8490",
"url": "https://theoremdb.org/statements/P8490",
"note": null,
"sort_order": 0
},
{
"kind": "formalization",
"citation": "Source formal statement (not locally verified)",
"url": "https://theoremdb.org/statements/P8490",
"not... | {
"source_record": "P8490",
"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":"2026-08-17T18:52:26+00:00","display_standard":{"editorial_review":"submitter_supplied","formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":8490,"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": "2026-09-08T03:33:07.199Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"e1bdcc98-4e1f-4372-9c7d-6eee6f437e8a","format_issues":null,"number":6908} |
1.0.0 | problem:theoremdb-p8492 | problem | theoremdb-p8492 | Transcendence of the natural logarithm of pi | {
"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 $\log$ denote the natural logarithm and let $\pi$ denote the ratio of the circumference of a circle to its diameter. A real number $\alpha$ is said to be transcendental over the rational numbers $\mathbb{Q}$ if there is no non-zero polynomial $P(x)$ with rational coefficients such that $P(\alpha) = 0$. Prove that $... | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P8492",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P8492",
"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 rlog_pi_transcendental : Transcendental ℚ (log π) | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem rlog_pi_transcendental : Transcendental ℚ (log π)",
"target": null,
"source_url": "https://theoremdb.org/statements/P8492",
"format": "declaration-or-fragment",
"sha256": "051d214a89310785c91412921ab22693af9596687ad66cf4e833504aaa8d11ea",
"license... | [] | TheoremDB P8492 · 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 P8492",
"url": "https://theoremdb.org/statements/P8492",
"note": null,
"sort_order": 0
},
{
"kind": "formalization",
"citation": "Source formal statement (not locally verified)",
"url": "https://theoremdb.org/statements/P8492",
"not... | {
"source_record": "P8492",
"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":"2026-08-17T18:52:28+00:00","display_standard":{"editorial_review":"submitter_supplied","formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":8492,"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": "2026-09-08T03:33:07.017Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"ce360d3c-8217-45d3-8cb9-df27b6a8b16c","format_issues":null,"number":6867} |
1.0.0 | problem:theoremdb-p8494 | problem | theoremdb-p8494 | Ramsey Size Linearity of the Graph $H_5$ | {
"main": "math",
"sub": "combinatorics",
"slug": "combinatorics",
"name": "Combinatorics",
"original_discipline": "mathematics"
} | [] | [
{
"code": "05",
"basis": "community-map",
"evidence": "Inferred from community field: combinatorics"
}
] | Let $H_5$ denote the simple graph on the vertex set $\{0,1,2,3,4\}$ obtained from the 5-cycle $C_5$ by adding the two vertex-disjoint chords $\{0,2\}$ and $\{1,3\}$. A graph $G$ is said to be *Ramsey size linear* if there exists a constant $c > 0$ such that for every graph $H$ with $m$ edges and no isolated vertices, t... | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P8494",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P8494",
"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_567.parts.iii : answer(sorry) ↔ IsRamseySizeLinear H5 | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem erdos_567.parts.iii : answer(sorry) ↔ IsRamseySizeLinear H5",
"target": null,
"source_url": "https://theoremdb.org/statements/P8494",
"format": "declaration-or-fragment",
"sha256": "c1de861659752c8c3b7164dddb99a4edd1c72bc8b3b272a9c1e7c284dc0d4f9a",
... | [] | TheoremDB P8494 · 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 P8494",
"url": "https://theoremdb.org/statements/P8494",
"note": null,
"sort_order": 0
},
{
"kind": "formalization",
"citation": "Source formal statement (not locally verified)",
"url": "https://theoremdb.org/statements/P8494",
"not... | {
"source_record": "P8494",
"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":"2026-08-17T18:52:32+00:00","display_standard":{"editorial_review":"submitter_supplied","formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":8494,"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": "2026-09-08T03:33:05.409Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"7911c0a2-b063-4fc8-8bad-e1b86ceeb91f","format_issues":null,"number":6521} |
1.0.0 | problem:theoremdb-p8496 | problem | theoremdb-p8496 | Eventual periodicity of consecutive differences in Ulam's sequence | {
"main": "math",
"sub": "combinatorics",
"slug": "combinatorics",
"name": "Combinatorics",
"original_discipline": "mathematics"
} | [] | [
{
"code": "05",
"basis": "community-map",
"evidence": "Inferred from community field: combinatorics"
}
] | The Ulam sequence is the unique sequence of positive integers $(a(n))_{n \geq 0}$ defined by $a(0) = 1$, $a(1) = 2$, and for each $n \geq 2$, the term $a(n)$ is the least integer greater than $a(n-1)$ that can be expressed as $a(i) + a(j)$ for exactly one pair of indices $(i,j)$ with $i < j < n$. Define the difference ... | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P8496",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P8496",
"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_342.parts.ii :
answer(sorry) ↔
∀ a : ℕ → ℕ, IsUlamSequence a →
let d (n : ℕ) : ℤ := a (n + 1) - a n
∃ p > 0, ∀ᶠ m in atTop, d (m + p) = d m | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem erdos_342.parts.ii :\n answer(sorry) ↔\n ∀ a : ℕ → ℕ, IsUlamSequence a →\n let d (n : ℕ) : ℤ := a (n + 1) - a n\n ∃ p > 0, ∀ᶠ m in atTop, d (m + p) = d m",
"target": null,
"source_url": "https://theoremdb.org/statements/P8496",
"form... | [] | TheoremDB P8496 · 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 P8496",
"url": "https://theoremdb.org/statements/P8496",
"note": null,
"sort_order": 0
},
{
"kind": "formalization",
"citation": "Source formal statement (not locally verified)",
"url": "https://theoremdb.org/statements/P8496",
"not... | {
"source_record": "P8496",
"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":"2026-08-17T18:52:38+00:00","display_standard":{"formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":8496,"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": "2026-09-08T03:32:56.783Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"e9c948f4-1bb8-4007-af31-011a69f1e02a","format_issues":null,"number":4814} |
1.0.0 | problem:theoremdb-p8498 | problem | theoremdb-p8498 | Existence of a Hadamard matrix of order 668 | {
"main": "math",
"sub": "algebra",
"slug": "algebra",
"name": "Algebra",
"original_discipline": "mathematics"
} | [] | [] | A Hadamard matrix of order $n$ is an $n \times n$ matrix with entries in $\{1, -1\}$ whose determinant has absolute value equal to $n^{n/2}$. Equivalently, it is an $n \times n$ matrix with entries in $\{1, -1\}$ satisfying $M^{\mathsf{T}} M = n I_n$, where $I_n$ denotes the $n \times n$ identity matrix. It is known th... | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P8498",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P8498",
"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 HadamardConjecture.variants.«167» : ∃ M, IsHadamard (n := 4 * 167) M | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem HadamardConjecture.variants.«167» : ∃ M, IsHadamard (n := 4 * 167) M",
"target": null,
"source_url": "https://theoremdb.org/statements/P8498",
"format": "declaration-or-fragment",
"sha256": "e8353868f8446ccb6a1ace9f5375c6d1a84e32878b47e5a0b0a2e0931a29... | [] | TheoremDB P8498 · 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 P8498",
"url": "https://theoremdb.org/statements/P8498",
"note": null,
"sort_order": 0
},
{
"kind": "formalization",
"citation": "Source formal statement (not locally verified)",
"url": "https://theoremdb.org/statements/P8498",
"not... | {
"source_record": "P8498",
"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":"2026-08-17T18:52:47+00:00","display_standard":{"editorial_review":"submitter_supplied","formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":8498,"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": "2026-09-08T03:32:58.449Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"f5c874ee-f7e6-4362-92f4-f0e019d52f06","format_issues":null,"number":5185} |
1.0.0 | problem:theoremdb-p8500 | problem | theoremdb-p8500 | Unboundedness of the limsup of prime-index deviations in symmetric 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 $n \geq 2$, define $f(n)$ to be the minimum value of $p_{n+i} + p_{n-i}$ over all integers $i$ with $0 < i < n$, where $p_k$ denotes the $k$th prime number. Determine whether
$$
\limsup_{n \to \infty} \bigl(f(n) - 2p_n\bigr) = +\infty.
$$
That is, does the quantity $f(n) - 2p_n$ attain arbitrarily ... | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P8500",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P8500",
"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_454 : answer(sorry) ↔ limsup (fun n => (f n - 2 * n.nth Prime : ℕ∞)) atTop = ⊤ | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem erdos_454 : answer(sorry) ↔ limsup (fun n => (f n - 2 * n.nth Prime : ℕ∞)) atTop = ⊤",
"target": null,
"source_url": "https://theoremdb.org/statements/P8500",
"format": "declaration-or-fragment",
"sha256": "7f9d8879f894b6d2343bd60a61a1f01d76928c895690... | [] | TheoremDB P8500 · 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 P8500",
"url": "https://theoremdb.org/statements/P8500",
"note": null,
"sort_order": 0
},
{
"kind": "formalization",
"citation": "Source formal statement (not locally verified)",
"url": "https://theoremdb.org/statements/P8500",
"not... | {
"source_record": "P8500",
"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":"2026-08-17T18:52:54+00:00","display_standard":{"editorial_review":"submitter_supplied","formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":8500,"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": "2026-09-08T03:33:07.171Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"635f9b54-a889-45c2-a0ab-278651e44fd4","format_issues":null,"number":6902} |
1.0.0 | problem:theoremdb-p8502 | problem | theoremdb-p8502 | Erdős–Hall conjecture on the small-o rate for sums of least products | {
"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 $n$, let $t_2(n)$ denote the least positive integer $m$ such that $n$ divides the product $m(m+1)$. Is it true that for every real number $c$ with $c < \log 2$, the sum $\sum_{n \leq x} t_2(n)$ is $o\left(\frac{x^2}{(\log x)^c}\right)$ as $x \to \infty$? In other words, does
$$
\sum_{n \leq x} t... | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P8502",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P8502",
"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_394.variants.hall_conjecture :
∀ c < Real.log 2, (fun x ↦ ∑ n ∈ Icc 1 ⌊x⌋₊,
(t 2 n : ℝ)) =o[atTop]
(fun x ↦ x ^ 2 / (Real.log x) ^ c) | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem erdos_394.variants.hall_conjecture :\n ∀ c < Real.log 2, (fun x ↦ ∑ n ∈ Icc 1 ⌊x⌋₊,\n (t 2 n : ℝ)) =o[atTop]\n (fun x ↦ x ^ 2 / (Real.log x) ^ c)",
"target": null,
"source_url": "https://theoremdb.org/statements/P8502",
"format": "declaration-or-... | [] | TheoremDB P8502 · 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 P8502",
"url": "https://theoremdb.org/statements/P8502",
"note": null,
"sort_order": 0
},
{
"kind": "formalization",
"citation": "Source formal statement (not locally verified)",
"url": "https://theoremdb.org/statements/P8502",
"not... | {
"source_record": "P8502",
"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":"2026-08-17T18:53:04+00:00","display_standard":{"editorial_review":"submitter_supplied","formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":8502,"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": "2026-09-08T03:32:56.682Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"6729a72e-69c8-4227-a897-baac722bed38","format_issues":null,"number":4792} |
1.0.0 | problem:theoremdb-p8504 | problem | theoremdb-p8504 | Andrica's Conjecture on Prime Gaps | {
"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 natural number $n$, let $p_n$ denote the $n$-th prime number, where the primes are enumerated in increasing order as $p_1 = 2, p_2 = 3, p_3 = 5, \ldots$. Prove that for all $n \geq 1$, the inequality
$$
\sqrt{p_{n+1}} - \sqrt{p_n} < 1
$$
holds. | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P8504",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P8504",
"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 andrica_conjecture (n : ℕ) :
Real.sqrt ((n+1).nth Nat.Prime) - Real.sqrt (n.nth Nat.Prime) < 1 | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem andrica_conjecture (n : ℕ) :\n Real.sqrt ((n+1).nth Nat.Prime) - Real.sqrt (n.nth Nat.Prime) < 1",
"target": null,
"source_url": "https://theoremdb.org/statements/P8504",
"format": "declaration-or-fragment",
"sha256": "031c1a122f29726176ef0594a27bd... | [] | TheoremDB P8504 · 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 P8504",
"url": "https://theoremdb.org/statements/P8504",
"note": null,
"sort_order": 0
},
{
"kind": "formalization",
"citation": "Source formal statement (not locally verified)",
"url": "https://theoremdb.org/statements/P8504",
"not... | {
"source_record": "P8504",
"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":"2026-08-17T18:53:13+00:00","display_standard":{"editorial_review":"submitter_supplied","formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":8504,"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": "2026-09-08T03:32:49.637Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"56e5727b-c84d-4138-918f-cc7948f84264","format_issues":null,"number":3251} |
1.0.0 | problem:theoremdb-p8506 | problem | theoremdb-p8506 | Infinitude of Consecutive Equal Totient Values | {
"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, which counts the number of integers from $1$ to $n$ that are coprime to $n$. Determine whether the set $\{n \in \mathbb{N} \mid \varphi(n) = \varphi(n+1)\}$ is infinite. | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P8506",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P8506",
"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_1003 : answer(sorry) ↔ Set.Infinite {n | φ n = φ (n + 1)} | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem erdos_1003 : answer(sorry) ↔ Set.Infinite {n | φ n = φ (n + 1)}",
"target": null,
"source_url": "https://theoremdb.org/statements/P8506",
"format": "declaration-or-fragment",
"sha256": "a7bba242db7ea53c19d36a572145d605d602fe0f4f202277071ec47af60d8a26"... | [] | TheoremDB P8506 · 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 P8506",
"url": "https://theoremdb.org/statements/P8506",
"note": null,
"sort_order": 0
},
{
"kind": "formalization",
"citation": "Source formal statement (not locally verified)",
"url": "https://theoremdb.org/statements/P8506",
"not... | {
"source_record": "P8506",
"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":"2026-08-17T18:53:20+00:00","display_standard":{"editorial_review":"submitter_supplied","formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":8506,"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": "2026-09-08T03:33:01.137Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"61ee4f7b-75d5-4ec1-a22b-213039039c41","format_issues":null,"number":5665} |
1.0.0 | problem:theoremdb-p8508 | problem | theoremdb-p8508 | Sufficient Conditions for Bounded Gaps in Difference 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"
}
] | For a set $A \subseteq \mathbb{N}$, let $D(A)$ denote the set of integers $d$ such that the equation $a_1 - a_2 = d$ has infinitely many solutions with $a_1, a_2 \in A$. A set $S \subseteq \mathbb{Z}$ has bounded gaps if there exists a positive integer $M$ such that every interval $[z, z+M)$ with $z \in \mathbb{Z}$ con... | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P8508",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P8508",
"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_332 (A : Set ℕ) : (answer(sorry) : Set ℕ → Prop) A → HasBoundedGaps (D_A A) | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem erdos_332 (A : Set ℕ) : (answer(sorry) : Set ℕ → Prop) A → HasBoundedGaps (D_A A)",
"target": null,
"source_url": "https://theoremdb.org/statements/P8508",
"format": "declaration-or-fragment",
"sha256": "f8d04d8b3ae5ca879322c2e6a557a0766d93f77ba6f9fa3... | [] | TheoremDB P8508 · 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 P8508",
"url": "https://theoremdb.org/statements/P8508",
"note": null,
"sort_order": 0
},
{
"kind": "formalization",
"citation": "Source formal statement (not locally verified)",
"url": "https://theoremdb.org/statements/P8508",
"not... | {
"source_record": "P8508",
"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":"2026-08-17T18:53:29+00:00","display_standard":{"editorial_review":"submitter_supplied","formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":8508,"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": "2026-09-08T03:33:06.163Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"3b01a28d-c8c7-4cbf-b4b1-11f834575c06","format_issues":null,"number":6684} |
1.0.0 | problem:theoremdb-p8510 | problem | theoremdb-p8510 | Maximum Number of Affine Translates of {0,1,3} | {
"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 $n$, let $M(n)$ denote the maximum number of affine translates of the set $\{0,1,3\}$ that can be contained in a set $A$ of $n$ integers. More precisely, for a finite set $A \subset \mathbb{Z}$ with $|A| = n$, count the number of ordered pairs $(x,y) \in A \times A$ with $x \neq y$ such that $... | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P8510",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P8510",
"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_24 : ∀ n, max013AffineTranslates n = answer(sorry) | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem green_24 : ∀ n, max013AffineTranslates n = answer(sorry)",
"target": null,
"source_url": "https://theoremdb.org/statements/P8510",
"format": "declaration-or-fragment",
"sha256": "146cd0f12f3a4768d09a5479183065241e8d4d6a4687435e448e9f78167f7685",
"... | [] | TheoremDB P8510 · 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 P8510",
"url": "https://theoremdb.org/statements/P8510",
"note": null,
"sort_order": 0
},
{
"kind": "formalization",
"citation": "Source formal statement (not locally verified)",
"url": "https://theoremdb.org/statements/P8510",
"not... | {
"source_record": "P8510",
"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":"2026-08-17T18:53:36+00:00","display_standard":{"formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":8510,"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": "2026-09-08T03:33:03.804Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"8f949d50-189f-4bbb-80f7-007092aff636","format_issues":null,"number":6175} |
1.0.0 | problem:theoremdb-p8512 | problem | theoremdb-p8512 | Erdős's Problem on Thin Sumsets with Large Lower Density | {
"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 set $A \subseteq \mathbb{N}$, let $A + A = \{a + b : a, b \in A\}$ denote its sumset. The lower density of a set $S \subseteq \mathbb{N}$ is defined as $\underline{d}(S) = \liminf_{N \to \infty} \frac{|S \cap \{1, 2, \ldots, N\}|}{N}$. For $n \in \mathbb{N}$, define the representation function $r_A(n) = |\{(a, b)... | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P8512",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P8512",
"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_749 : answer(sorry) ↔ ∀ ε > (0 : ℝ),
∃ A : Set ℕ, 1 - ε ≤ lowerDensity (A + A) ∧
((Nat.cast (R := ℝ) ∘ sumRep A) ≪ (fun n => (1: ℝ))) | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem erdos_749 : answer(sorry) ↔ ∀ ε > (0 : ℝ),\n ∃ A : Set ℕ, 1 - ε ≤ lowerDensity (A + A) ∧\n ((Nat.cast (R := ℝ) ∘ sumRep A) ≪ (fun n => (1: ℝ)))",
"target": null,
"source_url": "https://theoremdb.org/statements/P8512",
"format": "declaration-or-fragm... | [] | TheoremDB P8512 · 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 P8512",
"url": "https://theoremdb.org/statements/P8512",
"note": null,
"sort_order": 0
},
{
"kind": "formalization",
"citation": "Source formal statement (not locally verified)",
"url": "https://theoremdb.org/statements/P8512",
"not... | {
"source_record": "P8512",
"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":"2026-08-17T18:53:43+00:00","display_standard":{"editorial_review":"submitter_supplied","formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":8512,"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": "2026-09-08T03:32:56.563Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"66e1a8f2-ccfe-4145-b2a7-0325c83110c4","format_issues":null,"number":4767} |
1.0.0 | problem:theoremdb-p8514 | problem | theoremdb-p8514 | Existence of a SIC-POVM in dimension 56 | {
"main": "math",
"sub": "algebra",
"slug": "algebra",
"name": "Algebra",
"original_discipline": "mathematics"
} | [] | [] | A state vector in the $d$-dimensional complex Hilbert space $\mathbb{C}^d$ is a unit vector. The squared overlap between two state vectors $\varphi$ and $\psi$ is the squared magnitude of their inner product, denoted $\lvert\langle\varphi|\psi\rangle\rvert^2$. A family of $N$ state vectors in $\mathbb{C}^d$ has constan... | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P8514",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P8514",
"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_56 : answer(sorry) ↔ HasSICPOVM 56 | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem hasSICPOVM_56 : answer(sorry) ↔ HasSICPOVM 56",
"target": null,
"source_url": "https://theoremdb.org/statements/P8514",
"format": "declaration-or-fragment",
"sha256": "e1e6ee824130c0c61f23c20a86d6e8b79db4c972bf4070997b351d8988d8de95",
"license": "... | [] | TheoremDB P8514 · 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 P8514",
"url": "https://theoremdb.org/statements/P8514",
"note": null,
"sort_order": 0
},
{
"kind": "formalization",
"citation": "Source formal statement (not locally verified)",
"url": "https://theoremdb.org/statements/P8514",
"not... | {
"source_record": "P8514",
"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":"2026-08-17T18:53:52+00:00","display_standard":{"editorial_review":"submitter_supplied","formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":8514,"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": "2026-09-08T03:32:58.487Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"912c8607-4bf4-4556-851f-cb28131ae1a2","format_issues":null,"number":5193} |
1.0.0 | problem:theoremdb-p8516 | problem | theoremdb-p8516 | The Littlewood conjecture on simultaneous Diophantine approximation | {
"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 two real numbers $\alpha$ and $\beta$, define the distance to the nearest integer of a real number $x$ by $\|x\| = \min(|x - \lfloor x \rfloor|, |x - \lceil x \rceil|)$. Determine whether it is true that $\displaystyle\liminf_{n\to\infty} \, n \cdot \|n\alpha\| \cdot \|n\beta\| = 0.$. | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P8516",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P8516",
"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 littlewood_conjecture (α β : ℝ) :
atTop.liminf (fun (n : ℕ) ↦ n * distToNearestInt (n * α) * distToNearestInt (n * β)) = 0 | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem littlewood_conjecture (α β : ℝ) :\n atTop.liminf (fun (n : ℕ) ↦ n * distToNearestInt (n * α) * distToNearestInt (n * β)) = 0",
"target": null,
"source_url": "https://theoremdb.org/statements/P8516",
"format": "declaration-or-fragment",
"sha256": "2... | [] | TheoremDB P8516 · 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 P8516",
"url": "https://theoremdb.org/statements/P8516",
"note": null,
"sort_order": 0
},
{
"kind": "formalization",
"citation": "Source formal statement (not locally verified)",
"url": "https://theoremdb.org/statements/P8516",
"not... | {
"source_record": "P8516",
"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":"2026-08-17T18:53:59+00:00","display_standard":{"formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":8516,"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": "2026-09-08T03:33:06.635Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"dce49b02-810a-41b4-a54d-8d500799961a","format_issues":null,"number":6785} |
1.0.0 | problem:theoremdb-p8518 | problem | theoremdb-p8518 | Consecutive 3-full 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"
}
] | A positive integer $n$ is called $k$-full if for every prime $p$ dividing $n$, the exponent of $p$ in the prime factorization of $n$ is at least $k$. Does there exist a positive integer $n$ such that both $n$ and $n+1$ are $3$-full? | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P8518",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P8518",
"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_366.variants.weaker : answer(sorry) ↔
∃ n > 0, (3).Full n ∧ (3).Full (n + 1) | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem erdos_366.variants.weaker : answer(sorry) ↔\n ∃ n > 0, (3).Full n ∧ (3).Full (n + 1)",
"target": null,
"source_url": "https://theoremdb.org/statements/P8518",
"format": "declaration-or-fragment",
"sha256": "7bdb0fb0aeced4e858264fec6288f5a7ed4f30117... | [] | TheoremDB P8518 · 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 P8518",
"url": "https://theoremdb.org/statements/P8518",
"note": null,
"sort_order": 0
},
{
"kind": "formalization",
"citation": "Source formal statement (not locally verified)",
"url": "https://theoremdb.org/statements/P8518",
"not... | {
"source_record": "P8518",
"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":"2026-08-17T18:54:06+00:00","display_standard":{"editorial_review":"submitter_supplied","formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":8518,"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": "2026-09-08T03:32:52.601Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"2fd89074-adba-42d0-ac98-921fc6a806eb","format_issues":null,"number":3895} |
1.0.0 | problem:theoremdb-p8520 | problem | theoremdb-p8520 | Erdős's Conjecture on Products 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 an integer $N \geq 2$, we ask whether $N$ can be represented as a ratio of products of $k$ consecutive positive integers:
$$
N = \frac{(m+1)(m+2)\cdots(m+k)}{(n+1)(n+2)\cdots(n+k)}
$$
for some integers $k \geq 2$, $n \geq 0$, and $m \geq n+k$. Does this representation exist for every integer $N \geq 2$? | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P8520",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P8520",
"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_686 :
answer(sorry) ↔ ∀ N ≥ (2 : ℕ), ∃ᵉ (k ≥ 2) (n : ℕ) (m ≥ n + k),
(N : ℚ) = (∏ i ∈ Finset.Icc 1 k, (m + i)) / (∏ i ∈ Finset.Icc 1 k, (n + i)) | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem erdos_686 :\n answer(sorry) ↔ ∀ N ≥ (2 : ℕ), ∃ᵉ (k ≥ 2) (n : ℕ) (m ≥ n + k),\n (N : ℚ) = (∏ i ∈ Finset.Icc 1 k, (m + i)) / (∏ i ∈ Finset.Icc 1 k, (n + i))",
"target": null,
"source_url": "https://theoremdb.org/statements/P8520",
"format": "declara... | [] | TheoremDB P8520 · 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 P8520",
"url": "https://theoremdb.org/statements/P8520",
"note": null,
"sort_order": 0
},
{
"kind": "formalization",
"citation": "Source formal statement (not locally verified)",
"url": "https://theoremdb.org/statements/P8520",
"not... | {
"source_record": "P8520",
"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":"2026-08-17T18:54:08+00:00","display_standard":{"editorial_review":"submitter_supplied","formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":8520,"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": "2026-09-08T03:32:56.356Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"f2bfc9ba-3baf-4542-990d-b5c964f238b9","format_issues":null,"number":4723} |
1.0.0 | problem:theoremdb-p8522 | problem | theoremdb-p8522 | Asymptotic Density of Divisor Sum 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"
}
] | For a positive integer $t$, define the divisor sum set $\mathcal{D}(t)$ to be the set of natural numbers $n$ for which $t$ can be expressed as a sum of distinct divisors of $n$. That is,
$$
\mathcal{D}(t) = \left\{ n \in \mathbb{N} : \exists\, S \subseteq \{ d \in \mathbb{N} : d \mid n \},\; t = \sum_{d \in S} d \righ... | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P8522",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P8522",
"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_859 :
∃ c₁ > 0, ∃ c₂ > (0 : ℝ), ∃ d : ℕ → ℝ, (∀ t > 0, (DivisorSumSet t).HasDensity (d t)) ∧
(fun (t : ℕ) ↦ d t) ~[atTop] (fun t ↦ c₁ / Real.log t ^ c₂) | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem erdos_859 :\n ∃ c₁ > 0, ∃ c₂ > (0 : ℝ), ∃ d : ℕ → ℝ, (∀ t > 0, (DivisorSumSet t).HasDensity (d t)) ∧\n (fun (t : ℕ) ↦ d t) ~[atTop] (fun t ↦ c₁ / Real.log t ^ c₂)",
"target": null,
"source_url": "https://theoremdb.org/statements/P8522",
"format": ... | [] | TheoremDB P8522 · 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 P8522",
"url": "https://theoremdb.org/statements/P8522",
"note": null,
"sort_order": 0
},
{
"kind": "formalization",
"citation": "Source formal statement (not locally verified)",
"url": "https://theoremdb.org/statements/P8522",
"not... | {
"source_record": "P8522",
"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":"2026-08-17T18:54:09+00:00","display_standard":{"editorial_review":"submitter_supplied","formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":8522,"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": "2026-09-08T03:32:49.997Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"344765c8-dbb8-44d6-b3a5-6430e7ad183f","format_issues":null,"number":3329} |
1.0.0 | problem:theoremdb-p8524 | problem | theoremdb-p8524 | Infinitude of primary pseudoperfect 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 $n$ is called a primary pseudoperfect number if there exists a non-empty finite set $P$ of distinct prime numbers such that
$$
\sum_{p \in P} \frac{1}{p} = 1 - \frac{1}{n}.
$$
Prove that there are infinitely many primary pseudoperfect numbers. | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P8524",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P8524",
"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_313.variants.primary_pseudoperfect_are_infinite :
Set.Infinite {n | IsPrimaryPseudoperfect n} | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem erdos_313.variants.primary_pseudoperfect_are_infinite :\n Set.Infinite {n | IsPrimaryPseudoperfect n}",
"target": null,
"source_url": "https://theoremdb.org/statements/P8524",
"format": "declaration-or-fragment",
"sha256": "bce51587e22197265ecd8aba... | [] | TheoremDB P8524 · 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 P8524",
"url": "https://theoremdb.org/statements/P8524",
"note": null,
"sort_order": 0
},
{
"kind": "formalization",
"citation": "Source formal statement (not locally verified)",
"url": "https://theoremdb.org/statements/P8524",
"not... | {
"source_record": "P8524",
"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":"2026-08-17T18:54:11+00:00","display_standard":{"editorial_review":"submitter_supplied","formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":8524,"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": "2026-09-08T03:33:01.190Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"a43ffcff-03d9-4fc3-80d6-e27e17e2834d","format_issues":null,"number":5677} |
1.0.0 | problem:theoremdb-p8526 | problem | theoremdb-p8526 | Density ratio inequality for multiples of a finite set | {
"main": "math",
"sub": "combinatorics",
"slug": "combinatorics",
"name": "Combinatorics",
"original_discipline": "mathematics"
} | [] | [
{
"code": "05",
"basis": "community-map",
"evidence": "Inferred from community field: combinatorics"
}
] | Let $A$ be a nonempty finite set of natural numbers with $0 \notin A$ and $1 \notin A$. Define
$$
B = \{n \geq 1 : a \mid n \text{ for some } a \in A\}.
$$
For positive integers $n$ and $m$ with $m > n \geq \max(A)$, does the inequality
$$
\frac{|B \cap [1,m]|}{m} < 2 \cdot \frac{|B \cap [1,n]|}{n}
$$
always hold? | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P8526",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P8526",
"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_488 : answer(sorry) ↔ ∀ (A : Finset ℕ), A.Nonempty →
-- These are needed for the reasons outlined here: https://github.com/google-deepmind/formal-conjectures/pull/256
0 ∉ A → 1 ∉ A →
letI B := {n ≥ 1 | ∃ a ∈ A, a ∣ n}
∀ᵉ (n : ℕ) (m > n), A.max ≤ n →
((Finset.Icc 1 m).filter (· ∈ B)).... | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem erdos_488 : answer(sorry) ↔ ∀ (A : Finset ℕ), A.Nonempty →\n -- These are needed for the reasons outlined here: https://github.com/google-deepmind/formal-conjectures/pull/256\n 0 ∉ A → 1 ∉ A →\n letI B := {n ≥ 1 | ∃ a ∈ A, a ∣ n}\n ∀ᵉ (n : ℕ) (m > n), A.m... | [] | TheoremDB P8526 · 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 P8526",
"url": "https://theoremdb.org/statements/P8526",
"note": null,
"sort_order": 0
},
{
"kind": "formalization",
"citation": "Source formal statement (not locally verified)",
"url": "https://theoremdb.org/statements/P8526",
"not... | {
"source_record": "P8526",
"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":"2026-08-17T18:54:12+00:00","display_standard":{"editorial_review":"submitter_supplied","formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":8526,"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": "2026-09-08T03:32:53.123Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"55a4c497-fc9c-466b-9192-a6c8074cd895","format_issues":null,"number":4008} |
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