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1.0.0 | problem:theoremdb-p8528 | problem | theoremdb-p8528 | Erdős Problem 124: Sums of Distinct Powers with Nonzero Exponent Shift | {
"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 $k$ be a nonzero integer, and let $D$ be a finite set of integers, each at least $3$, such that
$$
\sum_{d \in D} \frac{1}{d-1} \geq 1
$$
and $\gcd(D) = 1$. For each $d \in D$ and each integer $k \geq 0$, let $S(d,k)$ denote the set of natural numbers that can be written as
$$
\sum_{i \in s} d^i
$$
for some fin... | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P8528",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P8528",
"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... | lemma erdos124.ne_zero : answer(sorry) ↔
∀ k ≠ 0, ∀ D : Finset ℕ, (∀ d ∈ D, 3 ≤ d) → 1 ≤ ∑ d ∈ D, (d - 1 : ℚ)⁻¹ → D.gcd id = 1 →
∀ᶠ n in atTop, n ∈ ∑ d ∈ D, sumsOfDistinctPowers d k | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "lemma erdos124.ne_zero : answer(sorry) ↔\n ∀ k ≠ 0, ∀ D : Finset ℕ, (∀ d ∈ D, 3 ≤ d) → 1 ≤ ∑ d ∈ D, (d - 1 : ℚ)⁻¹ → D.gcd id = 1 →\n ∀ᶠ n in atTop, n ∈ ∑ d ∈ D, sumsOfDistinctPowers d k",
"target": null,
"source_url": "https://theoremdb.org/statements/P8528",... | [] | TheoremDB P8528 · 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 P8528",
"url": "https://theoremdb.org/statements/P8528",
"note": null,
"sort_order": 0
},
{
"kind": "formalization",
"citation": "Source formal statement (not locally verified)",
"url": "https://theoremdb.org/statements/P8528",
"not... | {
"source_record": "P8528",
"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":"2026-08-17T18:54:14+00:00","display_standard":{"formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":8528,"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": "2026-09-08T03:32:55.126Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"82cb04b7-ed2b-4c60-8793-6010621cd24d","format_issues":null,"number":4452} |
1.0.0 | problem:theoremdb-p8530 | problem | theoremdb-p8530 | Existence of a positive natural density for primitive-root 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"
}
] | Let $a$ be an integer that is not a perfect square and not equal to $-1$. For a prime $p$, let $a$ be regarded as an element of the finite field $\mathbb{Z}/p\mathbb{Z}$ via reduction modulo $p$ (when $p\nmid a$). We say that $a$ is a primitive root modulo $p$ if the multiplicative order of $a$ in $(\mathbb{Z}/p\mathbb... | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P8530",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P8530",
"citation": null,
"doi": null,
"arxiv_id": null,
"license": "Selected packets label their content CC0-1.0; original references retain their own terms",
"notes": "Selected problem... | theorem artin_primitive_roots.parts.i (a : ℤ) (ha : ¬IsSquare a) (ha' : a ≠ -1) :
∃ x > 0, (S a).HasDensity x {p | p.Prime} | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem artin_primitive_roots.parts.i (a : ℤ) (ha : ¬IsSquare a) (ha' : a ≠ -1) :\n ∃ x > 0, (S a).HasDensity x {p | p.Prime}",
"target": null,
"source_url": "https://theoremdb.org/statements/P8530",
"format": "declaration-or-fragment",
"sha256": "556d4487... | [] | TheoremDB P8530 · 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 P8530",
"url": "https://theoremdb.org/statements/P8530",
"note": null,
"sort_order": 0
},
{
"kind": "formalization",
"citation": "Source formal statement (not locally verified)",
"url": "https://theoremdb.org/statements/P8530",
"not... | {
"source_record": "P8530",
"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":"2026-08-17T18:54:15+00:00","display_standard":{"editorial_review":"submitter_supplied","formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":8530,"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": "2026-09-08T03:32:58.472Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"3feb936f-031b-4a7f-a6d6-56e6b211dd26","format_issues":null,"number":5190} |
1.0.0 | problem:theoremdb-p8532 | problem | theoremdb-p8532 | Density-zero complement membership in the difference set of the greedy Sidon sequence | {
"main": "math",
"sub": "combinatorics",
"slug": "combinatorics",
"name": "Combinatorics",
"original_discipline": "mathematics"
} | [] | [
{
"code": "05",
"basis": "community-map",
"evidence": "Inferred from community field: combinatorics"
}
] | Let $A = \{1, 2, 4, 8, 13, 21, 31, 45, 66, 81, 97, \ldots\}$ be the greedy Sidon sequence, defined by starting with $1$ and iteratively including the next smallest integer that preserves the Sidon property (that is, there are no non-trivial solutions to $a + b = c + d$). Let $A - A = \{a - b : a, b \in A\}$ denote the ... | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P8532",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P8532",
"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_340.variants.co_density_zero_sub : answer(sorry) ↔
∃ S : Set ℕ, S.HasDensity 0 ∧ ∀ n ∈ Sᶜ, n ∈ Set.range greedySidon - Set.range greedySidon | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem erdos_340.variants.co_density_zero_sub : answer(sorry) ↔\n ∃ S : Set ℕ, S.HasDensity 0 ∧ ∀ n ∈ Sᶜ, n ∈ Set.range greedySidon - Set.range greedySidon",
"target": null,
"source_url": "https://theoremdb.org/statements/P8532",
"format": "declaration-or-fra... | [] | TheoremDB P8532 · 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 P8532",
"url": "https://theoremdb.org/statements/P8532",
"note": null,
"sort_order": 0
},
{
"kind": "formalization",
"citation": "Source formal statement (not locally verified)",
"url": "https://theoremdb.org/statements/P8532",
"not... | {
"source_record": "P8532",
"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":"2026-08-17T18:54:19+00:00","display_standard":{"editorial_review":"submitter_supplied","formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":8532,"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": "2026-09-08T03:32:53.132Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"2f202ead-f8a9-4aa0-930d-b6563a92c597","format_issues":null,"number":4010} |
1.0.0 | problem:theoremdb-p8534 | problem | theoremdb-p8534 | Prime pairs in dyadic intervals with half-prime companion | {
"main": "math",
"sub": "number theory",
"slug": "number-theory",
"name": "Number theory",
"original_discipline": "mathematics"
} | [] | [
{
"code": "11",
"basis": "community-map",
"evidence": "Inferred from community field: number-theory"
}
] | For a natural number $n$, we say that $n$ is a prime number if $n > 1$ and $n$ has no positive divisors other than $1$ and $n$. Determine whether there exists a natural number $N_0$ such that for every natural number $N$ with $N_0 \leq N$, there exists a prime number $p$ satisfying $N \leq p \leq 2N$ and such that $\fr... | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P8534",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P8534",
"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_287.variants.prime_conjecture : answer(sorry) ↔
∃ N₀ : ℕ, ∀ N : ℕ, N₀ ≤ N →
∃ p : ℕ, Nat.Prime p ∧ N ≤ p ∧ p ≤ 2 * N ∧ Nat.Prime ((p + 1) / 2) | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem erdos_287.variants.prime_conjecture : answer(sorry) ↔\n ∃ N₀ : ℕ, ∀ N : ℕ, N₀ ≤ N →\n ∃ p : ℕ, Nat.Prime p ∧ N ≤ p ∧ p ≤ 2 * N ∧ Nat.Prime ((p + 1) / 2)",
"target": null,
"source_url": "https://theoremdb.org/statements/P8534",
"format": "declarati... | [] | TheoremDB P8534 · 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 P8534",
"url": "https://theoremdb.org/statements/P8534",
"note": null,
"sort_order": 0
},
{
"kind": "formalization",
"citation": "Source formal statement (not locally verified)",
"url": "https://theoremdb.org/statements/P8534",
"not... | {
"source_record": "P8534",
"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":"2026-08-17T18:54:20+00:00","display_standard":{"formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":8534,"public_status":{"evidence":{"hasFormalization":false,"hasInformalProof":false,"hasRecordedWork":false,"solutionRecord":null},"lean":{"kernelAccepted":false,"label":"","reaso... | [
"formalization-contains-placeholder",
"formalization-not-compiled-during-export",
"current-status-unassessed",
"source-full-text-not-present"
] | [] | {
"created_at": "2026-09-08T03:32:15.935Z",
"updated_at": "2026-09-08T03:32:15.935Z",
"published_at": "2026-09-08T03:33:05.124Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"bb2175c7-a88b-4af2-9541-9d10b5552afd","format_issues":null,"number":6459} |
1.0.0 | problem:theoremdb-p8536 | problem | theoremdb-p8536 | Erdős's squarefree–power-of-two sum problem for integers not divisible by four | {
"main": "math",
"sub": "number theory",
"slug": "number-theory",
"name": "Number theory",
"original_discipline": "mathematics"
} | [] | [
{
"code": "11",
"basis": "community-map",
"evidence": "Inferred from community field: number-theory"
}
] | Let $n$ be a natural number such that $1 < n$ and $4$ does not divide $n$. A natural number is called squarefree if it is not divisible by the square of any prime number. Does there exist a squarefree natural number $k$ and a non-negative integer $l$ such that $n = k + 2^l$? | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P8536",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P8536",
"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_11.variants.not_four_dvd (n : ℕ) (hn : ¬ 4 ∣ n) (hn' : 1 < n) :
∃ k l : ℕ , Squarefree k ∧ n = k + 2^l | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem erdos_11.variants.not_four_dvd (n : ℕ) (hn : ¬ 4 ∣ n) (hn' : 1 < n) :\n ∃ k l : ℕ , Squarefree k ∧ n = k + 2^l",
"target": null,
"source_url": "https://theoremdb.org/statements/P8536",
"format": "declaration-or-fragment",
"sha256": "65a339b0e5a00d3... | [] | TheoremDB P8536 · 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 P8536",
"url": "https://theoremdb.org/statements/P8536",
"note": null,
"sort_order": 0
},
{
"kind": "formalization",
"citation": "Source formal statement (not locally verified)",
"url": "https://theoremdb.org/statements/P8536",
"not... | {
"source_record": "P8536",
"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":"2026-08-17T18:54:21+00:00","display_standard":{"editorial_review":"submitter_supplied","formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":8536,"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": "2026-09-08T03:32:56.585Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"62f6b37a-1301-49c7-96e6-ba0947e8862f","format_issues":null,"number":4772} |
1.0.0 | problem:theoremdb-p8538 | problem | theoremdb-p8538 | Infinitude of perfect 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 perfect if it equals the sum of its proper positive divisors (that is, all positive divisors of the number excluding the number itself). For example, 6 is perfect because its proper divisors are 1, 2, and 3, and $1 + 2 + 3 = 6$; similarly, 28 is perfect because $1 + 2 + 4 + 7 + 14 = 28$. De... | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P8538",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P8538",
"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 infinitely_many_perfect :
answer(sorry) ↔ {n : ℕ | Perfect n}.Infinite | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem infinitely_many_perfect :\n answer(sorry) ↔ {n : ℕ | Perfect n}.Infinite",
"target": null,
"source_url": "https://theoremdb.org/statements/P8538",
"format": "declaration-or-fragment",
"sha256": "11626748d41af076fd7530e6133a53c4cd6ffb3696a0958fec9b9... | [] | TheoremDB P8538 · 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 P8538",
"url": "https://theoremdb.org/statements/P8538",
"note": null,
"sort_order": 0
},
{
"kind": "formalization",
"citation": "Source formal statement (not locally verified)",
"url": "https://theoremdb.org/statements/P8538",
"not... | {
"source_record": "P8538",
"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":"2026-08-17T18:54:22+00:00","display_standard":{"editorial_review":"submitter_supplied","formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":8538,"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": "2026-09-08T03:33:01.186Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"f83beef3-abaa-4f16-a350-31d41057239e","format_issues":null,"number":5676} |
1.0.0 | problem:theoremdb-p8540 | problem | theoremdb-p8540 | Growth rate of convex prime sequences | {
"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 $(q_n)_{n=0}^{\infty}$ be a strictly increasing sequence of prime numbers such that the successive differences are nondecreasing, i.e., $q_{n+2} - q_{n+1} \geq q_{n+1} - q_n$ for all $n \geq 0$. Must it be that $\displaystyle\lim_{n \to \infty} \frac{q_n}{n^2} = \infty$? | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P8540",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P8540",
"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_455: answer(sorry) ↔ ∀ q : ℕ → ℕ, StrictMono q →
(∀ n, (q n).Prime ∧ q (n + 2) - q (n + 1) ≥ q (n + 1) - q n) →
Tendsto (fun n : ℕ => (q n : ℝ) / n ^ 2) atTop atTop | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem erdos_455: answer(sorry) ↔ ∀ q : ℕ → ℕ, StrictMono q →\n (∀ n, (q n).Prime ∧ q (n + 2) - q (n + 1) ≥ q (n + 1) - q n) →\n Tendsto (fun n : ℕ => (q n : ℝ) / n ^ 2) atTop atTop",
"target": null,
"source_url": "https://theoremdb.org/statements/P8540",
... | [] | TheoremDB P8540 · 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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},
{
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"url": "https://theoremdb.org/statements/P8540",
"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":"2026-08-17T18:54:26+00:00","display_standard":{"editorial_review":"submitter_supplied","formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":8540,"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": "2026-09-08T03:32:59.502Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"43120b4a-b570-4cf3-b120-6e689835f0de","format_issues":null,"number":5409} |
1.0.0 | problem:theoremdb-p8542 | problem | theoremdb-p8542 | Transcendence of the Catalan constant | {
"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 Catalan constant $G$ is defined by the convergent series
$$
G = \sum_{n=0}^{\infty} \frac{(-1)^n}{(2n+1)^2} = 1 - \frac{1}{3^2} + \frac{1}{5^2} - \frac{1}{7^2} + \cdots.
$$
A real number is said to be transcendental if it is not algebraic over the rational numbers $\mathbb{Q}$; equivalently, it is not a root of... | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P8542",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P8542",
"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 transcendental_catalanConstant : Transcendental ℚ catalanConstant | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem transcendental_catalanConstant : Transcendental ℚ catalanConstant",
"target": null,
"source_url": "https://theoremdb.org/statements/P8542",
"format": "declaration-or-fragment",
"sha256": "bef0a59922c5b24fae015a7deed915b32160c66133884606c3ba923aa9ffbd6... | [] | TheoremDB P8542 · 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/P8542",
"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":"2026-08-17T18:54:28+00:00","display_standard":{"editorial_review":"submitter_supplied","formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":8542,"public_status":{"evidence":{"hasFormalization":false,"hasInformalProof":false,"hasRecordedWork":false,"solutionRecord":null},"lean":{... | [
"formalization-not-compiled-during-export",
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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": "2026-09-08T03:33:06.998Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"681fbb58-4550-4f09-accf-9a2ca487eedb","format_issues":null,"number":6863} |
1.0.0 | problem:theoremdb-p8544 | problem | theoremdb-p8544 | Ramsey size linearity of the 3-dimensional hypercube | {
"main": "math",
"sub": "combinatorics",
"slug": "combinatorics",
"name": "Combinatorics",
"original_discipline": "mathematics"
} | [] | [
{
"code": "05",
"basis": "community-map",
"evidence": "Inferred from community field: combinatorics"
}
] | A graph $G$ is said to be Ramsey size linear if there exists a constant $c = c(G)$ such that, for every graph $H$ with $m$ edges and no isolated vertices, the size Ramsey number satisfies $\hat{r}(G,H) \leq c \cdot m$. The size Ramsey number $\hat{r}(G,H)$ is the minimum number of edges in a graph $F$ such that every r... | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P8544",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P8544",
"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.i : answer(sorry) ↔ IsRamseySizeLinear Q3 | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem erdos_567.parts.i : answer(sorry) ↔ IsRamseySizeLinear Q3",
"target": null,
"source_url": "https://theoremdb.org/statements/P8544",
"format": "declaration-or-fragment",
"sha256": "0305afa1762eb09ecf75c680a13c6e00ae075a301019fea8aab3cb1c6b20a260",
... | [] | TheoremDB P8544 · 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/P8544",
"note": null,
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},
{
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"citation": "Source formal statement (not locally verified)",
"url": "https://theoremdb.org/statements/P8544",
"not... | {
"source_record": "P8544",
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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":"2026-08-17T18:54:29+00:00","display_standard":{"editorial_review":"submitter_supplied","formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":8544,"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": "2026-09-08T03:33:05.399Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"eacdf31e-3d17-4092-9b39-bca2645ac8e5","format_issues":null,"number":6519} |
1.0.0 | problem:theoremdb-p8546 | problem | theoremdb-p8546 | Lower bound for the arithmetic sum associated with the classical theta 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 integers $h$ and $k$ with $k \geq 1$, define the alternating sum
$$
S'(h, k) = \sum_{j=1}^{k-1} (-1)^{j + 1 + \lfloor hj/k \rfloor},
$$
where $\lfloor \cdot \rfloor$ denotes the floor function. For a positive integer $k$, define the total sum
$$
S(k) = \sum_{h=1}^{k-1} S'(h, k).
$$
Let $k$ be an odd prime with... | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P8546",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P8546",
"citation": null,
"doi": null,
"arxiv_id": null,
"license": "Selected packets label their content CC0-1.0; original references retain their own terms",
"notes": "Selected problem... | theorem conjecture_4_2 (k : ℕ) (hprim : k.Prime) (hodd : Odd k) (hgt : k > 233) : 2 * k < S k | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem conjecture_4_2 (k : ℕ) (hprim : k.Prime) (hodd : Odd k) (hgt : k > 233) : 2 * k < S k",
"target": null,
"source_url": "https://theoremdb.org/statements/P8546",
"format": "declaration-or-fragment",
"sha256": "a20da72139fbd2789ae7d46db43b43af6f43dc6caf3... | [] | TheoremDB P8546 · 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 P8546",
"url": "https://theoremdb.org/statements/P8546",
"note": null,
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},
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"citation": "Source formal statement (not locally verified)",
"url": "https://theoremdb.org/statements/P8546",
"not... | {
"source_record": "P8546",
"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":"2026-08-17T18:54:31+00:00","display_standard":{"editorial_review":"submitter_supplied","formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":8546,"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": "2026-09-08T03:33:03.443Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"32d264ab-ae9f-4994-855d-f49fdb75c1d1","format_issues":null,"number":6097} |
1.0.0 | problem:theoremdb-p8548 | problem | theoremdb-p8548 | Asymptotic growth of minimal integers whose divisor sums represent n | {
"main": "math",
"sub": "number theory",
"slug": "number-theory",
"name": "Number theory",
"original_discipline": "mathematics"
} | [] | [
{
"code": "11",
"basis": "community-map",
"evidence": "Inferred from community field: number-theory"
}
] | For each positive integer $n$, define $f(n)$ to be the smallest positive integer $m$ such that $n$ equals the sum of the $k$ smallest divisors of $m$ for some integer $k \geq 1$, if such an $m$ exists; otherwise $f(n) = 0$. Here the divisors of $m$ are listed in increasing order, and the $k$ smallest divisors means the... | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P8548",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P8548",
"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_1054.parts.i : answer(sorry) ↔ (fun n ↦ (f n : ℝ)) =o[atTop] (fun n ↦ (n : ℝ)) | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem erdos_1054.parts.i : answer(sorry) ↔ (fun n ↦ (f n : ℝ)) =o[atTop] (fun n ↦ (n : ℝ))",
"target": null,
"source_url": "https://theoremdb.org/statements/P8548",
"format": "declaration-or-fragment",
"sha256": "9d66407c023e46eb5a901d1811c8e8a0f4c7e772a456... | [] | TheoremDB P8548 · 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 P8548",
"url": "https://theoremdb.org/statements/P8548",
"note": null,
"sort_order": 0
},
{
"kind": "formalization",
"citation": "Source formal statement (not locally verified)",
"url": "https://theoremdb.org/statements/P8548",
"not... | {
"source_record": "P8548",
"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":"2026-08-17T18:54:33+00:00","display_standard":{"editorial_review":"submitter_supplied","formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":8548,"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": "2026-09-08T03:32:50.140Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"a1f82d29-577e-4d4a-8281-931e478a163f","format_issues":null,"number":3359} |
1.0.0 | problem:theoremdb-p8550 | problem | theoremdb-p8550 | The Lebesgue–Nagell equation for odd prime exponents | {
"main": "math",
"sub": "number theory",
"slug": "number-theory",
"name": "Number theory",
"original_discipline": "mathematics"
} | [] | [
{
"code": "11",
"basis": "community-map",
"evidence": "Inferred from community field: number-theory"
}
] | Let $p$ be an odd prime number, and let $x$ and $y$ be integers. Consider the Diophantine equation $ x^2 - 2 = y^p $. Are the only solutions given by $(x, y) = (1, -1)$ and $(x, y) = (-1, -1)$? Equivalently, prove or disprove that for every odd prime $p$ and all integers $x$ and $y$, the equation $ x^2 - 2 = y^p $ hold... | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P8550",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P8550",
"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 lebesgue_nagell (p : ℕ) (hp : p.Prime) (hodd : Odd p) (x y : ℤ) :
x ^ 2 - 2 = y ^ p ↔ (x = 1 ∨ x = -1) ∧ y = -1 | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem lebesgue_nagell (p : ℕ) (hp : p.Prime) (hodd : Odd p) (x y : ℤ) :\n x ^ 2 - 2 = y ^ p ↔ (x = 1 ∨ x = -1) ∧ y = -1",
"target": null,
"source_url": "https://theoremdb.org/statements/P8550",
"format": "declaration-or-fragment",
"sha256": "968237c1bea7... | [] | TheoremDB P8550 · 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 P8550",
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"url": "https://theoremdb.org/statements/P8550",
"not... | {
"source_record": "P8550",
"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":"2026-08-17T18:54:36+00:00","display_standard":{"formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":8550,"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": "2026-09-08T03:33:06.608Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"7b100e5c-45fb-471c-89ce-e03c5175fea6","format_issues":null,"number":6779} |
1.0.0 | problem:theoremdb-p8552 | problem | theoremdb-p8552 | Infinitude of Amicable Pairs | {
"main": "math",
"sub": "number theory",
"slug": "number-theory",
"name": "Number theory",
"original_discipline": "mathematics"
} | [] | [
{
"code": "11",
"basis": "community-map",
"evidence": "Inferred from community field: number-theory"
}
] | For positive integers $a$ and $b$, define the sum-of-divisors function $\sigma(n) = \sum_{d \mid n} d$. An amicable pair is a pair of distinct positive integers $(a,b)$ such that $\sigma(a) = \sigma(b) = a + b$. Are there infinitely many amicable pairs? | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P8552",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P8552",
"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_830.parts.i : answer(sorry) ↔ {(a, b) | IsAmicable a b}.Infinite | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem erdos_830.parts.i : answer(sorry) ↔ {(a, b) | IsAmicable a b}.Infinite",
"target": null,
"source_url": "https://theoremdb.org/statements/P8552",
"format": "declaration-or-fragment",
"sha256": "bfb6993fd2223541877e3888023110e6979537f6e3444c4ee10a2ba26c... | [] | TheoremDB P8552 · 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 P8552",
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"citation": "Source formal statement (not locally verified)",
"url": "https://theoremdb.org/statements/P8552",
"not... | {
"source_record": "P8552",
"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":"2026-08-17T18:54:37+00:00","display_standard":{"editorial_review":"submitter_supplied","formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":8552,"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": "2026-09-08T03:33:01.081Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"020f3c78-ce9d-4ac4-b9f1-fb9f65a462ec","format_issues":null,"number":5653} |
1.0.0 | problem:theoremdb-p8554 | problem | theoremdb-p8554 | Eventual Constancy of the Hofstadter Q-Sequence Variant | {
"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"
}
] | Define a function $f: \mathbb{Z}^+ \to \mathbb{Z}^+$ by $f(1) = f(2) = 1$ and, for all integers $n > 2$,
$$
f(n) = f(n - f(n-1)) + f(n - f(n-2)).
$$
Note that it is not known whether $f(n)$ is well-defined for all $n$. Determine whether $f$ is eventually constant, i.e., whether there exist $N, c \in \mathbb{Z}^+$ s... | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P8554",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P8554",
"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_422.variants.eventually_const : answer(sorry) ↔ EventuallyConst f atTop | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem erdos_422.variants.eventually_const : answer(sorry) ↔ EventuallyConst f atTop",
"target": null,
"source_url": "https://theoremdb.org/statements/P8554",
"format": "declaration-or-fragment",
"sha256": "df08a0304dde0f6cb28e025c8673220f901fdf24e3e4ab3cc27... | [] | TheoremDB P8554 · 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 P8554",
"url": "https://theoremdb.org/statements/P8554",
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},
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"citation": "Source formal statement (not locally verified)",
"url": "https://theoremdb.org/statements/P8554",
"not... | {
"source_record": "P8554",
"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":"2026-08-17T18:54:39+00:00","display_standard":{"formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":8554,"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": "2026-09-08T03:32:56.742Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"9b1ba7fc-e0d3-4469-b11c-43042746b87f","format_issues":null,"number":4805} |
1.0.0 | problem:theoremdb-p8556 | problem | theoremdb-p8556 | Infinitude of Leinster Groups | {
"main": "math",
"sub": "algebra",
"slug": "algebra",
"name": "Algebra",
"original_discipline": "mathematics"
} | [] | [] | A finite group is called a Leinster group if the sum of the orders of all its normal subgroups equals twice the order of the group. Determine whether there exist infinitely many non-isomorphic finite groups that are Leinster groups. Equivalently, does there exist a natural number $n$ such that every Leinster group has ... | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P8556",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P8556",
"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 infinitely_many_leinster_groups : answer(sorry) ↔
¬∃ n : ℕ, ∀ G : Type, ∀ (_ : Group G) (_ : Fintype G),
IsLeinster G → Fintype.card G < n | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem infinitely_many_leinster_groups : answer(sorry) ↔\n ¬∃ n : ℕ, ∀ G : Type, ∀ (_ : Group G) (_ : Fintype G),\n IsLeinster G → Fintype.card G < n",
"target": null,
"source_url": "https://theoremdb.org/statements/P8556",
"format": "declaration-or-frag... | [] | TheoremDB P8556 · 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 P8556",
"url": "https://theoremdb.org/statements/P8556",
"note": null,
"sort_order": 0
},
{
"kind": "formalization",
"citation": "Source formal statement (not locally verified)",
"url": "https://theoremdb.org/statements/P8556",
"not... | {
"source_record": "P8556",
"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":"2026-08-17T18:54:43+00:00","display_standard":{"editorial_review":"submitter_supplied","formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":8556,"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": "2026-09-08T03:33:01.163Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"652bc41b-98e0-4992-abe0-66467d56327e","format_issues":null,"number":5671} |
1.0.0 | problem:theoremdb-p8558 | problem | theoremdb-p8558 | Asymptotic growth rate of the maximum size of a minimal vanishing signed unit fraction set | {
"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 $c(N)$ denote the largest cardinality of a subset $A \subseteq \{1, 2, \ldots, N\}$ for which there exists a function $\delta: A \to \{-1, 1\}$ such that
$$
\sum_{n \in A} \frac{\delta(n)}{n} = 0
$$
and such that for every nonempty proper subset $A' \subsetneq A$,
$$
\sum_{n \in... | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P8558",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P8558",
"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_319.variants.isTheta (N : ℕ) (c : ℕ → ℝ)
(h : ∀ N, IsGreatest
{ (#A : ℝ) | (A) (_ : A ⊆ Finset.Icc 1 N)
(_ : ∃ δ : ℕ → ℤˣ, ∑ n ∈ A, (δ n : ℚ) / n = 0 ∧
∀ A' ⊂ A, A'.Nonempty → ∑ n ∈ A', (δ n : ℚ) / n ≠ 0) } (c N)) :
c =Θ[atTop] (answer(sorry) : ℕ → ℝ) | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem erdos_319.variants.isTheta (N : ℕ) (c : ℕ → ℝ)\n (h : ∀ N, IsGreatest\n { (#A : ℝ) | (A) (_ : A ⊆ Finset.Icc 1 N)\n (_ : ∃ δ : ℕ → ℤˣ, ∑ n ∈ A, (δ n : ℚ) / n = 0 ∧\n ∀ A' ⊂ A, A'.Nonempty → ∑ n ∈ A', (δ n : ℚ) / n ≠ 0) } (c N)) :\n c =Θ[atTop] (a... | [] | TheoremDB P8558 · 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 P8558",
"url": "https://theoremdb.org/statements/P8558",
"note": null,
"sort_order": 0
},
{
"kind": "formalization",
"citation": "Source formal statement (not locally verified)",
"url": "https://theoremdb.org/statements/P8558",
"not... | {
"source_record": "P8558",
"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":"2026-08-17T18:54:47+00:00","display_standard":{"editorial_review":"submitter_supplied","formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":8558,"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": "2026-09-08T03:32:50.287Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"dcdc14fa-8175-4909-a83a-0a607cdcea2e","format_issues":null,"number":3389} |
1.0.0 | problem:theoremdb-p8560 | problem | theoremdb-p8560 | Existence of an absolutely maximally entangled state AME(11,10) | {
"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$ and $d \geq 2$ be positive integers. An $n$-partite pure quantum state $\psi$ with local dimension $d$ is called absolutely maximally entangled, denoted $\mathrm{AME}(n,d)$, if for every subset of at most $\lfloor n/2 \rfloor$ parties, the reduced density matrix of $\psi$ on that subset is maximally mixe... | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P8560",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P8560",
"citation": null,
"doi": null,
"arxiv_id": null,
"license": "Selected packets label their content CC0-1.0; original references retain their own terms",
"notes": "Selected problem... | theorem ame_11_10_open :
answer(sorry) ↔ ExistsAME 11 10 | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem ame_11_10_open :\n answer(sorry) ↔ ExistsAME 11 10",
"target": null,
"source_url": "https://theoremdb.org/statements/P8560",
"format": "declaration-or-fragment",
"sha256": "941f0a0fb35b02d2bf85e870e5f84063513e102d10b26f18488d28ffb24e9f63",
"lic... | [] | TheoremDB P8560 · 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 P8560",
"url": "https://theoremdb.org/statements/P8560",
"note": null,
"sort_order": 0
},
{
"kind": "formalization",
"citation": "Source formal statement (not locally verified)",
"url": "https://theoremdb.org/statements/P8560",
"not... | {
"source_record": "P8560",
"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":"2026-08-17T18:54:49+00:00","display_standard":{"editorial_review":"submitter_supplied","formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":8560,"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": "2026-09-08T03:32:58.604Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"d42dc4c5-061e-4df5-901b-fb61a89770fd","format_issues":null,"number":5217} |
1.0.0 | problem:theoremdb-p8562 | problem | theoremdb-p8562 | Asymptotic lower bound for sums of fewer than k kth 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 positive integers $k$ and $m$ with $1 \leq m < k$, let $f_{k,m}(x)$ denote the number of integers $n \leq x$ that can be expressed as the sum of $m$ nonnegative $k$th powers. Determine whether there exists a positive constant $C$ (depending only on $k$ and $m$) such that $f_{k,m}(x) \geq C x^{m/k}$ holds for all su... | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P8562",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P8562",
"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_323.parts.ii :
answer(sorry) ↔ ∀ k m : ℕ, 1 ≤ m → m < k →
(fun (x : ℕ) ↦ (x : ℝ) ^ ((m : ℝ) / (k : ℝ))) =O[atTop] (fun (x : ℕ) ↦ (f k m x : ℝ)) | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem erdos_323.parts.ii :\n answer(sorry) ↔ ∀ k m : ℕ, 1 ≤ m → m < k →\n (fun (x : ℕ) ↦ (x : ℝ) ^ ((m : ℝ) / (k : ℝ))) =O[atTop] (fun (x : ℕ) ↦ (f k m x : ℝ))",
"target": null,
"source_url": "https://theoremdb.org/statements/P8562",
"format": "declarat... | [] | TheoremDB P8562 · 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 P8562",
"url": "https://theoremdb.org/statements/P8562",
"note": null,
"sort_order": 0
},
{
"kind": "formalization",
"citation": "Source formal statement (not locally verified)",
"url": "https://theoremdb.org/statements/P8562",
"not... | {
"source_record": "P8562",
"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":"2026-08-17T18:54:52+00:00","display_standard":{"editorial_review":"submitter_supplied","formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":8562,"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": "2026-09-08T03:32:50.306Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"576e99a9-04c5-43cd-a591-6cab7fa143e5","format_issues":null,"number":3393} |
1.0.0 | problem:theoremdb-p8564 | problem | theoremdb-p8564 | Irrationality of the sum 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. Determine whether $e + \pi$ is irrational. That is, does there exist no rational number $p/q$ with integers $p$ and $q \neq 0$ such that $e + \pi = p/q$? | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P8564",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P8564",
"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_plus_pi :
answer(sorry) ↔ Irrational (e + π) | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem irrational_e_plus_pi :\n answer(sorry) ↔ Irrational (e + π)",
"target": null,
"source_url": "https://theoremdb.org/statements/P8564",
"format": "declaration-or-fragment",
"sha256": "6623cd8c4a1c8f373ffc445e0fda93da23b4a030063737513269cb12656bd467",... | [] | TheoremDB P8564 · 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 P8564",
"url": "https://theoremdb.org/statements/P8564",
"note": null,
"sort_order": 0
},
{
"kind": "formalization",
"citation": "Source formal statement (not locally verified)",
"url": "https://theoremdb.org/statements/P8564",
"not... | {
"source_record": "P8564",
"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":"2026-08-17T18:54:55+00:00","display_standard":{"editorial_review":"submitter_supplied","formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":8564,"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": "2026-09-08T03:33:01.835Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"66d4e5e7-6f4a-4935-b094-d35d6eda47ed","format_issues":null,"number":5745} |
1.0.0 | problem:theoremdb-p8566 | problem | theoremdb-p8566 | Natural density of indices with nondecreasing consecutive 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"
}
] | Let $p_n$ denote the $n$-th prime number, and define the prime gap at index $n$ to be $g_n = p_{n+1} - p_n$. Determine whether the set of indices $n$ for which $g_n \leq g_{n+1}$ has natural density equal to $\frac{1}{2}$. | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P8566",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P8566",
"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_218.variants.le : {n | primeGap n ≤ primeGap (n + 1)}.HasDensity <| 1 / 2 | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem erdos_218.variants.le : {n | primeGap n ≤ primeGap (n + 1)}.HasDensity <| 1 / 2",
"target": null,
"source_url": "https://theoremdb.org/statements/P8566",
"format": "declaration-or-fragment",
"sha256": "3361f69531d6a1f324e4a086db4d8e49eb38ebe743d2d8bde... | [] | TheoremDB P8566 · 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 P8566",
"url": "https://theoremdb.org/statements/P8566",
"note": null,
"sort_order": 0
},
{
"kind": "formalization",
"citation": "Source formal statement (not locally verified)",
"url": "https://theoremdb.org/statements/P8566",
"not... | {
"source_record": "P8566",
"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":"2026-08-17T18:55:05+00:00","display_standard":{"editorial_review":"submitter_supplied","formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":8566,"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": "2026-09-08T03:33:04.198Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"4f6b9d00-d0e3-42d3-a4f7-0a65e143cb26","format_issues":null,"number":6260} |
1.0.0 | problem:theoremdb-p8568 | problem | theoremdb-p8568 | Asymptotic growth of minimal covering densities for Hamming balls by linear subspaces | {
"main": "math",
"sub": "combinatorics",
"slug": "combinatorics",
"name": "Combinatorics",
"original_discipline": "mathematics"
} | [] | [
{
"code": "05",
"basis": "community-map",
"evidence": "Inferred from community field: combinatorics"
}
] | For a positive integer $n$, let $\mathbb{F}_2^n$ denote the $n$-dimensional vector space over the field with two elements. For a non-negative integer $r$, the Hamming ball $H(r)$ in $\mathbb{F}_2^n$ is the set of all vectors whose Hamming norm (the number of non-zero coordinates) is at most $r$. A covering subspace for... | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P8568",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P8568",
"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_40 : answer(sorry) ↔ Tendsto f atTop (𝓝 ⊤) | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem green_40 : answer(sorry) ↔ Tendsto f atTop (𝓝 ⊤)",
"target": null,
"source_url": "https://theoremdb.org/statements/P8568",
"format": "declaration-or-fragment",
"sha256": "489c4593669de780170befa119c163c88ea4327b0422d02f07786dd48f0bcf3f",
"license... | [] | TheoremDB P8568 · 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 P8568",
"url": "https://theoremdb.org/statements/P8568",
"note": null,
"sort_order": 0
},
{
"kind": "formalization",
"citation": "Source formal statement (not locally verified)",
"url": "https://theoremdb.org/statements/P8568",
"not... | {
"source_record": "P8568",
"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":"2026-08-17T18:55:10+00:00","display_standard":{"editorial_review":"submitter_supplied","formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":8568,"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": "2026-09-08T03:32:50.126Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"21c3bd16-b0ba-4200-852e-7810dddb7cb4","format_issues":null,"number":3356} |
1.0.0 | problem:theoremdb-p8570 | problem | theoremdb-p8570 | Long blocks of consecutive primes with uniformly large 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"
}
] | Let $c_1,c_2>0$. Prove or disprove that, for every sufficiently large real number $x$, there are integers $m\ge1$ and $k>c_1\log x$ such that the consecutive primes $p_m,p_{m+1},\ldots,p_{m+k-1}$ all satisfy $p_{m+i}\le x$, and
$$
p_{m+i+1}-p_{m+i}>c_2\qquad(0\le i<k-1).
$$
. | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P8570",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P8570",
"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_238 : answer(sorry) ↔ ∀ᵉ (c₁ > 0) (c₂ > 0), ∀ᶠ (x : ℝ) in atTop, ∃ (k : ℕ),
c₁ * log x < k ∧ ∃ f : Fin k → ℕ, ∃ m, (∀ i, f i ≤ x ∧ f i = (m + i.1).nth Nat.Prime)
∧ ∀ i : Fin (k - 1), c₂ < primeGap (m + i.1) | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem erdos_238 : answer(sorry) ↔ ∀ᵉ (c₁ > 0) (c₂ > 0), ∀ᶠ (x : ℝ) in atTop, ∃ (k : ℕ),\n c₁ * log x < k ∧ ∃ f : Fin k → ℕ, ∃ m, (∀ i, f i ≤ x ∧ f i = (m + i.1).nth Nat.Prime)\n ∧ ∀ i : Fin (k - 1), c₂ < primeGap (m + i.1)",
"target": null,
"source_url": "htt... | [] | TheoremDB P8570 · 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 P8570",
"url": "https://theoremdb.org/statements/P8570",
"note": null,
"sort_order": 0
},
{
"kind": "formalization",
"citation": "Source formal statement (not locally verified)",
"url": "https://theoremdb.org/statements/P8570",
"not... | {
"source_record": "P8570",
"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":"2026-08-17T18:55:22+00:00","display_standard":{"editorial_review":"submitter_supplied","formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":8570,"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": "2026-09-08T03:33:03.340Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"c76c5c0c-fe09-49ec-b37f-7f22298965f7","format_issues":null,"number":6074} |
1.0.0 | problem:theoremdb-p8572 | problem | theoremdb-p8572 | Transcendence of π^(π^π) | {
"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 *transcendental* if it is not a root of any nonzero polynomial with rational coefficients. Let π denote the ratio of the circumference of a circle to its diameter. Prove that the number $\pi^{\pi^{\pi}}$ is transcendental. | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P8572",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P8572",
"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_pow_pi_pow_pi_transcendental : Transcendental ℚ (π ^ (π ^ π)) | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem pi_pow_pi_pow_pi_transcendental : Transcendental ℚ (π ^ (π ^ π))",
"target": null,
"source_url": "https://theoremdb.org/statements/P8572",
"format": "declaration-or-fragment",
"sha256": "8b7a140e7e8baf1ed29fc176c0278a9a7433ac153db09a091ae6f2db97396feb... | [] | TheoremDB P8572 · 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 P8572",
"url": "https://theoremdb.org/statements/P8572",
"note": null,
"sort_order": 0
},
{
"kind": "formalization",
"citation": "Source formal statement (not locally verified)",
"url": "https://theoremdb.org/statements/P8572",
"not... | {
"source_record": "P8572",
"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":"2026-08-17T18:55:36+00:00","display_standard":{"editorial_review":"submitter_supplied","formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":8572,"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": "2026-09-08T03:33:07.031Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"19ad251c-1505-49ea-8859-796efb3016f9","format_issues":null,"number":6870} |
1.0.0 | problem:theoremdb-p8574 | problem | theoremdb-p8574 | Lower Bound on Largest Induced Tree Size in Connected Graphs | {
"main": "math",
"sub": "combinatorics",
"slug": "combinatorics",
"name": "Combinatorics",
"original_discipline": "mathematics"
} | [] | [
{
"code": "05",
"basis": "community-map",
"evidence": "Inferred from community field: combinatorics"
}
] | Let $G$ be a finite simple connected graph. The girth of $G$ is the length of the shortest cycle in $G$, or $0$ if $G$ is acyclic. For a vertex $v$ of $G$, the independence number of the neighborhood of $v$, denoted $l(v)$, is the maximum size of an independent set in the subgraph induced by the neighbors of $v$. The l... | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P8574",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P8574",
"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 conjecture141 (G : SimpleGraph α) [DecidableRel G.Adj] (h : G.Connected) :
(G.girth / 2 : ℤ) - 1 + ((Finset.univ.sup (indepNeighborsCard G) : ℕ) : ℤ) ≤
(largestInducedTreeSize G : ℤ) | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem conjecture141 (G : SimpleGraph α) [DecidableRel G.Adj] (h : G.Connected) :\n (G.girth / 2 : ℤ) - 1 + ((Finset.univ.sup (indepNeighborsCard G) : ℕ) : ℤ) ≤\n (largestInducedTreeSize G : ℤ)",
"target": null,
"source_url": "https://theoremdb.org/statements/... | [] | TheoremDB P8574 · 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 P8574",
"url": "https://theoremdb.org/statements/P8574",
"note": null,
"sort_order": 0
},
{
"kind": "formalization",
"citation": "Source formal statement (not locally verified)",
"url": "https://theoremdb.org/statements/P8574",
"not... | {
"source_record": "P8574",
"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":"2026-08-17T18:55:50+00:00","display_standard":{"formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":8574,"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": "2026-09-08T03:33:03.555Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"8b8406c3-1807-4f25-8233-5c64a813be99","format_issues":null,"number":6120} |
1.0.0 | problem:theoremdb-p8576 | problem | theoremdb-p8576 | Erdős–Rado sunflower threshold growth rate | {
"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 $f(n,k)$ denote the smallest integer such that every family of $n$-element sets with at least $f(n,k)$ members contains $k$ sets whose pairwise intersections are all equal (such a configuration is called a $k$-sunflower). Does there exist a function $c \colon \mathbb{N} \to \mathb... | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P8576",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P8576",
"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_20 : answer(sorry) ↔ ∃ (c : ℕ → ℕ), ∀ n k, n > 0 → f n k < (c k) ^ n | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem erdos_20 : answer(sorry) ↔ ∃ (c : ℕ → ℕ), ∀ n k, n > 0 → f n k < (c k) ^ n",
"target": null,
"source_url": "https://theoremdb.org/statements/P8576",
"format": "declaration-or-fragment",
"sha256": "f9a8825c93d9dcb1f1fa02e1dd8f45d742730cec4c610775578780... | [] | TheoremDB P8576 · 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 P8576",
"url": "https://theoremdb.org/statements/P8576",
"note": null,
"sort_order": 0
},
{
"kind": "formalization",
"citation": "Source formal statement (not locally verified)",
"url": "https://theoremdb.org/statements/P8576",
"not... | {
"source_record": "P8576",
"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":"2026-08-17T18:56:04+00:00","display_standard":{"editorial_review":"submitter_supplied","formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":8576,"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": "2026-09-08T03:32:56.696Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"109c79b5-c0e3-4f63-b847-07426f79def1","format_issues":null,"number":4795} |
1.0.0 | problem:theoremdb-p8578 | problem | theoremdb-p8578 | Erdős Problem 835 on Johnson Graph Chromatic Numbers | {
"main": "math",
"sub": "combinatorics",
"slug": "combinatorics",
"name": "Combinatorics",
"original_discipline": "mathematics"
} | [] | [
{
"code": "05",
"basis": "community-map",
"evidence": "Inferred from community field: combinatorics"
}
] | For a positive integer $n$ and a nonnegative integer $k$ with $k \leq n$, the Johnson graph $J(n, k)$ is the graph whose vertices are the $k$-element subsets of $\{1, 2, \ldots, n\}$, with two vertices adjacent when their corresponding subsets intersect in exactly $k-1$ elements. The chromatic number of a graph is the ... | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P8578",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P8578",
"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_835.variants.johnson : (∃ l,
-- making sure k > 2
letI k := l + 3
J(2 * k, k).chromaticNumber = k + 1) ↔ answer(sorry) | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem erdos_835.variants.johnson : (∃ l,\n -- making sure k > 2\n letI k := l + 3\n J(2 * k, k).chromaticNumber = k + 1) ↔ answer(sorry)",
"target": null,
"source_url": "https://theoremdb.org/statements/P8578",
"format": "declaration-or-fragment",
... | [] | TheoremDB P8578 · 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 P8578",
"url": "https://theoremdb.org/statements/P8578",
"note": null,
"sort_order": 0
},
{
"kind": "formalization",
"citation": "Source formal statement (not locally verified)",
"url": "https://theoremdb.org/statements/P8578",
"not... | {
"source_record": "P8578",
"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":"2026-08-17T18:56:11+00:00","display_standard":{"formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":8578,"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": "2026-09-08T03:32:56.102Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"92461f14-5868-4b0b-b420-883725208de3","format_issues":null,"number":4667} |
1.0.0 | problem:theoremdb-p8580 | problem | theoremdb-p8580 | Classification of Absolutely Maximally Entangled Pure States | {
"main": "math",
"sub": "combinatorics",
"slug": "combinatorics",
"name": "Combinatorics",
"original_discipline": "mathematics"
} | [] | [
{
"code": "05",
"basis": "community-map",
"evidence": "Inferred from community field: combinatorics"
}
] | Let $n$ and $d$ be integers with $n \ge 2$ and $d \ge 2$. An $n$-partite pure state $\psi$ with local dimension $d$ is called absolutely maximally entangled (AME) if, for every subset of at most $\lfloor n/2 \rfloor$ parties, the reduced density matrix of $\psi$ on that subset is maximally mixed. Determine the exact se... | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P8580",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P8580",
"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 oqp_35 :
{nd : ℕ × ℕ | 2 ≤ nd.1 ∧ 2 ≤ nd.2 ∧ ExistsAME nd.1 nd.2} = answer(sorry) | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem oqp_35 :\n {nd : ℕ × ℕ | 2 ≤ nd.1 ∧ 2 ≤ nd.2 ∧ ExistsAME nd.1 nd.2} = answer(sorry)",
"target": null,
"source_url": "https://theoremdb.org/statements/P8580",
"format": "declaration-or-fragment",
"sha256": "8de40a083a74a92522e4d79cf9a65413783f41f06a... | [] | TheoremDB P8580 · 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 P8580",
"url": "https://theoremdb.org/statements/P8580",
"note": null,
"sort_order": 0
},
{
"kind": "formalization",
"citation": "Source formal statement (not locally verified)",
"url": "https://theoremdb.org/statements/P8580",
"not... | {
"source_record": "P8580",
"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":"2026-08-17T18:56:14+00:00","display_standard":{"formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":8580,"public_status":{"evidence":{"hasFormalization":false,"hasInformalProof":false,"hasRecordedWork":false,"solutionRecord":null},"lean":{"kernelAccepted":false,"label":"","reaso... | [
"formalization-contains-placeholder",
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"current-status-unassessed",
"source-full-text-not-present"
] | [] | {
"created_at": "2026-09-08T03:32:15.935Z",
"updated_at": "2026-09-08T03:32:15.935Z",
"published_at": "2026-09-08T03:32:51.749Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"d57904f0-e080-4a6d-be4d-8e0b52b56f29","format_issues":null,"number":3706} |
1.0.0 | problem:theoremdb-p8582 | problem | theoremdb-p8582 | Erdős problem on sums of $r$-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"
}
] | For an integer $r \ge 2$, a positive integer $n$ is called $r$-powerful if for every prime $p$ dividing $n$, we have $p^r \mid n$. In other words, every prime divisor of $n$ appears with multiplicity at least $r$. For each $r \ge 3$, let $E_r$ denote the set of natural numbers that can be expressed as the sum of at mos... | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P8582",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P8582",
"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_940 :
answer(sorry) ↔ ∀ r ≥ 3,
{n : ℕ | ∃ (S : Multiset ℕ), S.card ≤ r ∧ (∀ s ∈ S, r.Full s) ∧ n = S.sum}.HasDensity 0 | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem erdos_940 :\n answer(sorry) ↔ ∀ r ≥ 3,\n {n : ℕ | ∃ (S : Multiset ℕ), S.card ≤ r ∧ (∀ s ∈ S, r.Full s) ∧ n = S.sum}.HasDensity 0",
"target": null,
"source_url": "https://theoremdb.org/statements/P8582",
"format": "declaration-or-fragment",
"sh... | [] | TheoremDB P8582 · 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 P8582",
"url": "https://theoremdb.org/statements/P8582",
"note": null,
"sort_order": 0
},
{
"kind": "formalization",
"citation": "Source formal statement (not locally verified)",
"url": "https://theoremdb.org/statements/P8582",
"not... | {
"source_record": "P8582",
"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":"2026-08-17T18:56:18+00:00","display_standard":{"editorial_review":"submitter_supplied","formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":8582,"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": "2026-09-08T03:32:56.297Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"05c8ac21-ce8f-4873-8434-356914f43b84","format_issues":null,"number":4710} |
1.0.0 | problem:theoremdb-p8584 | problem | theoremdb-p8584 | Lower bound for the size of a set whose sumset contains many squares | {
"main": "math",
"sub": "number theory",
"slug": "number-theory",
"name": "Number theory",
"original_discipline": "mathematics"
} | [] | [
{
"code": "11",
"basis": "community-map",
"evidence": "Inferred from community field: number-theory"
}
] | For a set $A$ of natural numbers, define the sumset $A + A = \{a + a' : a, a' \in A\}$. Determine whether the following holds: there exists a function $f : \mathbb{N} \to \mathbb{R}$ with $\lim_{n \to \infty} f(n) = 0$ such that for every integer $n \geq 1$ and every finite set $A \subseteq \mathbb{N}$, if the sumset $... | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P8584",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P8584",
"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_61 :
answer(sorry) ↔
∃ f : ℕ → ℝ, Tendsto f atTop (𝓝 0) ∧
∀ n : ℕ, n ≥ 1 → ∀ (A : Finset ℕ),
(Finset.Icc 1 n).image (· ^ 2) ⊆ A + A →
(n : ℝ) ^ (1 - f n) ≤ A.card | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem green_61 :\n answer(sorry) ↔\n ∃ f : ℕ → ℝ, Tendsto f atTop (𝓝 0) ∧\n ∀ n : ℕ, n ≥ 1 → ∀ (A : Finset ℕ),\n (Finset.Icc 1 n).image (· ^ 2) ⊆ A + A →\n (n : ℝ) ^ (1 - f n) ≤ A.card",
"target": null,
"source_url": "https://the... | [] | TheoremDB P8584 · 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 P8584",
"url": "https://theoremdb.org/statements/P8584",
"note": null,
"sort_order": 0
},
{
"kind": "formalization",
"citation": "Source formal statement (not locally verified)",
"url": "https://theoremdb.org/statements/P8584",
"not... | {
"source_record": "P8584",
"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":"2026-08-17T18:56:19+00:00","display_standard":{"formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":8584,"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": "2026-09-08T03:33:03.493Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"d6888a7e-864b-4fb0-8772-33ffedfa6060","format_issues":null,"number":6107} |
1.0.0 | problem:theoremdb-p8586 | problem | theoremdb-p8586 | Transcendence of gamma values at reciprocal integers | {
"main": "math",
"sub": "analysis",
"slug": "analysis",
"name": "Analysis",
"original_discipline": "mathematics"
} | [] | [] | Let $n$ be a natural number with $n \geq 2$. The gamma function $\Gamma$ is defined for positive real numbers $x$ by $\Gamma(x) = \int_0^{\infty} t^{x-1} e^{-t} \, dt$. A real number is said to be transcendental over the rational numbers if it is not a root of any non-zero polynomial with rational coefficients. Prove t... | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P8586",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P8586",
"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 transcendental_gamma_one_div (n : ℕ) (hn : 2 ≤ n) : Transcendental ℚ (1 / n : ℝ).Gamma | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem transcendental_gamma_one_div (n : ℕ) (hn : 2 ≤ n) : Transcendental ℚ (1 / n : ℝ).Gamma",
"target": null,
"source_url": "https://theoremdb.org/statements/P8586",
"format": "declaration-or-fragment",
"sha256": "79ac450cdacf344a204e68ce6d48042a3ccb5d0371... | [] | TheoremDB P8586 · 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 P8586",
"url": "https://theoremdb.org/statements/P8586",
"note": null,
"sort_order": 0
},
{
"kind": "formalization",
"citation": "Source formal statement (not locally verified)",
"url": "https://theoremdb.org/statements/P8586",
"not... | {
"source_record": "P8586",
"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":"2026-08-17T18:56:19+00:00","display_standard":{"editorial_review":"submitter_supplied","formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":8586,"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": "2026-09-08T03:33:06.988Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"4d62004f-0ac8-4991-b5e6-6b301bee5951","format_issues":null,"number":6861} |
1.0.0 | problem:theoremdb-p8588 | problem | theoremdb-p8588 | Erdős Problem 25: Existence of Logarithmic Density for Size-Dependent Congruence Avoidance 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 $n_1 < n_2 < \dots$ be an arbitrary strictly increasing sequence of positive integers, and for each $i$ let $a_i$ be an integer residue. Define $A$ to be the set of natural numbers $x$ such that for every index $i$, either $x < n_i$ or $x \not\equiv a_i \pmod{n_i}$. Must the logarithmic density of $A$ exist? | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P8588",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P8588",
"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_25 : answer(sorry) ↔
∀ (seq_n : ℕ → ℕ) (seq_a : ℕ → ℤ), (∀ i, 0 < seq_n i) → StrictMono seq_n →
∃ d, Set.HasLogDensity
{ x : ℕ | ∀ i, (x : ℤ) < seq_n i ∨ ¬((x : ℤ) ≡ seq_a i [ZMOD seq_n i]) } d | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem erdos_25 : answer(sorry) ↔\n ∀ (seq_n : ℕ → ℕ) (seq_a : ℕ → ℤ), (∀ i, 0 < seq_n i) → StrictMono seq_n →\n ∃ d, Set.HasLogDensity\n { x : ℕ | ∀ i, (x : ℤ) < seq_n i ∨ ¬((x : ℤ) ≡ seq_a i [ZMOD seq_n i]) } d",
"target": null,
"source_url": "https... | [] | TheoremDB P8588 · 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 P8588",
"url": "https://theoremdb.org/statements/P8588",
"note": null,
"sort_order": 0
},
{
"kind": "formalization",
"citation": "Source formal statement (not locally verified)",
"url": "https://theoremdb.org/statements/P8588",
"not... | {
"source_record": "P8588",
"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":"2026-08-17T18:56:28+00:00","display_standard":{"formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":8588,"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": "2026-09-08T03:32:55.262Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"68ea220a-0827-4267-a127-ea4f1e1eb7de","format_issues":null,"number":4482} |
1.0.0 | problem:theoremdb-p8590 | problem | theoremdb-p8590 | Upper bound for the minimum gcd of binomial coefficients | {
"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 $f(n) = \min_{1 < k \le n/2} \gcd\bigl(n, \binom{n}{k}\bigr)$ for integers $n \ge 4$. Is it true that for every real number $A > 0$, there exists a real constant $C > 0$ such that $f(n) \le C \cdot n/(\log n)^A$ holds for every composite integer $n > 1$? | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P8590",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P8590",
"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_700.parts.iii :
answer(sorry) ↔ (∀ A : ℝ, 0 < A → ∃ C : ℝ, 0 < C ∧ ∀ n : ℕ, ¬ n.Prime → 1 < n →
(f n : ℝ) ≤ C * (n : ℝ) / (Real.log n) ^ A) | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem erdos_700.parts.iii :\n answer(sorry) ↔ (∀ A : ℝ, 0 < A → ∃ C : ℝ, 0 < C ∧ ∀ n : ℕ, ¬ n.Prime → 1 < n →\n (f n : ℝ) ≤ C * (n : ℝ) / (Real.log n) ^ A)",
"target": null,
"source_url": "https://theoremdb.org/statements/P8590",
"format": "declaration-... | [] | TheoremDB P8590 · 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 P8590",
"url": "https://theoremdb.org/statements/P8590",
"note": null,
"sort_order": 0
},
{
"kind": "formalization",
"citation": "Source formal statement (not locally verified)",
"url": "https://theoremdb.org/statements/P8590",
"not... | {
"source_record": "P8590",
"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":"2026-08-17T18:56:29+00:00","display_standard":{"editorial_review":"submitter_supplied","formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":8590,"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": "2026-09-08T03:33:07.352Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"13fa7219-b90c-4557-94d8-752114ca1088","format_issues":null,"number":6941} |
1.0.0 | problem:theoremdb-p8592 | problem | theoremdb-p8592 | Maximal number of mutually unbiased bases in dimension 14 | {
"main": "math",
"sub": "combinatorics",
"slug": "combinatorics",
"name": "Combinatorics",
"original_discipline": "mathematics"
} | [] | [
{
"code": "05",
"basis": "community-map",
"evidence": "Inferred from community field: combinatorics"
}
] | Let $\mathbb{C}^{14}$ denote the 14-dimensional complex Hilbert space with its standard inner product. An orthonormal basis of $\mathbb{C}^{14}$ is a set of 14 unit vectors that are pairwise orthogonal. Two orthonormal bases $\mathcal{B} = \{e_0, \dots, e_{13}\}$ and $\mathcal{B}' = \{f_0, \dots, f_{13}\}$ are called m... | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P8592",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P8592",
"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_dim14 :
IsMaxMUBCount 14 (answer(sorry)) | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem mutuallyUnbiasedBases_dim14 :\n IsMaxMUBCount 14 (answer(sorry))",
"target": null,
"source_url": "https://theoremdb.org/statements/P8592",
"format": "declaration-or-fragment",
"sha256": "215e3c4374c704684193eaa9ad079db77cf708dfcfb6f1e22037bf986677f... | [] | TheoremDB P8592 · 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 P8592",
"url": "https://theoremdb.org/statements/P8592",
"note": null,
"sort_order": 0
},
{
"kind": "formalization",
"citation": "Source formal statement (not locally verified)",
"url": "https://theoremdb.org/statements/P8592",
"not... | {
"source_record": "P8592",
"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":"2026-08-17T18:56:36+00:00","display_standard":{"formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":8592,"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": "2026-09-08T03:33:03.656Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"63b1085e-4e7f-46af-a7bd-afb6fcc63439","format_issues":null,"number":6141} |
1.0.0 | problem:theoremdb-p8594 | problem | theoremdb-p8594 | Rationality of the Erdős prime-factor LCM reciprocal sum | {
"main": "math",
"sub": "number theory",
"slug": "number-theory",
"name": "Number theory",
"original_discipline": "mathematics"
} | [] | [
{
"code": "11",
"basis": "community-map",
"evidence": "Inferred from community field: number-theory"
}
] | Let $P$ be a finite set of primes with $|P| \ge 2$. Let $\{a_1 < a_2 < \dots\}$ be the strictly increasing sequence of all positive integers whose prime factors all belong to $P$, where by convention $1$ is included as it has no prime divisors. For each $n \ge 1$, let $[a_1, \ldots, a_n]$ denote the least common multip... | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P8594",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P8594",
"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_269.variants.rational : answer(sorry) ↔
∀ᵉ (P : Finset ℕ) (h : ∀ p ∈ P, p.Prime) (h_card : P.card ≥ 2),
∃ (q : ℚ), q = (series (P : Set ℕ)) | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem erdos_269.variants.rational : answer(sorry) ↔\n ∀ᵉ (P : Finset ℕ) (h : ∀ p ∈ P, p.Prime) (h_card : P.card ≥ 2),\n ∃ (q : ℚ), q = (series (P : Set ℕ))",
"target": null,
"source_url": "https://theoremdb.org/statements/P8594",
"format": "declaration-or... | [] | TheoremDB P8594 · 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 P8594",
"url": "https://theoremdb.org/statements/P8594",
"note": null,
"sort_order": 0
},
{
"kind": "formalization",
"citation": "Source formal statement (not locally verified)",
"url": "https://theoremdb.org/statements/P8594",
"not... | {
"source_record": "P8594",
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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":"2026-08-17T18:56:39+00:00","display_standard":{"editorial_review":"submitter_supplied","formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":8594,"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": "2026-09-08T03:33:05.435Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"7e4f7178-82bb-4db6-9864-a9d05cf8ea10","format_issues":null,"number":6527} |
1.0.0 | problem:theoremdb-p8596 | problem | theoremdb-p8596 | Characterization of composite integers with minimal binomial GCD equal to n divided by largest prime factor | {
"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
$$
f(n) = \min_{1 < k \le n/2} \gcd\!\left(n, \binom{n}{k}\right),
$$
where the minimum is taken over integers $k$ satisfying $1 < k \le n/2$, and define $P(n)$ to be the largest prime factor of $n$ (with $P(n) = 0$ if $n \le 1$). Let $n$ be a composite integer with $n > 1$. Determi... | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P8596",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P8596",
"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_700.parts.i (n : ℕ) (hn : ¬ n.Prime) (hn1 : 1 < n) :
f n = n / P n ↔ answer(sorry) | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem erdos_700.parts.i (n : ℕ) (hn : ¬ n.Prime) (hn1 : 1 < n) :\n f n = n / P n ↔ answer(sorry)",
"target": null,
"source_url": "https://theoremdb.org/statements/P8596",
"format": "declaration-or-fragment",
"sha256": "032cc2164caa114c773777aad4c71469f5a... | [] | TheoremDB P8596 · 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 P8596",
"url": "https://theoremdb.org/statements/P8596",
"note": null,
"sort_order": 0
},
{
"kind": "formalization",
"citation": "Source formal statement (not locally verified)",
"url": "https://theoremdb.org/statements/P8596",
"not... | {
"source_record": "P8596",
"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":"2026-08-17T18:56:44+00:00","display_standard":{"formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":8596,"public_status":{"evidence":{"hasFormalization":false,"hasInformalProof":false,"hasRecordedWork":false,"solutionRecord":null},"lean":{"kernelAccepted":false,"label":"","reaso... | [
"formalization-contains-placeholder",
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"current-status-unassessed",
"source-full-text-not-present"
] | [] | {
"created_at": "2026-09-08T03:32:15.935Z",
"updated_at": "2026-09-08T03:32:15.935Z",
"published_at": "2026-09-08T03:32:51.063Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"dbe45b4b-eb8c-4af1-b040-f3b681af3c17","format_issues":null,"number":3553} |
1.0.0 | problem:theoremdb-p8598 | problem | theoremdb-p8598 | The limit of the minimum overlap quotient | {
"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$, consider all partitions of $\{1, 2, \dots, 2n\}$ into two disjoint $n$-element subsets $A$ and $B$. For integers $a \in A$ and $b \in B$, define the overlap of $A$ and $B$ at difference $k \in \mathbb{Z}$ to be the number of pairs $(a,b)$ with $a - b = k$. Define the maximum overlap $M(A,... | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P8598",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P8598",
"citation": null,
"doi": null,
"arxiv_id": null,
"license": "Selected packets label their content CC0-1.0; original references retain their own terms",
"notes": "Selected problem... | theorem erdos_36 : atTop.Tendsto MinOverlapQuotient (𝓝 answer(sorry)) | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem erdos_36 : atTop.Tendsto MinOverlapQuotient (𝓝 answer(sorry))",
"target": null,
"source_url": "https://theoremdb.org/statements/P8598",
"format": "declaration-or-fragment",
"sha256": "478eae385bcfb98ef27535fedf38048cfb8e313143286b90f50009c11e7290d1",... | [] | TheoremDB P8598 · 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 P8598",
"url": "https://theoremdb.org/statements/P8598",
"note": null,
"sort_order": 0
},
{
"kind": "formalization",
"citation": "Source formal statement (not locally verified)",
"url": "https://theoremdb.org/statements/P8598",
"not... | {
"source_record": "P8598",
"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":"2026-08-17T18:56:53+00:00","display_standard":{"formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":8598,"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": "2026-09-08T03:33:06.617Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"359613f7-6ab7-48bd-b3da-5591c39ccdb9","format_issues":null,"number":6781} |
1.0.0 | problem:theoremdb-p8600 | problem | theoremdb-p8600 | The Juggler Conjecture | {
"main": "math",
"sub": "number theory",
"slug": "number-theory",
"name": "Number theory",
"original_discipline": "mathematics"
} | [] | [
{
"code": "11",
"basis": "community-map",
"evidence": "Inferred from community field: number-theory"
}
] | For a positive integer $n$, define the juggler step as follows: if $n$ is even, replace it with $\left\lfloor n^{1/2} \right\rfloor$; if $n$ is odd, replace it with $\left\lfloor n^{3/2} \right\rfloor$. Here $\lfloor x \rfloor$ denotes the greatest integer less than or equal to $x$. Starting from any positive integer $... | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P8600",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P8600",
"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 juggler_conjecture (n : ℕ) (hn : n > 0) : ∃ m, jugglerStep^[m] n = 1 | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem juggler_conjecture (n : ℕ) (hn : n > 0) : ∃ m, jugglerStep^[m] n = 1",
"target": null,
"source_url": "https://theoremdb.org/statements/P8600",
"format": "declaration-or-fragment",
"sha256": "5b51778b5d8682deaf981b1bc951c82ed9270e9988e18e2a8eada9c64bdf... | [] | TheoremDB P8600 · 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 P8600",
"url": "https://theoremdb.org/statements/P8600",
"note": null,
"sort_order": 0
},
{
"kind": "formalization",
"citation": "Source formal statement (not locally verified)",
"url": "https://theoremdb.org/statements/P8600",
"not... | {
"source_record": "P8600",
"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":"2026-08-17T18:56:58+00:00","display_standard":{"formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":8600,"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": "2026-09-08T03:33:06.571Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"1c1b0728-7668-45de-8e58-d4a9068bfe52","format_issues":null,"number":6771} |
1.0.0 | problem:theoremdb-p8602 | problem | theoremdb-p8602 | Existence of Eleven Consecutive Primes in Arithmetic Progression | {
"main": "math",
"sub": "combinatorics",
"slug": "combinatorics",
"name": "Combinatorics",
"original_discipline": "mathematics"
} | [] | [
{
"code": "05",
"basis": "community-map",
"evidence": "Inferred from community field: combinatorics"
}
] | A set $s$ of natural numbers is called a prime progression of length $l$ if there exists a natural number $a$ such that $s$ has cardinality $l$ and $s$ equals the set of the first $l$ primes starting from the $a$-th prime; that is, $s = \{p_{a+n} : n \in \mathbb{N},\ n < l\}$ where $p_m$ denotes the $m$-th prime number... | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P8602",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P8602",
"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.eleven : answer(sorry) ↔
∃ (s : Set ℕ), s.IsAPAndPrimeProgressionOfLength 11 | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem erdos_141.variants.eleven : answer(sorry) ↔\n ∃ (s : Set ℕ), s.IsAPAndPrimeProgressionOfLength 11",
"target": null,
"source_url": "https://theoremdb.org/statements/P8602",
"format": "declaration-or-fragment",
"sha256": "a66fff3cc35cbce5dd5455933005... | [] | TheoremDB P8602 · 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 P8602",
"url": "https://theoremdb.org/statements/P8602",
"note": null,
"sort_order": 0
},
{
"kind": "formalization",
"citation": "Source formal statement (not locally verified)",
"url": "https://theoremdb.org/statements/P8602",
"not... | {
"source_record": "P8602",
"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":"2026-08-17T18:57:09+00:00","display_standard":{"editorial_review":"submitter_supplied","formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":8602,"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": "2026-09-08T03:32:58.743Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"c5d84996-6ce4-4571-aeed-ac62eea2bed6","format_issues":null,"number":5246} |
1.0.0 | problem:theoremdb-p8604 | problem | theoremdb-p8604 | Irrationality of the Erdős–Straus divisor function series | {
"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$, define the real number
$$
S_k \coloneqq \sum_{n=1}^{\infty} \frac{\sigma_k(n)}{n!},
$$
where $\sigma_k(n)$ denotes the sum of the $k$-th powers of the positive divisors of $n$, that is, $\sigma_k(n) = \sum_{d \mid n} d^k$. Determine whether $S_k$ is irrational for every integer $k \geq ... | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P8604",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P8604",
"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_252 :
answer(sorry) ↔ ∀ k ≥ 1, Irrational (erdos_252_sum k) | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem erdos_252 :\n answer(sorry) ↔ ∀ k ≥ 1, Irrational (erdos_252_sum k)",
"target": null,
"source_url": "https://theoremdb.org/statements/P8604",
"format": "declaration-or-fragment",
"sha256": "c3ced52cc6d0e824271c82ae1264d33b3615e0c0c856dd6d2c8326a389... | [] | TheoremDB P8604 · 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 P8604",
"url": "https://theoremdb.org/statements/P8604",
"note": null,
"sort_order": 0
},
{
"kind": "formalization",
"citation": "Source formal statement (not locally verified)",
"url": "https://theoremdb.org/statements/P8604",
"not... | {
"source_record": "P8604",
"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":"2026-08-17T18:57:11+00:00","display_standard":{"editorial_review":"submitter_supplied","formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":8604,"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": "2026-09-08T03:33:01.752Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"83d99d32-07d4-4a46-b02e-0fa31513ad96","format_issues":null,"number":5727} |
1.0.0 | problem:theoremdb-p8606 | problem | theoremdb-p8606 | Lower bound for the van der Waerden number W(3, 23) | {
"main": "math",
"sub": "combinatorics",
"slug": "combinatorics",
"name": "Combinatorics",
"original_discipline": "mathematics"
} | [] | [
{
"code": "05",
"basis": "community-map",
"evidence": "Inferred from community field: combinatorics"
}
] | For positive integers $k$ and $r$, the 2-color van der Waerden number $W(k, r)$ is defined as the least positive integer $N$ such that every coloring of $\{1, 2, \ldots, N\}$ with two colors contains either a monochromatic arithmetic progression of length $k$ in the first color or a monochromatic arithmetic progression... | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P8606",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P8606",
"citation": null,
"doi": null,
"arxiv_id": null,
"license": "Selected packets label their content CC0-1.0; original references retain their own terms",
"notes": "Selected problem... | theorem W_3_23_lower : answer(sorry) ↔ W 3 23 ≥ 516 | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem W_3_23_lower : answer(sorry) ↔ W 3 23 ≥ 516",
"target": null,
"source_url": "https://theoremdb.org/statements/P8606",
"format": "declaration-or-fragment",
"sha256": "0ddf8e70b24c9accd1b0ba85d2d53cfd0d5ecaf23078ac685942b0662db0ff87",
"license": "Se... | [] | TheoremDB P8606 · 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 P8606",
"url": "https://theoremdb.org/statements/P8606",
"note": null,
"sort_order": 0
},
{
"kind": "formalization",
"citation": "Source formal statement (not locally verified)",
"url": "https://theoremdb.org/statements/P8606",
"not... | {
"source_record": "P8606",
"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":"2026-08-17T18:57:26+00:00","display_standard":{"formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":8606,"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": "2026-09-08T03:33:03.533Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"2934d16b-5d45-422b-bd42-65c4239011be","format_issues":null,"number":6115} |
1.0.0 | problem:theoremdb-p8608 | problem | theoremdb-p8608 | Asymptotic lower bound for differences of consecutive diagonal Ramsey numbers | {
"main": "math",
"sub": "combinatorics",
"slug": "combinatorics",
"name": "Combinatorics",
"original_discipline": "mathematics"
} | [] | [
{
"code": "05",
"basis": "community-map",
"evidence": "Inferred from community field: combinatorics"
}
] | Let $R(n)$ denote the diagonal Ramsey number $R(n,n)$, defined as the smallest integer $N$ such that every red-blue edge-coloring of the complete graph $K_N$ on $N$ vertices contains a monochromatic complete subgraph $K_n$ on $n$ vertices. Determine whether the difference of consecutive diagonal Ramsey numbers satisfie... | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P8608",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P8608",
"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_812.parts.ii :
answer(sorry) ↔
(fun n : ℕ ↦ (R (n + 1) : ℝ) - (R n : ℝ)) ≫ (fun n : ℕ ↦ (n : ℝ) ^ 2) | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem erdos_812.parts.ii :\n answer(sorry) ↔\n (fun n : ℕ ↦ (R (n + 1) : ℝ) - (R n : ℝ)) ≫ (fun n : ℕ ↦ (n : ℝ) ^ 2)",
"target": null,
"source_url": "https://theoremdb.org/statements/P8608",
"format": "declaration-or-fragment",
"sha256": "23c91d4e33... | [] | TheoremDB P8608 · 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 P8608",
"url": "https://theoremdb.org/statements/P8608",
"note": null,
"sort_order": 0
},
{
"kind": "formalization",
"citation": "Source formal statement (not locally verified)",
"url": "https://theoremdb.org/statements/P8608",
"not... | {
"source_record": "P8608",
"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":"2026-08-17T18:57:28+00:00","display_standard":{"editorial_review":"submitter_supplied","formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":8608,"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": "2026-09-08T03:32:50.302Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"59d9488a-33e0-4ddc-9502-f865bbe1053b","format_issues":null,"number":3392} |
1.0.0 | problem:theoremdb-p8610 | problem | theoremdb-p8610 | Existence of taxicab numbers for fifth powers with two summands | {
"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 natural numbers $k$, $m$, and $n$, a *taxicab number* is the smallest positive integer $x$ that can be expressed as a sum of $m$ positive $k$th powers in at least $n$ distinct ways, where the representations share no common terms. Specifically, $x$ is a *candidate* for a taxicab number for $k, m, n$ if there exists... | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P8610",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P8610",
"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 taxicab_for_5_2_n : answer(sorry) ↔ ∃ n : ℕ, n ≥ 2 ∧ (∃ x : ℕ, IsTaxicabFor 5 2 n x) | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem taxicab_for_5_2_n : answer(sorry) ↔ ∃ n : ℕ, n ≥ 2 ∧ (∃ x : ℕ, IsTaxicabFor 5 2 n x)",
"target": null,
"source_url": "https://theoremdb.org/statements/P8610",
"format": "declaration-or-fragment",
"sha256": "217d2348e703c7a9403b06708dbb6c817d8a1f4cd807... | [] | TheoremDB P8610 · 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 P8610",
"url": "https://theoremdb.org/statements/P8610",
"note": null,
"sort_order": 0
},
{
"kind": "formalization",
"citation": "Source formal statement (not locally verified)",
"url": "https://theoremdb.org/statements/P8610",
"not... | {
"source_record": "P8610",
"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":"2026-08-17T18:57:39+00:00","display_standard":{"editorial_review":"submitter_supplied","formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":8610,"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": "2026-09-08T03:32:58.830Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"2d1fab9f-a516-47f5-bec0-5bce694d453c","format_issues":null,"number":5264} |
1.0.0 | problem:theoremdb-p8612 | problem | theoremdb-p8612 | Erdős's Problem on Rational Reciprocal Sums of Integer Sequences | {
"main": "math",
"sub": "analysis",
"slug": "analysis",
"name": "Analysis",
"original_discipline": "mathematics"
} | [] | [] | Let $(a_n)_{n \ge 0}$ be a strictly increasing sequence of natural numbers such that $\lim_{n \to \infty} \frac{a_n}{a_{n-1}^2} = 1$ and the series $\sum_{n=0}^{\infty} \frac{1}{a_n}$ converges to a rational number. Prove that for all sufficiently large $n$, the sequence satisfies the exact recurrence $a_n = a_{n-1}^2 ... | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P8612",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P8612",
"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_243 (a : ℕ → ℕ) (ha₀ : StrictMono a)
(ha₁ : Tendsto (fun n ↦ (a n : ℝ) / a (n - 1) ^ 2) atTop (𝓝 1))
(ha₂ : Summable ((1 : ℚ) / a ·)) :
∀ᶠ n in atTop, a n = a (n - 1) ^ 2 - a (n - 1) + 1 | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem erdos_243 (a : ℕ → ℕ) (ha₀ : StrictMono a)\n (ha₁ : Tendsto (fun n ↦ (a n : ℝ) / a (n - 1) ^ 2) atTop (𝓝 1))\n (ha₂ : Summable ((1 : ℚ) / a ·)) :\n ∀ᶠ n in atTop, a n = a (n - 1) ^ 2 - a (n - 1) + 1",
"target": null,
"source_url": "https://theorem... | [] | TheoremDB P8612 · 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 P8612",
"url": "https://theoremdb.org/statements/P8612",
"note": null,
"sort_order": 0
},
{
"kind": "formalization",
"citation": "Source formal statement (not locally verified)",
"url": "https://theoremdb.org/statements/P8612",
"not... | {
"source_record": "P8612",
"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":"2026-08-17T18:57:40+00:00","display_standard":{"editorial_review":"submitter_supplied","formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":8612,"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": "2026-09-08T03:32:56.531Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"d4494b89-ec45-4ca6-9a5b-3499cb83243e","format_issues":null,"number":4760} |
1.0.0 | problem:theoremdb-p8614 | problem | theoremdb-p8614 | Hall's Conjecture on the Separation of Squares and Cubes | {
"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 real number $C > 0$ be called a Hall constant for an exponent $e > 0$ if for all integers $x$ and $y$ with $y^2 \neq x^3$, the inequality
$$
|y^2 - x^3| > C \cdot |x|^e
$$
holds. Hall's conjecture asserts that there exists a Hall constant for the exponent $\frac{1}{2}$. | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P8614",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P8614",
"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 hall_conjecture : HallConjectureExp 2⁻¹ | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem hall_conjecture : HallConjectureExp 2⁻¹",
"target": null,
"source_url": "https://theoremdb.org/statements/P8614",
"format": "declaration-or-fragment",
"sha256": "3284adef02af7135e4d606bd2c75372f5c9fc96ea577b3fd7abca04807baab45",
"license": "Select... | [] | TheoremDB P8614 · 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 P8614",
"url": "https://theoremdb.org/statements/P8614",
"note": null,
"sort_order": 0
},
{
"kind": "formalization",
"citation": "Source formal statement (not locally verified)",
"url": "https://theoremdb.org/statements/P8614",
"not... | {
"source_record": "P8614",
"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":"2026-08-17T18:57:54+00:00","display_standard":{"editorial_review":"submitter_supplied","formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":8614,"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": "2026-09-08T03:32:59.554Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"9fac38e7-99de-496c-add4-9da70ecf64d7","format_issues":null,"number":5420} |
1.0.0 | problem:theoremdb-p8616 | problem | theoremdb-p8616 | Existence of Odd Weird 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 natural number $n$ is called abundant if the sum of its proper divisors exceeds $n$. A natural number $n$ is called weird if it is abundant, but no subset of its proper divisors sums to exactly $n$. Determine whether there exists an odd weird number. | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P8616",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P8616",
"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_470.parts.i : answer(sorry) ↔ ∃ n : ℕ, n.Weird ∧ Odd n | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem erdos_470.parts.i : answer(sorry) ↔ ∃ n : ℕ, n.Weird ∧ Odd n",
"target": null,
"source_url": "https://theoremdb.org/statements/P8616",
"format": "declaration-or-fragment",
"sha256": "6b9db7720c426dd4ec23a8fb8d60d218a9833c98dea7d381e38cedc2d543fd7e",
... | [] | TheoremDB P8616 · 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 P8616",
"url": "https://theoremdb.org/statements/P8616",
"note": null,
"sort_order": 0
},
{
"kind": "formalization",
"citation": "Source formal statement (not locally verified)",
"url": "https://theoremdb.org/statements/P8616",
"not... | {
"source_record": "P8616",
"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":"2026-08-17T18:57:54+00:00","display_standard":{"editorial_review":"submitter_supplied","formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":8616,"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": "2026-09-08T03:32:58.793Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"6db4fb8e-25a7-4d0c-9735-382958fb3575","format_issues":null,"number":5256} |
1.0.0 | problem:theoremdb-p8618 | problem | theoremdb-p8618 | Infinitude of Prime Pell 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"
}
] | The Pell numbers $P_n$ are defined by $P_0 = 0$, $P_1 = 1$, and $P_{n+2} = 2P_{n+1} + P_n$ for all $n \geq 0$. Prove that the set $\{n \in \mathbb{N} : P_n \text{ is prime}\}$ is infinite. | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P8618",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P8618",
"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_pellNumber_primes : Infinite {n : ℕ | Prime (pellNumber n)} | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem infinite_pellNumber_primes : Infinite {n : ℕ | Prime (pellNumber n)}",
"target": null,
"source_url": "https://theoremdb.org/statements/P8618",
"format": "declaration-or-fragment",
"sha256": "97e78763ab2105aa9e7e8610fab4ced28faa3c63afe04c1d05d463dde48d... | [] | TheoremDB P8618 · 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 P8618",
"url": "https://theoremdb.org/statements/P8618",
"note": null,
"sort_order": 0
},
{
"kind": "formalization",
"citation": "Source formal statement (not locally verified)",
"url": "https://theoremdb.org/statements/P8618",
"not... | {
"source_record": "P8618",
"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":"2026-08-17T18:58:06+00:00","display_standard":{"editorial_review":"submitter_supplied","formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":8618,"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": "2026-09-08T03:33:01.200Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"c648541d-4d2f-4346-8789-d3c75c275277","format_issues":null,"number":5679} |
1.0.0 | problem:theoremdb-p8620 | problem | theoremdb-p8620 | Asymptotic constant for minimum restricted difference bases | {
"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$, an $N$-perfect ruler is a finite set $A \subseteq \mathbb{N}$ of marks such that every positive integer $k \leq N$ can be expressed as a difference $k = a_1 - a_0$ with $a_0, a_1 \in A$. A restricted difference basis with respect to $N$ is an $N$-perfect ruler contained in $\{0, 1, \dots,... | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P8620",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P8620",
"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... | lemma erdos170 : Filter.Tendsto (fun N => F N / √N) Filter.atTop (𝓝 answer(sorry)) | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "lemma erdos170 : Filter.Tendsto (fun N => F N / √N) Filter.atTop (𝓝 answer(sorry))",
"target": null,
"source_url": "https://theoremdb.org/statements/P8620",
"format": "declaration-or-fragment",
"sha256": "07208e0481ecf48b7963bef02e2d00585a44c9cf2d6dd038f5eec... | [] | TheoremDB P8620 · 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 P8620",
"url": "https://theoremdb.org/statements/P8620",
"note": null,
"sort_order": 0
},
{
"kind": "formalization",
"citation": "Source formal statement (not locally verified)",
"url": "https://theoremdb.org/statements/P8620",
"not... | {
"source_record": "P8620",
"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":"2026-08-17T18:58:14+00:00","display_standard":{"editorial_review":"submitter_supplied","formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":8620,"public_status":{"evidence":{"hasFormalization":false,"hasInformalProof":false,"hasRecordedWork":false,"solutionRecord":null},"lean":{... | [
"formalization-contains-placeholder",
"formalization-not-compiled-during-export",
"current-status-unassessed",
"source-full-text-not-present"
] | [] | {
"created_at": "2026-09-08T03:32:15.935Z",
"updated_at": "2026-09-08T03:32:15.935Z",
"published_at": "2026-09-08T03:32:49.958Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"f8fe11d9-6449-418c-a15e-c3f37bcd097f","format_issues":null,"number":3321} |
1.0.0 | problem:theoremdb-p8622 | problem | theoremdb-p8622 | Asymptotic growth of the least non-consecutive-sum 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"
}
] | Define a sequence of natural numbers $A = (A_0, A_1, A_2, \ldots)$ to be $\textit{good for}$ $n$ if it satisfies the following three conditions: $A_0 = n$; the sequence is strictly monotone increasing; and for every non-negative integer $j$, the term $A_{j+1}$ is the least natural number that is greater than $A_j$ and ... | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P8622",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P8622",
"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_359.parts.i (A : ℕ → ℕ) (hA : IsGoodFor A 1) :
atTop.Tendsto (fun k ↦ (A k : ℝ) / k) atTop | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem erdos_359.parts.i (A : ℕ → ℕ) (hA : IsGoodFor A 1) :\n atTop.Tendsto (fun k ↦ (A k : ℝ) / k) atTop",
"target": null,
"source_url": "https://theoremdb.org/statements/P8622",
"format": "declaration-or-fragment",
"sha256": "b2bcd00325a51ec8945ad45eba9... | [] | TheoremDB P8622 · 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 P8622",
"url": "https://theoremdb.org/statements/P8622",
"note": null,
"sort_order": 0
},
{
"kind": "formalization",
"citation": "Source formal statement (not locally verified)",
"url": "https://theoremdb.org/statements/P8622",
"not... | {
"source_record": "P8622",
"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":"2026-08-17T18:58:22+00:00","display_standard":{"editorial_review":"submitter_supplied","formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":8622,"public_status":{"evidence":{"hasFormalization":false,"hasInformalProof":false,"hasRecordedWork":false,"solutionRecord":null},"lean":{... | [
"formalization-not-compiled-during-export",
"current-status-unassessed",
"source-full-text-not-present"
] | [] | {
"created_at": "2026-09-08T03:32:15.935Z",
"updated_at": "2026-09-08T03:32:15.935Z",
"published_at": "2026-09-08T03:32:50.197Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"6a4b5b6e-cb19-4358-9858-2fc98b9011ba","format_issues":null,"number":3371} |
1.0.0 | problem:theoremdb-p8624 | problem | theoremdb-p8624 | Quantitative bounds for the pyjama problem | {
"main": "math",
"sub": "geometry and topology",
"slug": "geometry-and-topology",
"name": "Geometry and topology",
"original_discipline": "mathematics"
} | [] | [] | For a real number $\varepsilon > 0$, the pyjama set of width $\varepsilon$ is the subset of the complex plane $\mathbb{C}$ consisting of all points whose real part lies within distance $\varepsilon$ of some integer; equivalently, $\{z \in \mathbb{C} : \exists k \in \mathbb{Z}, |\operatorname{Re}(z) - k| \leq \varepsilo... | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P8624",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P8624",
"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_41 :
∃ C : ℝ, C > 0 ∧ ∃ ε₀ > 0, ∀ ε ∈ Ioc 0 ε₀,
let ans := (answer(sorry) : ℝ)
(minCopies ε : ℝ) ≤ ans ∧ ans < Real.exp (Real.exp (Real.exp (ε ^ (-C)))) | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem green_41 :\n ∃ C : ℝ, C > 0 ∧ ∃ ε₀ > 0, ∀ ε ∈ Ioc 0 ε₀,\n let ans := (answer(sorry) : ℝ)\n (minCopies ε : ℝ) ≤ ans ∧ ans < Real.exp (Real.exp (Real.exp (ε ^ (-C))))",
"target": null,
"source_url": "https://theoremdb.org/statements/P8624",
"fo... | [] | TheoremDB P8624 · 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 P8624",
"url": "https://theoremdb.org/statements/P8624",
"note": null,
"sort_order": 0
},
{
"kind": "formalization",
"citation": "Source formal statement (not locally verified)",
"url": "https://theoremdb.org/statements/P8624",
"not... | {
"source_record": "P8624",
"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":"2026-08-17T18:58:24+00:00","display_standard":{"editorial_review":"submitter_supplied","formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":8624,"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": "2026-09-08T03:33:05.347Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"53cf062e-ff96-4544-84fa-a84ec6c2d5b4","format_issues":null,"number":6508} |
1.0.0 | problem:theoremdb-p8626 | problem | theoremdb-p8626 | Weisenberg's variant of Erdős problem 891 on prime factor 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"
}
] | Let $p_1, p_2, \ldots$ denote the sequence of prime numbers in increasing order, so that $p_1 = 2$. For an integer $m \geq 1$, let $\omega(m)$ denote the number of distinct prime factors of $m$. For an integer $k \geq 2$, let $P_k = p_1 p_2 \cdots p_k = \prod_{i=0}^{k-1} p_{i+1}$. Determine whether there exist infinite... | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P8626",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P8626",
"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_891.variants.weisenberg (k : ℕ) (hk : k ≥ 2) :
∃ᶠ n in atTop,
∀ m ∈ Ico n (n + (∏ i ∈ range k, i.nth Nat.Prime) - 1),
ω m ≤ k | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem erdos_891.variants.weisenberg (k : ℕ) (hk : k ≥ 2) :\n ∃ᶠ n in atTop,\n ∀ m ∈ Ico n (n + (∏ i ∈ range k, i.nth Nat.Prime) - 1),\n ω m ≤ k",
"target": null,
"source_url": "https://theoremdb.org/statements/P8626",
"format": "declaration-or-frag... | [] | TheoremDB P8626 · 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 P8626",
"url": "https://theoremdb.org/statements/P8626",
"note": null,
"sort_order": 0
},
{
"kind": "formalization",
"citation": "Source formal statement (not locally verified)",
"url": "https://theoremdb.org/statements/P8626",
"not... | {
"source_record": "P8626",
"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":"2026-08-17T18:58:40+00:00","display_standard":{"editorial_review":"submitter_supplied","formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":8626,"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": "2026-09-08T03:33:07.473Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"d8595c9d-47dc-489e-b6e4-73f211c7eb83","format_issues":null,"number":6967} |
1.0.0 | problem:theoremdb-p8628 | problem | theoremdb-p8628 | De Giorgi's conjecture in dimension 7 | {
"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 = 7$. Consider a function $u : \mathbb{R}^n \to \mathbb{R}$. 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) + u(x) - u(x)^3 = 0$ for all $... | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P8628",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P8628",
"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_seven : (DeGiorgi_conclusion 7) | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem DeGiorgi_seven : (DeGiorgi_conclusion 7)",
"target": null,
"source_url": "https://theoremdb.org/statements/P8628",
"format": "declaration-or-fragment",
"sha256": "9c5b45fa946edb51a6f4453626dcbbc6ede8eaf4d9affe4bd78c83cdacdaf100",
"license": "Selec... | [] | TheoremDB P8628 · 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 P8628",
"url": "https://theoremdb.org/statements/P8628",
"note": null,
"sort_order": 0
},
{
"kind": "formalization",
"citation": "Source formal statement (not locally verified)",
"url": "https://theoremdb.org/statements/P8628",
"not... | {
"source_record": "P8628",
"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":"2026-08-17T18:58:43+00:00","display_standard":{"editorial_review":"submitter_supplied","formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":8628,"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": "2026-09-08T03:32:52.896Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"8c239d6c-32bb-4ef5-aa30-cf5ad9c55763","format_issues":null,"number":3959} |
1.0.0 | problem:theoremdb-p8630 | problem | theoremdb-p8630 | Existence of an Entire Transcendental Function with Dense Zero Sets for All Subsequence Iterated Derivatives | {
"main": "math",
"sub": "analysis",
"slug": "analysis",
"name": "Analysis",
"original_discipline": "mathematics"
} | [] | [] | A function $f : \mathbb{C} \to \mathbb{C}$ is called entire if it is complex differentiable at every point of $\mathbb{C}$. A function $f$ is transcendental over the polynomials if there is no nonzero polynomial $P \in \mathbb{C}[z, w]$ such that $P(z, f(z)) = 0$ for all $z \in \mathbb{C}$; equivalently, $f$ does not s... | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P8630",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P8630",
"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_906 : answer(sorry) ↔ ∃ f : ℂ → ℂ, Transcendental (Polynomial ℂ) f ∧
Differentiable ℂ f ∧ ∀ n : ℕ → ℕ, StrictMono n →
Dense { z | ∃ k, iteratedDeriv (n k) f z = 0 } | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem erdos_906 : answer(sorry) ↔ ∃ f : ℂ → ℂ, Transcendental (Polynomial ℂ) f ∧\n Differentiable ℂ f ∧ ∀ n : ℕ → ℕ, StrictMono n →\n Dense { z | ∃ k, iteratedDeriv (n k) f z = 0 }",
"target": null,
"source_url": "https://theoremdb.org/statements/P8630",
... | [] | TheoremDB P8630 · 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 P8630",
"url": "https://theoremdb.org/statements/P8630",
"note": null,
"sort_order": 0
},
{
"kind": "formalization",
"citation": "Source formal statement (not locally verified)",
"url": "https://theoremdb.org/statements/P8630",
"not... | {
"source_record": "P8630",
"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":"2026-08-17T18:58:49+00:00","display_standard":{"editorial_review":"submitter_supplied","formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":8630,"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": "2026-09-08T03:32:58.673Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"96de0103-6ec5-4391-8a71-b6e5e02ca653","format_issues":null,"number":5231} |
1.0.0 | problem:theoremdb-p8632 | problem | theoremdb-p8632 | Smale's mean value problem | {
"main": "math",
"sub": "algebra",
"slug": "algebra",
"name": "Algebra",
"original_discipline": "mathematics"
} | [] | [] | Let $p$ be a polynomial with complex coefficients of degree $d \geq 2$, and let $z \in \mathbb{C}$. A critical point of $p$ is a complex number $c$ such that $p'(c) = 0$, where $p'$ denotes the derivative of $p$. Does there exist a critical point $c$ of $p$ such that
$$
\frac{|p(z) - p(c)|}{|z - c|} \leq |p'(z)| \,? ... | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P8632",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P8632",
"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... | lemma mean_value_problem (p : Polynomial ℂ) (hp : 2 ≤ p.degree) (z : ℂ) (K : ℝ):
∃ c : ℂ, p.derivative.eval c = 0 ∧
‖p.eval z - p.eval c‖ / ‖z - c‖ ≤ ‖p.derivative.eval z‖ | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "lemma mean_value_problem (p : Polynomial ℂ) (hp : 2 ≤ p.degree) (z : ℂ) (K : ℝ):\n ∃ c : ℂ, p.derivative.eval c = 0 ∧\n ‖p.eval z - p.eval c‖ / ‖z - c‖ ≤ ‖p.derivative.eval z‖",
"target": null,
"source_url": "https://theoremdb.org/statements/P8632",
"form... | [] | TheoremDB P8632 · 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 P8632",
"url": "https://theoremdb.org/statements/P8632",
"note": null,
"sort_order": 0
},
{
"kind": "formalization",
"citation": "Source formal statement (not locally verified)",
"url": "https://theoremdb.org/statements/P8632",
"not... | {
"source_record": "P8632",
"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":"2026-08-17T18:58:51+00:00","display_standard":{"editorial_review":"submitter_supplied","formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":8632,"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": "2026-09-08T03:33:05.883Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"585451f3-a137-47e3-919c-05f62439d5fa","format_issues":null,"number":6623} |
1.0.0 | problem:theoremdb-p8634 | problem | theoremdb-p8634 | Maximum upper density of infinite Sidon sets | {
"main": "math",
"sub": "combinatorics",
"slug": "combinatorics",
"name": "Combinatorics",
"original_discipline": "mathematics"
} | [] | [
{
"code": "05",
"basis": "community-map",
"evidence": "Inferred from community field: combinatorics"
}
] | A set $A \subseteq \mathbb{N}$ of natural numbers is called a Sidon set if all pairwise sums of distinct elements are distinct; equivalently, for all $a, b, c, d \in A$ with $a \leq b$, $c \leq d$, and $\{a,b\} \neq \{c,d\}$, we have $a + b \neq c + d$. For a Sidon set $A$ and a positive integer $N$, define the square-... | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P8634",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P8634",
"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_329 : sSup {sidonUpperDensity A | (A : Set ℕ) (_ : IsSidon A)} =
answer(sorry) | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem erdos_329 : sSup {sidonUpperDensity A | (A : Set ℕ) (_ : IsSidon A)} =\n answer(sorry)",
"target": null,
"source_url": "https://theoremdb.org/statements/P8634",
"format": "declaration-or-fragment",
"sha256": "7428821e399404a52418ac5732daecdf27b0cbc... | [] | TheoremDB P8634 · 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 P8634",
"url": "https://theoremdb.org/statements/P8634",
"note": null,
"sort_order": 0
},
{
"kind": "formalization",
"citation": "Source formal statement (not locally verified)",
"url": "https://theoremdb.org/statements/P8634",
"not... | {
"source_record": "P8634",
"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":"2026-08-17T18:58:52+00:00","display_standard":{"formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":8634,"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": "2026-09-08T03:33:03.859Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"46f1fd32-2367-424b-ab4e-cc2a1afd620b","format_issues":null,"number":6186} |
1.0.0 | problem:theoremdb-p8636 | problem | theoremdb-p8636 | Voronovskaja-type asymptotic formula for Bézier–Bernstein operators with shape parameter | {
"main": "math",
"sub": "analysis",
"slug": "analysis",
"name": "Analysis",
"original_discipline": "mathematics"
} | [] | [] | Let $I = [0,1]$ and fix a real parameter $\alpha > 0$ with $\alpha \neq 1$. For each $n \in \mathbb{N}$, the Bernstein basis polynomial is defined by
$$
p_{n,j}(x) = \binom{n}{j} x^j (1-x)^{n-j}
$$
for $j = 0, 1, \ldots, n$, and the cumulative tail sum is
$$
J_{n,k}(x) = \sum_{j=k}^{n} p_{n,j}(x)
$$
for $k = 0,... | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P8636",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P8636",
"citation": null,
"doi": null,
"arxiv_id": null,
"license": "Selected packets label their content CC0-1.0; original references retain their own terms",
"notes": "Selected problem... | theorem voronovskaja_theorem.bezier_bernstein_operators.variants.answer_smoothness
(α : ℝ) (hα_pos : 0 < α) (hα : α ≠ 1) :
let p : ℕ × ((ℝ → ℝ) → ℝ → ℝ) := answer(sorry)
let m := p.1
let limitFormula := p.2
∀ (f : ℝ → ℝ) (x : ℝ), x ∈ I → ContDiffOn ℝ m f I →
Tendsto (fun n : ℕ => Real.sqrt n *... | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem voronovskaja_theorem.bezier_bernstein_operators.variants.answer_smoothness\n (α : ℝ) (hα_pos : 0 < α) (hα : α ≠ 1) :\n let p : ℕ × ((ℝ → ℝ) → ℝ → ℝ) := answer(sorry)\n let m := p.1\n let limitFormula := p.2\n ∀ (f : ℝ → ℝ) (x : ℝ), x ∈ I → ContDiffOn ℝ... | [] | TheoremDB P8636 · 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 P8636",
"url": "https://theoremdb.org/statements/P8636",
"note": null,
"sort_order": 0
},
{
"kind": "formalization",
"citation": "Source formal statement (not locally verified)",
"url": "https://theoremdb.org/statements/P8636",
"not... | {
"source_record": "P8636",
"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":"2026-08-17T18:58:57+00:00","display_standard":{"editorial_review":"submitter_supplied","formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":8636,"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": "2026-09-08T03:33:07.435Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"02254a8a-30c7-4de5-8aaa-2b9dd7ff9a89","format_issues":null,"number":6959} |
1.0.0 | problem:theoremdb-p8638 | problem | theoremdb-p8638 | Non-existence of monochromatic quantum graphs with six vertices | {
"main": "math",
"sub": "combinatorics",
"slug": "combinatorics",
"name": "Combinatorics",
"original_discipline": "mathematics"
} | [] | [
{
"code": "05",
"basis": "community-map",
"evidence": "Inferred from community field: combinatorics"
}
] | Let $N = 6$. A quantum graph with $N$ vertices and $D$ colours over a coefficient domain is specified by a weight function $W$ that assigns a value to each edge-coloured pair of vertices. For each assignment of colours to the $N$ vertices, the perfect-matching sum is defined as the sum over all perfect matchings of the... | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P8638",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P8638",
"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_ge3 :
answer(sorry) ↔
∀ D : Nat, D ≥ 3 →
¬ ∃ W : WeightsN 6 D ℂ, EqSystemN 6 D W | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem eqSystem6_no_solution_ge3 :\n answer(sorry) ↔\n ∀ D : Nat, D ≥ 3 →\n ¬ ∃ W : WeightsN 6 D ℂ, EqSystemN 6 D W",
"target": null,
"source_url": "https://theoremdb.org/statements/P8638",
"format": "declaration-or-fragment",
"sha256": "d0f14... | [] | TheoremDB P8638 · 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 P8638",
"url": "https://theoremdb.org/statements/P8638",
"note": null,
"sort_order": 0
},
{
"kind": "formalization",
"citation": "Source formal statement (not locally verified)",
"url": "https://theoremdb.org/statements/P8638",
"not... | {
"source_record": "P8638",
"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":"2026-08-17T18:58:58+00:00","display_standard":{"editorial_review":"submitter_supplied","formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":8638,"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": "2026-09-08T03:33:04.325Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"a090956f-3e38-4128-8d80-e569d59f5639","format_issues":null,"number":6287} |
1.0.0 | problem:theoremdb-p8640 | problem | theoremdb-p8640 | De Giorgi's conjecture in dimension six | {
"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 $n = 6$. Consider a function $u : \mathbb{R}^n \to \mathbb{R}$ that is a bounded classical solution to the Allen–Cahn equation $\Delta u + u - u^3 = 0$, meaning that $u$ 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) + u(x) ... | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P8640",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P8640",
"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_six : (DeGiorgi_conclusion 6) | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem DeGiorgi_six : (DeGiorgi_conclusion 6)",
"target": null,
"source_url": "https://theoremdb.org/statements/P8640",
"format": "declaration-or-fragment",
"sha256": "9bb0acf3ad6aded780e531be25dfb6d707ef328901cdf898cd056c6590841668",
"license": "Selecte... | [] | TheoremDB P8640 · 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 P8640",
"url": "https://theoremdb.org/statements/P8640",
"note": null,
"sort_order": 0
},
{
"kind": "formalization",
"citation": "Source formal statement (not locally verified)",
"url": "https://theoremdb.org/statements/P8640",
"not... | {
"source_record": "P8640",
"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":"2026-08-17T18:58:59+00:00","display_standard":{"editorial_review":"submitter_supplied","formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":8640,"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": "2026-09-08T03:32:52.915Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"3fa92fd2-90f2-4e1e-8868-01d5efa0a742","format_issues":null,"number":3963} |
1.0.0 | problem:theoremdb-p8642 | problem | theoremdb-p8642 | Lower bound for the count of amicable pairs up to x | {
"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 $\sigma(n)$ denote the sum of all positive divisors of $n$. Two positive integers $a$ and $b$ are called an amicable pair if $\sigma(a) = \sigma(b) = a + b$. For a real number $x \geq 1$, let $A(x)$ denote the number of amicable pairs $(a,b)$ with $1 \leq a \leq b \leq x$. Determine whet... | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P8642",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P8642",
"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_830.parts.ii : answer(sorry) ↔ ∃ o : ℝ → ℝ, o =o[atTop] (1 : ℝ → ℝ) ∧ ∀ᶠ x in atTop,
x ^ (1 - o x) < A x | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem erdos_830.parts.ii : answer(sorry) ↔ ∃ o : ℝ → ℝ, o =o[atTop] (1 : ℝ → ℝ) ∧ ∀ᶠ x in atTop,\n x ^ (1 - o x) < A x",
"target": null,
"source_url": "https://theoremdb.org/statements/P8642",
"format": "declaration-or-fragment",
"sha256": "d4538c027ece8... | [] | TheoremDB P8642 · 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 P8642",
"url": "https://theoremdb.org/statements/P8642",
"note": null,
"sort_order": 0
},
{
"kind": "formalization",
"citation": "Source formal statement (not locally verified)",
"url": "https://theoremdb.org/statements/P8642",
"not... | {
"source_record": "P8642",
"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":"2026-08-17T18:59:00+00:00","display_standard":{"editorial_review":"submitter_supplied","formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":8642,"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": "2026-09-08T03:33:03.455Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"7948eac0-6b0e-4af5-858a-d86677bcbd1f","format_issues":null,"number":6099} |
1.0.0 | problem:theoremdb-p8644 | problem | theoremdb-p8644 | Existence of an absolutely maximally entangled state AME(12,10) | {
"main": "math",
"sub": "combinatorics",
"slug": "combinatorics",
"name": "Combinatorics",
"original_discipline": "mathematics"
} | [] | [
{
"code": "05",
"basis": "community-map",
"evidence": "Inferred from community field: combinatorics"
}
] | A pure state on $n$ parties of local dimension $d$ is called absolutely maximally entangled (AME) if, for every subset of at most half of the parties, the corresponding reduced density matrix is maximally mixed. Does there exist an absolutely maximally entangled pure state on $n = 12$ parties with local dimension $d = ... | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P8644",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P8644",
"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_10_open :
answer(sorry) ↔ ExistsAME 12 10 | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem ame_12_10_open :\n answer(sorry) ↔ ExistsAME 12 10",
"target": null,
"source_url": "https://theoremdb.org/statements/P8644",
"format": "declaration-or-fragment",
"sha256": "dcc575e152502a381009276aec5fd941a9f9d095455341a8cd897168229c9eec",
"lic... | [] | TheoremDB P8644 · 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 P8644",
"url": "https://theoremdb.org/statements/P8644",
"note": null,
"sort_order": 0
},
{
"kind": "formalization",
"citation": "Source formal statement (not locally verified)",
"url": "https://theoremdb.org/statements/P8644",
"not... | {
"source_record": "P8644",
"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":"2026-08-17T18:59:03+00:00","display_standard":{"editorial_review":"submitter_supplied","formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":8644,"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": "2026-09-08T03:32:58.608Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"5aeca5ec-a1f2-4c59-8a73-6d68b89720a9","format_issues":null,"number":5218} |
1.0.0 | problem:theoremdb-p8646 | problem | theoremdb-p8646 | Infinitude of Fibonacci 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"
}
] | Let the Fibonacci sequence be defined by $F_0 = 0$, $F_1 = 1$, and $F_{m+2} = F_{m+1} + F_m$ for all $m \in \mathbb{N}$. A Fibonacci prime is a natural number that is both a Fibonacci number and a prime number. Determine whether the set of Fibonacci primes is infinite. | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P8646",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P8646",
"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 fib_primes_infinite : {n : ℕ | (∃ m : ℕ, m.fib = n) ∧ n.Prime}.Infinite | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem fib_primes_infinite : {n : ℕ | (∃ m : ℕ, m.fib = n) ∧ n.Prime}.Infinite",
"target": null,
"source_url": "https://theoremdb.org/statements/P8646",
"format": "declaration-or-fragment",
"sha256": "1e6046e152e5578ec83bf261a653d86d5cb82c04dc7e0079a140c4e78... | [] | TheoremDB P8646 · 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 P8646",
"url": "https://theoremdb.org/statements/P8646",
"note": null,
"sort_order": 0
},
{
"kind": "formalization",
"citation": "Source formal statement (not locally verified)",
"url": "https://theoremdb.org/statements/P8646",
"not... | {
"source_record": "P8646",
"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":"2026-08-17T18:59:05+00:00","display_standard":{"editorial_review":"submitter_supplied","formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":8646,"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": "2026-09-08T03:33:01.155Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"9f764d46-21e8-4f92-bbb4-c90914c8dade","format_issues":null,"number":5669} |
1.0.0 | problem:theoremdb-p8648 | problem | theoremdb-p8648 | Non-existence of integer-weighted monochromatic quantum graphs with six vertices and three colours | {
"main": "math",
"sub": "combinatorics",
"slug": "combinatorics",
"name": "Combinatorics",
"original_discipline": "mathematics"
} | [] | [
{
"code": "05",
"basis": "community-map",
"evidence": "Inferred from community field: combinatorics"
}
] | Let $N$ be a positive even integer and let $D$ be a positive integer. A quantum graph with $N$ vertices and $D$ colours consists of a complete graph on $N$ vertices together with an edge-weighting that assigns to each unordered pair of distinct vertices $\{i,j\}$ and each colour $c \in \{1,\dots,D\}$ a weight $W(i,j,c)... | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P8648",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P8648",
"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_int :
answer(sorry) ↔
¬ ∃ W : WeightsN 6 3 ℤ, EqSystemN 6 3 W | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem eqSystem6_no_solution_d3_int :\n answer(sorry) ↔\n ¬ ∃ W : WeightsN 6 3 ℤ, EqSystemN 6 3 W",
"target": null,
"source_url": "https://theoremdb.org/statements/P8648",
"format": "declaration-or-fragment",
"sha256": "14a633c5b667c89150d47bce7ceada... | [] | TheoremDB P8648 · 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 P8648",
"url": "https://theoremdb.org/statements/P8648",
"note": null,
"sort_order": 0
},
{
"kind": "formalization",
"citation": "Source formal statement (not locally verified)",
"url": "https://theoremdb.org/statements/P8648",
"not... | {
"source_record": "P8648",
"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":"2026-08-17T18:59:12+00:00","display_standard":{"editorial_review":"submitter_supplied","formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":8648,"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": "2026-09-08T03:33:04.280Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"8fee816a-8ba0-439d-a376-293a1f7e53e1","format_issues":null,"number":6278} |
1.0.0 | problem:theoremdb-p8650 | problem | theoremdb-p8650 | Almost Sure Convergence of the Normalized Root Count for Random \(\pm 1\) Coefficient Polynomials | {
"main": "math",
"sub": "algebra",
"slug": "algebra",
"name": "Algebra",
"original_discipline": "mathematics"
} | [] | [] | A sequence of Kac coefficients over a subset $S$ of a field $k$ is a countably infinite sequence of independent random variables, each uniformly distributed over $S$. Such a sequence determines a Kac polynomial of degree $n$ for each $n$, given by $\sum_{i=0}^{n} c_i z^i$ where the $c_i$ are the Kac coefficients. For a... | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P8650",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P8650",
"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_522 :
answer(sorry) ↔ ∀ {Ω : Type*} [MeasureSpace Ω] [IsProbabilityMeasure (ℙ : Measure Ω)]
(c : KacCoefficients ({-1, 1} : Set ℂ) Ω),
ℙ {ω | atTop.Tendsto (fun n : ℕ ↦ (2 * c.numRootsInUnitDisk n ω : ℝ) / n) (𝓝 1)} = 1 | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem erdos_522 :\n answer(sorry) ↔ ∀ {Ω : Type*} [MeasureSpace Ω] [IsProbabilityMeasure (ℙ : Measure Ω)]\n (c : KacCoefficients ({-1, 1} : Set ℂ) Ω),\n ℙ {ω | atTop.Tendsto (fun n : ℕ ↦ (2 * c.numRootsInUnitDisk n ω : ℝ) / n) (𝓝 1)} = 1",
"target": null,... | [] | TheoremDB P8650 · 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 P8650",
"url": "https://theoremdb.org/statements/P8650",
"note": null,
"sort_order": 0
},
{
"kind": "formalization",
"citation": "Source formal statement (not locally verified)",
"url": "https://theoremdb.org/statements/P8650",
"not... | {
"source_record": "P8650",
"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":"2026-08-17T18:59:22+00:00","display_standard":{"editorial_review":"submitter_supplied","formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":8650,"public_status":{"evidence":{"hasFormalization":false,"hasInformalProof":false,"hasRecordedWork":false,"solutionRecord":null},"lean":{... | [
"formalization-contains-placeholder",
"formalization-not-compiled-during-export",
"current-status-unassessed",
"source-full-text-not-present"
] | [] | {
"created_at": "2026-09-08T03:32:15.935Z",
"updated_at": "2026-09-08T03:32:15.935Z",
"published_at": "2026-09-08T03:32:49.545Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"86f0b09e-11e3-42c2-bbb1-769c50e67bc5","format_issues":null,"number":3231} |
1.0.0 | problem:theoremdb-p8652 | problem | theoremdb-p8652 | Asymptotic growth of minimal covering density with arbitrary subsets | {
"main": "math",
"sub": "combinatorics",
"slug": "combinatorics",
"name": "Combinatorics",
"original_discipline": "mathematics"
} | [] | [
{
"code": "05",
"basis": "community-map",
"evidence": "Inferred from community field: combinatorics"
}
] | For each pair of positive integers $n$ and $r$, let $\mathbb{F}_2^n$ denote the $n$-dimensional vector space over the field with two elements. The Hamming norm of a vector $x \in \mathbb{F}_2^n$, denoted $\|x\|_H$, is the number of nonzero coordinates of $x$. The Hamming ball of radius $r$ in $\mathbb{F}_2^n$, denoted ... | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P8652",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P8652",
"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_40.variants.arbitrary_subsets : answer(sorry) ↔ Tendsto f_tilde atTop (𝓝 ⊤) | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem green_40.variants.arbitrary_subsets : answer(sorry) ↔ Tendsto f_tilde atTop (𝓝 ⊤)",
"target": null,
"source_url": "https://theoremdb.org/statements/P8652",
"format": "declaration-or-fragment",
"sha256": "990515a1c262f39f7c6e64bcc5d25b810d5998aab0f8ec... | [] | TheoremDB P8652 · 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 P8652",
"url": "https://theoremdb.org/statements/P8652",
"note": null,
"sort_order": 0
},
{
"kind": "formalization",
"citation": "Source formal statement (not locally verified)",
"url": "https://theoremdb.org/statements/P8652",
"not... | {
"source_record": "P8652",
"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":"2026-08-17T18:59:33+00:00","display_standard":{"editorial_review":"submitter_supplied","formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":8652,"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": "2026-09-08T03:32:50.131Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"a8ca79b3-76ea-4d71-afc0-a5467d794046","format_issues":null,"number":3357} |
1.0.0 | problem:theoremdb-p8654 | problem | theoremdb-p8654 | Pretriangulated Categories That Are Not Triangulated | {
"main": "math",
"sub": "algebra",
"slug": "algebra",
"name": "Algebra",
"original_discipline": "mathematics"
} | [] | [] | A pretriangulated category is a preadditive category equipped with a shift functor by the integers and a class of distinguished triangles satisfying certain axioms. A triangulated category is a pretriangulated category in which the octahedral axiom holds. Determine whether every pretriangulated category is triangulated... | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P8654",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P8654",
"citation": null,
"doi": null,
"arxiv_id": null,
"license": "Selected packets label their content CC0-1.0; original references retain their own terms",
"notes": "Selected problem... | theorem mathoverflow_31809 : answer(sorry) ↔ (∀ (C : Type*) [Category C] [Preadditive C]
[HasZeroObject C] [HasShift C ℤ] [∀ (n : ℤ), (shiftFunctor C n).Additive]
[Pretriangulated C], IsTriangulated C) | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem mathoverflow_31809 : answer(sorry) ↔ (∀ (C : Type*) [Category C] [Preadditive C]\n [HasZeroObject C] [HasShift C ℤ] [∀ (n : ℤ), (shiftFunctor C n).Additive]\n [Pretriangulated C], IsTriangulated C)",
"target": null,
"source_url": "https://theoremdb.org/... | [] | TheoremDB P8654 · 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 P8654",
"url": "https://theoremdb.org/statements/P8654",
"note": null,
"sort_order": 0
},
{
"kind": "formalization",
"citation": "Source formal statement (not locally verified)",
"url": "https://theoremdb.org/statements/P8654",
"not... | {
"source_record": "P8654",
"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":"2026-08-17T18:59:38+00:00","display_standard":{"formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":8654,"public_status":{"evidence":{"hasFormalization":false,"hasInformalProof":false,"hasRecordedWork":false,"solutionRecord":null},"lean":{"kernelAccepted":false,"label":"","reaso... | [
"formalization-contains-placeholder",
"formalization-not-compiled-during-export",
"current-status-unassessed",
"source-full-text-not-present"
] | [] | {
"created_at": "2026-09-08T03:32:15.935Z",
"updated_at": "2026-09-08T03:32:15.935Z",
"published_at": "2026-09-08T03:33:05.040Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"357dc023-53ae-4ea6-8cbe-65d91cd1098b","format_issues":null,"number":6440} |
1.0.0 | problem:theoremdb-p8656 | problem | theoremdb-p8656 | Asymptotic growth of maximum 3-fold Sidon sets | {
"main": "math",
"sub": "combinatorics",
"slug": "combinatorics",
"name": "Combinatorics",
"original_discipline": "mathematics"
} | [] | [
{
"code": "05",
"basis": "community-map",
"evidence": "Inferred from community field: combinatorics"
}
] | For a positive integer $N$, let $f(N)$ denote the maximum cardinality of a subset $A \subseteq \{1, 2, \ldots, N\}$ such that all sums $a + b + c$ with $a, b, c \in A$ are distinct, except for the trivial coincidences arising from permutation of the summands. Does $f(N)$ satisfy the asymptotic formula
$$
f(N) \sim N^{... | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P8656",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P8656",
"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_241 :
answer(sorry) ↔ (fun N ↦ (f N 3 : ℝ)) ~[atTop] (fun N ↦ (N : ℝ) ^ ((1 : ℝ) / 3)) | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem erdos_241 :\n answer(sorry) ↔ (fun N ↦ (f N 3 : ℝ)) ~[atTop] (fun N ↦ (N : ℝ) ^ ((1 : ℝ) / 3))",
"target": null,
"source_url": "https://theoremdb.org/statements/P8656",
"format": "declaration-or-fragment",
"sha256": "6929af54a4b64d6aac0ff13c1704d92... | [] | TheoremDB P8656 · 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 P8656",
"url": "https://theoremdb.org/statements/P8656",
"note": null,
"sort_order": 0
},
{
"kind": "formalization",
"citation": "Source formal statement (not locally verified)",
"url": "https://theoremdb.org/statements/P8656",
"not... | {
"source_record": "P8656",
"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":"2026-08-17T18:59:51+00:00","display_standard":{"editorial_review":"submitter_supplied","formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":8656,"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": "2026-09-08T03:32:50.112Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"76c7eaba-d087-4ec0-9569-a4adfb3cf5b7","format_issues":null,"number":3353} |
1.0.0 | problem:theoremdb-p8658 | problem | theoremdb-p8658 | The Strong Sensitivity Conjecture | {
"main": "computer science",
"sub": "algorithms",
"slug": "algorithms",
"name": "Algorithms and data structures",
"original_discipline": "computer-science"
} | [] | [
{
"code": "68W",
"basis": "community-map",
"evidence": "Inferred from community field: algorithms"
}
] | Let $n$ be a natural number. A Boolean function on $n$ variables is a function $f : \{0,1\}^n \to \{0,1\}$. For $x \in \{0,1\}^n$ and $B \subseteq \{1, \ldots, n\}$, let $\operatorname{flip}(x, B)$ denote the element of $\{0,1\}^n$ obtained from $x$ by flipping all bits whose indices lie in $B$. The local sensitivity o... | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P8658",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P8658",
"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 strong_sensitivity_conjecture {n : ℕ} (f : (Fin n → Bool) → Bool) :
blockSensitivity f ≤ sensitivity f ^ 2 | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem strong_sensitivity_conjecture {n : ℕ} (f : (Fin n → Bool) → Bool) :\n blockSensitivity f ≤ sensitivity f ^ 2",
"target": null,
"source_url": "https://theoremdb.org/statements/P8658",
"format": "declaration-or-fragment",
"sha256": "be2ff4924268a7d19... | [] | TheoremDB P8658 · 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 P8658",
"url": "https://theoremdb.org/statements/P8658",
"note": null,
"sort_order": 0
},
{
"kind": "formalization",
"citation": "Source formal statement (not locally verified)",
"url": "https://theoremdb.org/statements/P8658",
"not... | {
"source_record": "P8658",
"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":"2026-08-17T18:59:52+00:00","display_standard":{"editorial_review":"submitter_supplied","formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":8658,"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": "2026-09-08T03:33:06.785Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"abfd18b6-4dc0-4af0-b8c7-103065807756","format_issues":null,"number":6817} |
1.0.0 | problem:theoremdb-p8660 | problem | theoremdb-p8660 | Density of hyperbolicity for Multibrot sets | {
"main": "math",
"sub": "dynamical systems",
"slug": "dynamical-systems",
"name": "Dynamical systems",
"original_discipline": "mathematics"
} | [] | [
{
"code": "37",
"basis": "community-map",
"evidence": "Inferred from community field: dynamical-systems"
}
] | For an integer $n \geq 2$, the Multibrot set of power $n$, denoted $\mathcal{M}_n$, is the set of all complex parameters $c$ for which the orbit of $0$ under iteration of the map $f_c(z) = z^n + c$ does not escape to infinity. Equivalently, $\mathcal{M}_n$ is the set of all $c \in \mathbb{C}$ such that $\|f_c^{[k]}(0)\... | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P8660",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P8660",
"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 density_of_hyperbolicity_general_exponent {n : ℕ} (hn : 2 ≤ n) :
multibrotSet n ⊆ closure {c | ∃ m z, IsAttractingCycle (fun z ↦ z ^ n + c) m z} | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem density_of_hyperbolicity_general_exponent {n : ℕ} (hn : 2 ≤ n) :\n multibrotSet n ⊆ closure {c | ∃ m z, IsAttractingCycle (fun z ↦ z ^ n + c) m z}",
"target": null,
"source_url": "https://theoremdb.org/statements/P8660",
"format": "declaration-or-fragm... | [] | TheoremDB P8660 · 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 P8660",
"url": "https://theoremdb.org/statements/P8660",
"note": null,
"sort_order": 0
},
{
"kind": "formalization",
"citation": "Source formal statement (not locally verified)",
"url": "https://theoremdb.org/statements/P8660",
"not... | {
"source_record": "P8660",
"imported_at": "2026-09-08",
"status_as_of": "2026-09-08",
"evidence_label": "Source candidate · not reviewed",
"note": "Full public directory statement. Source lane: backlog; source resolution: open. Source verification claims are attributed only; no local Lean verification has be... | {"created_at":"2026-08-17T19:00:04+00:00","display_standard":{"editorial_review":"submitter_supplied","formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":8660,"public_status":{"evidence":{"hasFormalization":false,"hasInformalProof":false,"hasRecordedWork":false,"solutionRecord":null},"lean":{... | [
"formalization-not-compiled-during-export",
"current-status-unassessed",
"source-full-text-not-present"
] | [] | {
"created_at": "2026-09-08T03:32:15.935Z",
"updated_at": "2026-09-08T03:32:15.935Z",
"published_at": "2026-09-08T03:32:53.069Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"78e92ad1-4a8b-45ae-813f-b8121929fd16","format_issues":null,"number":3997} |
1.0.0 | problem:theoremdb-p8664 | problem | theoremdb-p8664 | Giuga's formulation of the Agoh-Giuga conjecture | {
"main": "math",
"sub": "number theory",
"slug": "number-theory",
"name": "Number theory",
"original_discipline": "mathematics"
} | [] | [
{
"code": "11",
"basis": "community-map",
"evidence": "Inferred from community field: number-theory"
}
] | For every integer $p \geq 2$, determine whether $p$ is prime if and only if $p$ divides $1 + \sum_{i=1}^{p-1} i^{p-1}$. | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P8664",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P8664",
"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 agoh_giuga.variants.giuga : AgohGiugaSum | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem agoh_giuga.variants.giuga : AgohGiugaSum",
"target": null,
"source_url": "https://theoremdb.org/statements/P8664",
"format": "declaration-or-fragment",
"sha256": "41d93bcd1eb8a125951a8dcd4e4f32681b03100dd9cfc97a5c6c7a755bb65628",
"license": "Selec... | [] | TheoremDB P8664 · 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 P8664",
"url": "https://theoremdb.org/statements/P8664",
"note": null,
"sort_order": 0
},
{
"kind": "formalization",
"citation": "Source formal statement (not locally verified)",
"url": "https://theoremdb.org/statements/P8664",
"not... | {
"source_record": "P8664",
"imported_at": "2026-09-08",
"status_as_of": "2026-09-08",
"evidence_label": "Source candidate · not reviewed",
"note": "Full public directory statement. Source lane: backlog; source resolution: open. Source verification claims are attributed only; no local Lean verification has be... | {"created_at":"2026-08-17T19:03:38+00:00","display_standard":{"editorial_review":"submitter_supplied","formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":8664,"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": "2026-09-08T03:32:59.327Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"9960083c-d75e-4a7e-82be-63cd59e085ba","format_issues":null,"number":5371} |
1.0.0 | problem:theoremdb-p8666 | problem | theoremdb-p8666 | Transcendence of $e^{\pi^2}$ | {
"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 $\pi$ denote the ratio of the circumference of a circle to its diameter, and let $\exp(x) = e^x$ denote the exponential function with base $e$, where $e = \exp(1)$ is Euler's number. A real number is said to be transcendental if it is not a root of any non-zero polynomial with rational coefficients. Prove that $e^{... | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P8666",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P8666",
"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 exp_pow_pi_sq_transcendental : Transcendental ℚ (exp (π ^ 2)) | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem exp_pow_pi_sq_transcendental : Transcendental ℚ (exp (π ^ 2))",
"target": null,
"source_url": "https://theoremdb.org/statements/P8666",
"format": "declaration-or-fragment",
"sha256": "edeae182090f6cb2326698e60f38f990cec6b64c9db822303e60866a1aa3efc9",
... | [] | TheoremDB P8666 · 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 P8666",
"url": "https://theoremdb.org/statements/P8666",
"note": null,
"sort_order": 0
},
{
"kind": "formalization",
"citation": "Source formal statement (not locally verified)",
"url": "https://theoremdb.org/statements/P8666",
"not... | {
"source_record": "P8666",
"imported_at": "2026-09-08",
"status_as_of": "2026-09-08",
"evidence_label": "Source candidate · not reviewed",
"note": "Full public directory statement. Source lane: backlog; source resolution: open. Source verification claims are attributed only; no local Lean verification has be... | {"created_at":"2026-08-17T19:04:11+00:00","display_standard":{"editorial_review":"submitter_supplied","formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":8666,"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": "2026-09-08T03:33:06.977Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"84eb2eb2-6ba6-47a4-acc2-80aa304b9f8b","format_issues":null,"number":6859} |
1.0.0 | problem:theoremdb-p8668 | problem | theoremdb-p8668 | Conway's 99-Graph Problem | {
"main": "math",
"sub": "combinatorics",
"slug": "combinatorics",
"name": "Combinatorics",
"original_discipline": "mathematics"
} | [] | [
{
"code": "05",
"basis": "community-map",
"evidence": "Inferred from community field: combinatorics"
}
] | Let a graph be called locally linear if every edge belongs to exactly one triangle (that is, for every edge $\{u,v\}$, there is exactly one vertex $w$ such that $\{u,v,w\}$ forms a triangle, and no two triangles share an edge). Let a graph satisfy the non-edge diagonal property if every pair of non-adjacent vertices ha... | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P8668",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P8668",
"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 conway99Graph : answer(sorry) ↔ ∃ G : SimpleGraph (Fin 99),
G.LocallyLinear ∧ NonEdgesAreDiagonals G | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem conway99Graph : answer(sorry) ↔ ∃ G : SimpleGraph (Fin 99),\n G.LocallyLinear ∧ NonEdgesAreDiagonals G",
"target": null,
"source_url": "https://theoremdb.org/statements/P8668",
"format": "declaration-or-fragment",
"sha256": "92d2b4e719be146c241e083... | [] | TheoremDB P8668 · 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 P8668",
"url": "https://theoremdb.org/statements/P8668",
"note": null,
"sort_order": 0
},
{
"kind": "formalization",
"citation": "Source formal statement (not locally verified)",
"url": "https://theoremdb.org/statements/P8668",
"not... | {
"source_record": "P8668",
"imported_at": "2026-09-08",
"status_as_of": "2026-09-08",
"evidence_label": "Source candidate · not reviewed",
"note": "Full public directory statement. Source lane: backlog; source resolution: open. Source verification claims are attributed only; no local Lean verification has be... | {"created_at":"2026-08-17T19:04:17+00:00","display_standard":{"editorial_review":"submitter_supplied","formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":8668,"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": "2026-09-08T03:32:52.717Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"d590e516-1ad8-4ca0-a799-cf89c798cfbf","format_issues":null,"number":3920} |
1.0.0 | problem:theoremdb-p8670 | problem | theoremdb-p8670 | Termination of the sum-of-divisors iteration | {
"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$, define the iteration $f(n) = \sigma(n) - 1$, where $\sigma(n)$ denotes the sum of all positive divisors of $n$. Let $f^{[i]}$ denote the $i$-fold composition of $f$ with itself, so that $f^{[0]}(n) = n$ and $f^{[i+1]}(n) = f(f^{[i]}(n))$. Does there exist, for every integer $n > 1$, some non-... | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P8670",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P8670",
"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_termination (n : ℕ) (hn : n > 1) :
∃ i, (σ 1 · - 1)^[i] n |>.Prime | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem erdos_409.variants.sigma_termination (n : ℕ) (hn : n > 1) :\n ∃ i, (σ 1 · - 1)^[i] n |>.Prime",
"target": null,
"source_url": "https://theoremdb.org/statements/P8670",
"format": "declaration-or-fragment",
"sha256": "55cf33de8caf87ec8b530625f3fddb50... | [] | TheoremDB P8670 · 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 P8670",
"url": "https://theoremdb.org/statements/P8670",
"note": null,
"sort_order": 0
},
{
"kind": "formalization",
"citation": "Source formal statement (not locally verified)",
"url": "https://theoremdb.org/statements/P8670",
"not... | {
"source_record": "P8670",
"imported_at": "2026-09-08",
"status_as_of": "2026-09-08",
"evidence_label": "Source candidate · not reviewed",
"note": "Full public directory statement. Source lane: backlog; source resolution: open. Source verification claims are attributed only; no local Lean verification has be... | {"created_at":"2026-08-17T19:04:21+00:00","display_standard":{"formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":8670,"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": "2026-09-08T03:33:06.308Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"defb6324-783d-4d2f-b14e-fd4dd54d77c7","format_issues":null,"number":6715} |
1.0.0 | problem:theoremdb-p8672 | problem | theoremdb-p8672 | Maximal solution to the factorial product equation | {
"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 $S$ be the set of pairs $(n, (a_1, \ldots, a_k))$ consisting of a natural number $n$ and a list of natural numbers $a_1, \ldots, a_k$ such that:
$$
n! = a_1! \cdot a_2! \cdots a_k!,
$$
where $a_1 \geq a_2 \geq \cdots \geq a_k > 1$ and $a_1 < n - 1$. A pair satisfying these conditions is called a non-trivial solut... | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P8672",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P8672",
"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_373.variants.maximal_solution :
(16, [14, 5, 2]) ∈ S ∧ ∀ s ∈ S, s.fst ≤ 16 | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem erdos_373.variants.maximal_solution :\n (16, [14, 5, 2]) ∈ S ∧ ∀ s ∈ S, s.fst ≤ 16",
"target": null,
"source_url": "https://theoremdb.org/statements/P8672",
"format": "declaration-or-fragment",
"sha256": "cb78feed3e05bc442a19f1cdbe1da2483367927235d... | [] | TheoremDB P8672 · 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 P8672",
"url": "https://theoremdb.org/statements/P8672",
"note": null,
"sort_order": 0
},
{
"kind": "formalization",
"citation": "Source formal statement (not locally verified)",
"url": "https://theoremdb.org/statements/P8672",
"not... | {
"source_record": "P8672",
"imported_at": "2026-09-08",
"status_as_of": "2026-09-08",
"evidence_label": "Source candidate · not reviewed",
"note": "Full public directory statement. Source lane: backlog; source resolution: open. Source verification claims are attributed only; no local Lean verification has be... | {"created_at":"2026-08-17T19:04:29+00:00","display_standard":{"formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":8672,"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": "2026-09-08T03:33:03.661Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"9700fdf6-555c-4f9a-bca7-1d497c07378a","format_issues":null,"number":6142} |
1.0.0 | problem:theoremdb-p8674 | problem | theoremdb-p8674 | Irrationality of odd positive integer values of the Riemann zeta 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"
}
] | Let $\zeta(s)$ denote the Riemann zeta function, defined for $\Re(s) > 1$ by $\zeta(s) = \sum_{n=1}^{\infty} n^{-s}$ and extended by analytic continuation to $\mathbb{C} \setminus \{1\}$. A real number is called irrational if it is not equal to any ratio of two integers. For each positive integer $n$, determine whether... | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P8674",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P8674",
"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_odd (n : ℕ) (hn : 0 < n) :
∃ x, Irrational x ∧ riemannZeta (2 * n + 1) = x | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem irrational_odd (n : ℕ) (hn : 0 < n) :\n ∃ x, Irrational x ∧ riemannZeta (2 * n + 1) = x",
"target": null,
"source_url": "https://theoremdb.org/statements/P8674",
"format": "declaration-or-fragment",
"sha256": "919ffbacef4f883fbc71805e2b143273650861... | [] | TheoremDB P8674 · 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 P8674",
"url": "https://theoremdb.org/statements/P8674",
"note": null,
"sort_order": 0
},
{
"kind": "formalization",
"citation": "Source formal statement (not locally verified)",
"url": "https://theoremdb.org/statements/P8674",
"not... | {
"source_record": "P8674",
"imported_at": "2026-09-08",
"status_as_of": "2026-09-08",
"evidence_label": "Source candidate · not reviewed",
"note": "Full public directory statement. Source lane: backlog; source resolution: open. Source verification claims are attributed only; no local Lean verification has be... | {"created_at":"2026-08-17T19:04:31+00:00","display_standard":{"editorial_review":"submitter_supplied","formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":8674,"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": "2026-09-08T03:33:01.399Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"aaebbdb7-8097-438b-9108-abaa13b910a0","format_issues":null,"number":5722} |
1.0.0 | problem:theoremdb-p8676 | problem | theoremdb-p8676 | Erdős's intersecting subfamily conjecture for lower set systems | {
"main": "math",
"sub": "combinatorics",
"slug": "combinatorics",
"name": "Combinatorics",
"original_discipline": "mathematics"
} | [] | [
{
"code": "05",
"basis": "community-map",
"evidence": "Inferred from community field: combinatorics"
}
] | Let $X$ be a nonempty finite set. A family $\mathcal{F}$ of subsets of $X$ is called a lower set (or downset) if whenever $B \subseteq A \in \mathcal{F}$, we have $B \in \mathcal{F}$. A subfamily $\mathcal{F}' \subseteq \mathcal{F}$ is called intersecting if every pair of sets in $\mathcal{F}'$ has nonempty intersectio... | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P8676",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P8676",
"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_701 : answer(sorry) ↔ ∀ {X : Type} [Nonempty X] [Fintype X],
∀ (F : Set (Set X)), IsLowerSet F →
∃ x : X, ∀ᵉ (F' ⊆ F),
F'.Intersecting →
(#F' ≤ #{ A : Set X | A ∈ F ∧ x ∈ A }) | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem erdos_701 : answer(sorry) ↔ ∀ {X : Type} [Nonempty X] [Fintype X],\n ∀ (F : Set (Set X)), IsLowerSet F →\n ∃ x : X, ∀ᵉ (F' ⊆ F),\n F'.Intersecting →\n (#F' ≤ #{ A : Set X | A ∈ F ∧ x ∈ A })",
"target": null,
"source_url": "https://theo... | [] | TheoremDB P8676 · 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 P8676",
"url": "https://theoremdb.org/statements/P8676",
"note": null,
"sort_order": 0
},
{
"kind": "formalization",
"citation": "Source formal statement (not locally verified)",
"url": "https://theoremdb.org/statements/P8676",
"not... | {
"source_record": "P8676",
"imported_at": "2026-09-08",
"status_as_of": "2026-09-08",
"evidence_label": "Source candidate · not reviewed",
"note": "Full public directory statement. Source lane: backlog; source resolution: open. Source verification claims are attributed only; no local Lean verification has be... | {"created_at":"2026-08-17T19:04:35+00:00","display_standard":{"editorial_review":"submitter_supplied","formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":8676,"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": "2026-09-08T03:32:56.419Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"96a664f9-116b-4478-b7fb-e273ea795119","format_issues":null,"number":4736} |
1.0.0 | problem:theoremdb-p8678 | problem | theoremdb-p8678 | Uniqueness of the extremal progression in Rudin'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"
}
] | For an integer $N \geq 6$, let $Q(N; q, a)$ denote the number of perfect squares among the first $N$ terms of the arithmetic progression $\{qn + a : 0 \leq n < N\}$. A pair $(q, a)$ of natural numbers is called non-trivial if $q \geq 1$, $a \geq 1$, $\gcd(q, a) = 1$, and $(q, a) \neq (1, 1)$. Let $Q(N)$ denote the maxi... | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P8678",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P8678",
"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 rudins_conjecture_unique (N : ℕ) (hN : 6 ≤ N) (q a : ℕ)
(hqa : IsNontrivial q a) (hmax : Q N q a = Qmax N) : q = 24 | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem rudins_conjecture_unique (N : ℕ) (hN : 6 ≤ N) (q a : ℕ)\n (hqa : IsNontrivial q a) (hmax : Q N q a = Qmax N) : q = 24",
"target": null,
"source_url": "https://theoremdb.org/statements/P8678",
"format": "declaration-or-fragment",
"sha256": "afb61d93... | [] | TheoremDB P8678 · 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 P8678",
"url": "https://theoremdb.org/statements/P8678",
"note": null,
"sort_order": 0
},
{
"kind": "formalization",
"citation": "Source formal statement (not locally verified)",
"url": "https://theoremdb.org/statements/P8678",
"not... | {
"source_record": "P8678",
"imported_at": "2026-09-08",
"status_as_of": "2026-09-08",
"evidence_label": "Source candidate · not reviewed",
"note": "Full public directory statement. Source lane: backlog; source resolution: open. Source verification claims are attributed only; no local Lean verification has be... | {"created_at":"2026-08-17T19:04:38+00:00","display_standard":{"editorial_review":"submitter_supplied","formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":8678,"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": "2026-09-08T03:33:07.260Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"3ceb3e7b-1d0d-476f-a2ea-e31bdaa70cd3","format_issues":null,"number":6921} |
1.0.0 | problem:theoremdb-p8680 | problem | theoremdb-p8680 | Transcendence of lacunary binary series with rapidly growing exponents | {
"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_k)_{k=1}^{\infty}$ be a strictly increasing sequence of natural numbers such that
$$
\limsup_{k \to \infty} \frac{n_k}{k} = \infty.
$$
Is the real number
$$
\sum_{k=1}^{\infty} \frac{1}{2^{n_k}}
$$
transcendental over $\mathbb{Q}$? | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P8680",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P8680",
"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_247 : answer(sorry) ↔ ∀ (n : ℕ → ℕ), (StrictMono n) →
atTop.limsup (fun k => (n k / k.succ : EReal)) = ⊤ →
Transcendental ℚ (∑' k, (1 : ℝ) / 2 ^ n k) | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem erdos_247 : answer(sorry) ↔ ∀ (n : ℕ → ℕ), (StrictMono n) →\n atTop.limsup (fun k => (n k / k.succ : EReal)) = ⊤ →\n Transcendental ℚ (∑' k, (1 : ℝ) / 2 ^ n k)",
"target": null,
"source_url": "https://theoremdb.org/statements/P8680",
"format": "decl... | [] | TheoremDB P8680 · 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 P8680",
"url": "https://theoremdb.org/statements/P8680",
"note": null,
"sort_order": 0
},
{
"kind": "formalization",
"citation": "Source formal statement (not locally verified)",
"url": "https://theoremdb.org/statements/P8680",
"not... | {
"source_record": "P8680",
"imported_at": "2026-09-08",
"status_as_of": "2026-09-08",
"evidence_label": "Source candidate · not reviewed",
"note": "Full public directory statement. Source lane: backlog; source resolution: open. Source verification claims are attributed only; no local Lean verification has be... | {"created_at":"2026-08-17T19:04:41+00:00","display_standard":{"editorial_review":"submitter_supplied","formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":8680,"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": "2026-09-08T03:33:06.994Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"64f6bf59-c582-4e09-9485-335f32e42809","format_issues":null,"number":6862} |
1.0.0 | problem:theoremdb-p8682 | problem | theoremdb-p8682 | Primality of successors in sequence A067720 | {
"main": "math",
"sub": "number theory",
"slug": "number-theory",
"name": "Number theory",
"original_discipline": "mathematics"
} | [] | [
{
"code": "11",
"basis": "community-map",
"evidence": "Inferred from community field: number-theory"
}
] | Let $\varphi$ denote Euler's totient function. A positive integer $k$ is said to belong to sequence A067720 if $\varphi(k^2 + 1) = k \cdot \varphi(k + 1)$. Suppose $k$ belongs to sequence A067720 and $k \neq 8$. Prove that $k + 1$ is prime. | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P8682",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P8682",
"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 prime_add_one_of_a {k : ℕ} (h : A k) (hne : k ≠ 8) : (k + 1).Prime | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem prime_add_one_of_a {k : ℕ} (h : A k) (hne : k ≠ 8) : (k + 1).Prime",
"target": null,
"source_url": "https://theoremdb.org/statements/P8682",
"format": "declaration-or-fragment",
"sha256": "20dc45b5c5b39b53273c983dcae6ef241e35bf9c1d585e62d6321af3d167da... | [] | TheoremDB P8682 · 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 P8682",
"url": "https://theoremdb.org/statements/P8682",
"note": null,
"sort_order": 0
},
{
"kind": "formalization",
"citation": "Source formal statement (not locally verified)",
"url": "https://theoremdb.org/statements/P8682",
"not... | {
"source_record": "P8682",
"imported_at": "2026-09-08",
"status_as_of": "2026-09-08",
"evidence_label": "Source candidate · not reviewed",
"note": "Full public directory statement. Source lane: backlog; source resolution: open. Source verification claims are attributed only; no local Lean verification has be... | {"created_at":"2026-08-17T19:04:43+00:00","display_standard":{"formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":8682,"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": "2026-09-08T03:33:05.053Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"c6514635-7312-495f-a874-c3a72021ddaf","format_issues":null,"number":6443} |
1.0.0 | problem:theoremdb-p8684 | problem | theoremdb-p8684 | Non-existence of real monochromatic quantum graphs for even orders at least six and at least three colours | {
"main": "math",
"sub": "combinatorics",
"slug": "combinatorics",
"name": "Combinatorics",
"original_discipline": "mathematics"
} | [] | [
{
"code": "05",
"basis": "community-map",
"evidence": "Inferred from community field: combinatorics"
}
] | Let $N$ and $D$ be positive integers with $N \geq 6$ even and $D \geq 3$. Consider a complete graph on $N$ vertices whose edges are coloured with $D$ colours and assigned real weights. For each vertex colouring $\iota$ (an assignment of one of the $D$ colours to each vertex), define the \emph{perfect-matching sum} $S(\... | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P8684",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P8684",
"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_real :
answer(sorry) ↔
∀ N D : Nat, N ≥ 6 → Even N → D ≥ 3 →
¬ ∃ W : WeightsN N D ℝ, EqSystemN N D W | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem eqSystem_no_solution_ge6_ge3_real :\n answer(sorry) ↔\n ∀ N D : Nat, N ≥ 6 → Even N → D ≥ 3 →\n ¬ ∃ W : WeightsN N D ℝ, EqSystemN N D W",
"target": null,
"source_url": "https://theoremdb.org/statements/P8684",
"format": "declaration-or-frag... | [] | TheoremDB P8684 · 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 P8684",
"url": "https://theoremdb.org/statements/P8684",
"note": null,
"sort_order": 0
},
{
"kind": "formalization",
"citation": "Source formal statement (not locally verified)",
"url": "https://theoremdb.org/statements/P8684",
"not... | {
"source_record": "P8684",
"imported_at": "2026-09-08",
"status_as_of": "2026-09-08",
"evidence_label": "Source candidate · not reviewed",
"note": "Full public directory statement. Source lane: backlog; source resolution: open. Source verification claims are attributed only; no local Lean verification has be... | {"created_at":"2026-08-17T19:04:46+00:00","display_standard":{"editorial_review":"submitter_supplied","formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":8684,"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": "2026-09-08T03:33:04.410Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"c7c4687a-4506-4cdb-ba8f-0b11388501c0","format_issues":null,"number":6304} |
1.0.0 | problem:theoremdb-p8686 | problem | theoremdb-p8686 | Asymptotic ratio of the extremal triangle-covering numbers | {
"main": "math",
"sub": "combinatorics",
"slug": "combinatorics",
"name": "Combinatorics",
"original_discipline": "mathematics"
} | [] | [
{
"code": "05",
"basis": "community-map",
"evidence": "Inferred from community field: combinatorics"
}
] | For integers $n \geq 3$ and $r \geq 1$, let $e(n,r)$ denote the minimum number of edges in an $n$-vertex graph $G$ with the property that every edge of $G$ lies in at least one triangle, and some edge of $G$ lies in at least $r$ triangles. Let $r \geq 2$. Does the ratio $\frac{e(n,r+1)}{e(n,r)}$ converge to $1$ as $n \... | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P8686",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P8686",
"citation": null,
"doi": null,
"arxiv_id": null,
"license": "Selected packets label their content CC0-1.0; original references retain their own terms",
"notes": "Selected problem... | theorem erdos_600.parts.ii :
answer(sorry) ↔ ∀ r : ℕ, 2 ≤ r →
Tendsto (fun (n : ℕ) ↦ (eFunction n (r + 1) : ℝ) / (eFunction n r : ℝ)) atTop (𝓝 1) | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem erdos_600.parts.ii :\n answer(sorry) ↔ ∀ r : ℕ, 2 ≤ r →\n Tendsto (fun (n : ℕ) ↦ (eFunction n (r + 1) : ℝ) / (eFunction n r : ℝ)) atTop (𝓝 1)",
"target": null,
"source_url": "https://theoremdb.org/statements/P8686",
"format": "declaration-or-frag... | [] | TheoremDB P8686 · 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 P8686",
"url": "https://theoremdb.org/statements/P8686",
"note": null,
"sort_order": 0
},
{
"kind": "formalization",
"citation": "Source formal statement (not locally verified)",
"url": "https://theoremdb.org/statements/P8686",
"not... | {
"source_record": "P8686",
"imported_at": "2026-09-08",
"status_as_of": "2026-09-08",
"evidence_label": "Source candidate · not reviewed",
"note": "Full public directory statement. Source lane: backlog; source resolution: open. Source verification claims are attributed only; no local Lean verification has be... | {"created_at":"2026-08-17T19:04:49+00:00","display_standard":{"editorial_review":"submitter_supplied","formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":8686,"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": "2026-09-08T03:32:50.382Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"737f0fc9-4de7-4a60-9026-d070f42dd3b6","format_issues":null,"number":3408} |
1.0.0 | problem:theoremdb-p8688 | problem | theoremdb-p8688 | Prime case of Kurepa'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"
}
] | For a prime number $p > 2$, let the left factorial of $p$, denoted $!p$, be defined as $!p = \sum_{m=0}^{p-1} m! = 0! + 1! + 2! + \cdots + (p-1)!$. Prove that $!p \not\equiv 0 \pmod{p}$. | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P8688",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P8688",
"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 kurepa_conjecture.variants.prime (p : ℕ) (h_p : 2 < p) :
p.Prime → (!p : ℕ) % p ≠ 0 | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem kurepa_conjecture.variants.prime (p : ℕ) (h_p : 2 < p) :\n p.Prime → (!p : ℕ) % p ≠ 0",
"target": null,
"source_url": "https://theoremdb.org/statements/P8688",
"format": "declaration-or-fragment",
"sha256": "ac5f510cade76960dd8d0387a9c3c522718758eb... | [] | TheoremDB P8688 · 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 P8688",
"url": "https://theoremdb.org/statements/P8688",
"note": null,
"sort_order": 0
},
{
"kind": "formalization",
"citation": "Source formal statement (not locally verified)",
"url": "https://theoremdb.org/statements/P8688",
"not... | {
"source_record": "P8688",
"imported_at": "2026-09-08",
"status_as_of": "2026-09-08",
"evidence_label": "Source candidate · not reviewed",
"note": "Full public directory statement. Source lane: backlog; source resolution: open. Source verification claims are attributed only; no local Lean verification has be... | {"created_at":"2026-08-17T19:04:54+00:00","display_standard":{"formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":8688,"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": "2026-09-08T03:33:05.067Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"06ce6339-d0f7-4d33-95e0-e52364a3252f","format_issues":null,"number":6446} |
1.0.0 | problem:theoremdb-p8690 | problem | theoremdb-p8690 | Transcendence of $\pi^{\sqrt{2}}$ | {
"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 transcendental if it is not a root of any non-zero polynomial with rational coefficients. Let $\pi$ denote the ratio of the circumference of a circle to its diameter, and let $\sqrt{2}$ denote the positive real number whose square equals $2$. Determine whether $\pi^{\sqrt{2}}$ is transcenden... | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P8690",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P8690",
"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_pow_sqrt_two_transcendental : Transcendental ℚ (π ^ √2) | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem pi_pow_sqrt_two_transcendental : Transcendental ℚ (π ^ √2)",
"target": null,
"source_url": "https://theoremdb.org/statements/P8690",
"format": "declaration-or-fragment",
"sha256": "3bcf00009ffeca9a9614737a3f58baf9a0bb9f82c8578396dbfc16381bb039f1",
... | [] | TheoremDB P8690 · 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 P8690",
"url": "https://theoremdb.org/statements/P8690",
"note": null,
"sort_order": 0
},
{
"kind": "formalization",
"citation": "Source formal statement (not locally verified)",
"url": "https://theoremdb.org/statements/P8690",
"not... | {
"source_record": "P8690",
"imported_at": "2026-09-08",
"status_as_of": "2026-09-08",
"evidence_label": "Source candidate · not reviewed",
"note": "Full public directory statement. Source lane: backlog; source resolution: open. Source verification claims are attributed only; no local Lean verification has be... | {"created_at":"2026-08-17T19:04:58+00:00","display_standard":{"editorial_review":"submitter_supplied","formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":8690,"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": "2026-09-08T03:33:06.964Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"45351959-40e0-419b-b3f3-a746719bfb00","format_issues":null,"number":6856} |
1.0.0 | problem:theoremdb-p8692 | problem | theoremdb-p8692 | Finiteness of Unitary Perfect 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 unitary perfect number if it equals the sum of its proper unitary divisors. A proper unitary divisor of $n$ is a divisor $d$ of $n$ such that $1 \leq d < n$ and $d$ is coprime to $n/d$ (that is, $\gcd(d, n/d) = 1$). Are there only finitely many unitary perfect numbers? | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P8692",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P8692",
"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_1052 :
answer(sorry) ↔ {n | IsUnitaryPerfect n}.Finite | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem erdos_1052 :\n answer(sorry) ↔ {n | IsUnitaryPerfect n}.Finite",
"target": null,
"source_url": "https://theoremdb.org/statements/P8692",
"format": "declaration-or-fragment",
"sha256": "06a71fd7df0265d872abdb3ee86a7367cad2f5f1929ff1471d08af22c294e92... | [] | TheoremDB P8692 · 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 P8692",
"url": "https://theoremdb.org/statements/P8692",
"note": null,
"sort_order": 0
},
{
"kind": "formalization",
"citation": "Source formal statement (not locally verified)",
"url": "https://theoremdb.org/statements/P8692",
"not... | {
"source_record": "P8692",
"imported_at": "2026-09-08",
"status_as_of": "2026-09-08",
"evidence_label": "Source candidate · not reviewed",
"note": "Full public directory statement. Source lane: backlog; source resolution: open. Source verification claims are attributed only; no local Lean verification has be... | {"created_at":"2026-08-17T19:05:02+00:00","display_standard":{"editorial_review":"submitter_supplied","formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":8692,"public_status":{"evidence":{"hasFormalization":false,"hasInformalProof":false,"hasRecordedWork":false,"solutionRecord":null},"lean":{... | [
"formalization-contains-placeholder",
"formalization-not-compiled-during-export",
"current-status-unassessed",
"source-full-text-not-present"
] | [] | {
"created_at": "2026-09-08T03:32:15.935Z",
"updated_at": "2026-09-08T03:32:15.935Z",
"published_at": "2026-09-08T03:32:59.130Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"4aaf70e9-21b9-41ca-b2d9-638406230541","format_issues":null,"number":5329} |
1.0.0 | problem:theoremdb-p8694 | problem | theoremdb-p8694 | Upper density of the Ulam 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 defined as follows. Let a unique Ulam sum for a sequence $a : \mathbb{N} \to \mathbb{N}$ at stage $n$ be an integer $m$ that can be written uniquely as $a(i) + a(j)$ with $i < j < n$. The Ulam sequence is the sequence $a$ with $a(0) = 1$, $a(1) = 2$, and for... | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P8694",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P8694",
"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.iii :
answer(sorry) ↔
∀ a : ℕ → ℕ, IsUlamSequence a →
Set.upperDensity (Set.range a) = 0 | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem erdos_342.parts.iii :\n answer(sorry) ↔\n ∀ a : ℕ → ℕ, IsUlamSequence a →\n Set.upperDensity (Set.range a) = 0",
"target": null,
"source_url": "https://theoremdb.org/statements/P8694",
"format": "declaration-or-fragment",
"sha256": "5e2... | [] | TheoremDB P8694 · 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/P8694",
"note": null,
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},
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"citation": "Source formal statement (not locally verified)",
"url": "https://theoremdb.org/statements/P8694",
"not... | {
"source_record": "P8694",
"imported_at": "2026-09-08",
"status_as_of": "2026-09-08",
"evidence_label": "Source candidate · not reviewed",
"note": "Full public directory statement. Source lane: backlog; source resolution: open. Source verification claims are attributed only; no local Lean verification has be... | {"created_at":"2026-08-17T19:05:09+00:00","display_standard":{"editorial_review":"submitter_supplied","formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":8694,"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": "2026-09-08T03:33:07.408Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"275149b4-9e1a-478b-9953-6bcb1e240654","format_issues":null,"number":6953} |
1.0.0 | problem:theoremdb-p8696 | problem | theoremdb-p8696 | Lower bound on the largest induced forest size via residue and diameter | {
"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 simple connected graph. The \emph{Havel-Hakimi residue} of $G$, denoted $\mathrm{residue}(G)$, is the number of zeros remaining after iteratively applying the Havel-Hakimi reduction to the degree sequence of $G$: given a non-increasing degree sequence $d_1 \geq d_2 \geq \cdots \geq d_n$, remove $d_1$ and s... | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P8696",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P8696",
"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 conjecture61 (G : SimpleGraph α) [DecidableRel G.Adj] (h : G.Connected) :
(residue G : ℝ) + ⌈(G.diam : ℝ) / 3⌉ ≤ (G.largestInducedForestSize : ℝ) | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem conjecture61 (G : SimpleGraph α) [DecidableRel G.Adj] (h : G.Connected) :\n (residue G : ℝ) + ⌈(G.diam : ℝ) / 3⌉ ≤ (G.largestInducedForestSize : ℝ)",
"target": null,
"source_url": "https://theoremdb.org/statements/P8696",
"format": "declaration-or-frag... | [] | TheoremDB P8696 · 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 P8696",
"url": "https://theoremdb.org/statements/P8696",
"note": null,
"sort_order": 0
},
{
"kind": "formalization",
"citation": "Source formal statement (not locally verified)",
"url": "https://theoremdb.org/statements/P8696",
"not... | {
"source_record": "P8696",
"imported_at": "2026-09-08",
"status_as_of": "2026-09-08",
"evidence_label": "Source candidate · not reviewed",
"note": "Full public directory statement. Source lane: backlog; source resolution: open. Source verification claims are attributed only; no local Lean verification has be... | {"created_at":"2026-08-17T19:05:12+00:00","display_standard":{"formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":8696,"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": "2026-09-08T03:33:03.574Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"2e78e911-6c8c-4f26-929b-797ab6f34b14","format_issues":null,"number":6124} |
1.0.0 | problem:theoremdb-p8698 | problem | theoremdb-p8698 | Existence of Large Cosets in Iterated Sumsets over Finite Vector Spaces | {
"main": "math",
"sub": "combinatorics",
"slug": "combinatorics",
"name": "Combinatorics",
"original_discipline": "mathematics"
} | [] | [
{
"code": "05",
"basis": "community-map",
"evidence": "Inferred from community field: combinatorics"
}
] | Let $\mathbb{F}_2^n$ denote the $n$-dimensional vector space over the field with two elements. For a subset $A \subseteq \mathbb{F}_2^n$, the density of $A$ is $\alpha = |A| / 2^n$. For a positive integer $k$, the $k$-fold sumset $kA$ is the set of all sums of $k$ elements from $A$ with repetition allowed, that is, $kA... | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P8698",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P8698",
"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_50 : answer(sorry) ↔
∃ C > (0 : ℝ), ∀ n : ℕ, ∀ A : Finset (𝔽₂ n),
A.Nonempty →
let α : ℝ := A.dens
∃ (W : Submodule (ZMod 2) (𝔽₂ n)) (v : 𝔽₂ n),
v +ᵥ (W : Set (𝔽₂ n)) ⊆ ↑(10 • A) ∧
(n : ℝ) - C * Real.logb 2 (1 / α) ≤ Module.finrank (ZMod 2) W | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem green_50 : answer(sorry) ↔\n ∃ C > (0 : ℝ), ∀ n : ℕ, ∀ A : Finset (𝔽₂ n),\n A.Nonempty →\n let α : ℝ := A.dens\n ∃ (W : Submodule (ZMod 2) (𝔽₂ n)) (v : 𝔽₂ n),\n v +ᵥ (W : Set (𝔽₂ n)) ⊆ ↑(10 • A) ∧\n (n : ℝ) - C * Real.logb 2 (1 / α) ≤ Module... | [] | TheoremDB P8698 · 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 P8698",
"url": "https://theoremdb.org/statements/P8698",
"note": null,
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"kind": "formalization",
"citation": "Source formal statement (not locally verified)",
"url": "https://theoremdb.org/statements/P8698",
"not... | {
"source_record": "P8698",
"imported_at": "2026-09-08",
"status_as_of": "2026-09-08",
"evidence_label": "Source candidate · not reviewed",
"note": "Full public directory statement. Source lane: backlog; source resolution: open. Source verification claims are attributed only; no local Lean verification has be... | {"created_at":"2026-08-17T19:05:28+00:00","display_standard":{"editorial_review":"submitter_supplied","formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":8698,"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": "2026-09-08T03:32:58.757Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"e77b0b29-84ae-4c03-b301-9afd2202e94f","format_issues":null,"number":5249} |
1.0.0 | problem:theoremdb-p8700 | problem | theoremdb-p8700 | The Hadwiger–Nelson Problem | {
"main": "math",
"sub": "geometry and topology",
"slug": "geometry-and-topology",
"name": "Geometry and topology",
"original_discipline": "mathematics"
} | [] | [] | Let the unit-distance graph of the plane be the simple graph whose vertex set is $\mathbb{R}^2$ and in which two distinct points are adjacent if and only if their Euclidean distance equals $1$. The chromatic number of this graph, denoted $\chi(\mathbb{R}^2)$, is the smallest number of colors needed to color all points ... | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P8700",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P8700",
"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 HadwigerNelsonProblem :
χ(ℝ²) = answer(sorry) | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem HadwigerNelsonProblem :\n χ(ℝ²) = answer(sorry)",
"target": null,
"source_url": "https://theoremdb.org/statements/P8700",
"format": "declaration-or-fragment",
"sha256": "5dff17518a766eb48121e5fed635326de4752ba7f567fa395a49b668cdc142ad",
"licens... | [] | TheoremDB P8700 · 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 P8700",
"url": "https://theoremdb.org/statements/P8700",
"note": null,
"sort_order": 0
},
{
"kind": "formalization",
"citation": "Source formal statement (not locally verified)",
"url": "https://theoremdb.org/statements/P8700",
"not... | {
"source_record": "P8700",
"imported_at": "2026-09-08",
"status_as_of": "2026-09-08",
"evidence_label": "Source candidate · not reviewed",
"note": "Full public directory statement. Source lane: backlog; source resolution: open. Source verification claims are attributed only; no local Lean verification has be... | {"created_at":"2026-08-17T19:05:35+00:00","display_standard":{"formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":8700,"public_status":{"evidence":{"hasFormalization":false,"hasInformalProof":false,"hasRecordedWork":false,"solutionRecord":null},"lean":{"kernelAccepted":false,"label":"","reaso... | [
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"current-status-unassessed",
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"created_at": "2026-09-08T03:32:15.935Z",
"updated_at": "2026-09-08T03:32:15.935Z",
"published_at": "2026-09-08T03:33:06.525Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"15318cab-ed89-447e-bc17-b2033de54166","format_issues":null,"number":6761} |
1.0.0 | problem:theoremdb-p8702 | problem | theoremdb-p8702 | Asymptotic Growth of the Excess in Erdős's Composite 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) = \max\{m + p(m) \mid m < n \text{ and } m \text{ is composite}\},
$$
where $p(m)$ denotes the least prime divisor of $m$. Determine whether $F(n) - n \to \infty$ as $n \to \infty$. | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P8702",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P8702",
"citation": null,
"doi": null,
"arxiv_id": null,
"license": "Selected packets label their content CC0-1.0; original references retain their own terms",
"notes": "Selected problem... | theorem erdos_385.parts.ii : answer(sorry) ↔ atTop.Tendsto (fun n ↦ F n - n) atTop | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem erdos_385.parts.ii : answer(sorry) ↔ atTop.Tendsto (fun n ↦ F n - n) atTop",
"target": null,
"source_url": "https://theoremdb.org/statements/P8702",
"format": "declaration-or-fragment",
"sha256": "359b31d8cb73a787f682cd7643da004fa74de5d4f90e64a090aa73... | [] | TheoremDB P8702 · 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 P8702",
"url": "https://theoremdb.org/statements/P8702",
"note": null,
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"citation": "Source formal statement (not locally verified)",
"url": "https://theoremdb.org/statements/P8702",
"not... | {
"source_record": "P8702",
"imported_at": "2026-09-08",
"status_as_of": "2026-09-08",
"evidence_label": "Source candidate · not reviewed",
"note": "Full public directory statement. Source lane: backlog; source resolution: open. Source verification claims are attributed only; no local Lean verification has be... | {"created_at":"2026-08-17T19:05:52+00:00","display_standard":{"editorial_review":"submitter_supplied","formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":8702,"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": "2026-09-08T03:32:50.167Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"003af4db-70c4-4044-b794-d9cd7bc9a0b0","format_issues":null,"number":3365} |
1.0.0 | problem:theoremdb-p8704 | problem | theoremdb-p8704 | Infinitely many primes uniquely determined by their minimal totient preimage | {
"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 $m_n$ denote the smallest positive integer $k$ such that $n$ divides Euler's totient function $\phi(k)$. Determine whether there are infinitely many prime numbers $q$ such that $q - 1$ is the unique positive integer $n$ satisfying $m_n = q$. | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P8704",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P8704",
"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_456.parts.iii :
answer(sorry) ↔
{ q | q.Prime ∧ ∀ n, m n = q ↔ n = q - 1 }.Infinite | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem erdos_456.parts.iii :\n answer(sorry) ↔\n { q | q.Prime ∧ ∀ n, m n = q ↔ n = q - 1 }.Infinite",
"target": null,
"source_url": "https://theoremdb.org/statements/P8704",
"format": "declaration-or-fragment",
"sha256": "e36569ec46a318a89bf71d3a8cc... | [] | TheoremDB P8704 · 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 P8704",
"url": "https://theoremdb.org/statements/P8704",
"note": null,
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"citation": "Source formal statement (not locally verified)",
"url": "https://theoremdb.org/statements/P8704",
"not... | {
"source_record": "P8704",
"imported_at": "2026-09-08",
"status_as_of": "2026-09-08",
"evidence_label": "Source candidate · not reviewed",
"note": "Full public directory statement. Source lane: backlog; source resolution: open. Source verification claims are attributed only; no local Lean verification has be... | {"created_at":"2026-08-17T19:06:03+00:00","display_standard":{"editorial_review":"submitter_supplied","formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":8704,"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": "2026-09-08T03:33:01.001Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"cab1b7de-edc3-4e14-87bc-1173e9e417f3","format_issues":null,"number":5635} |
1.0.0 | problem:theoremdb-p8706 | problem | theoremdb-p8706 | The Union-Closed Sets Conjecture | {
"main": "math",
"sub": "combinatorics",
"slug": "combinatorics",
"name": "Combinatorics",
"original_discipline": "mathematics"
} | [] | [
{
"code": "05",
"basis": "community-map",
"evidence": "Inferred from community field: combinatorics"
}
] | Let $n$ be a nonempty finite type with decidable equality, and let $A$ be a finite set of finite subsets of $n$. A family $A$ is called union-closed if for every $X \in A$ and every $Y \in A$, the union $X \cup Y$ also belongs to $A$. Suppose that $A \neq \{\emptyset\}$ and that $A$ is union-closed. Does there exist an... | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P8706",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P8706",
"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 union_closed
[Nonempty n]
(h_ne_singleton_empty : A ≠ {∅})
(h_union_closed : IsUnionClosed A) :
∃ i : n, (1 / 2 : ℚ) * #A ≤ #{x ∈ A | i ∈ x} | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem union_closed\n [Nonempty n]\n (h_ne_singleton_empty : A ≠ {∅})\n (h_union_closed : IsUnionClosed A) :\n ∃ i : n, (1 / 2 : ℚ) * #A ≤ #{x ∈ A | i ∈ x}",
"target": null,
"source_url": "https://theoremdb.org/statements/P8706",
"format": "declarati... | [] | TheoremDB P8706 · 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 P8706",
"url": "https://theoremdb.org/statements/P8706",
"note": null,
"sort_order": 0
},
{
"kind": "formalization",
"citation": "Source formal statement (not locally verified)",
"url": "https://theoremdb.org/statements/P8706",
"not... | {
"source_record": "P8706",
"imported_at": "2026-09-08",
"status_as_of": "2026-09-08",
"evidence_label": "Source candidate · not reviewed",
"note": "Full public directory statement. Source lane: backlog; source resolution: open. Source verification claims are attributed only; no local Lean verification has be... | {"created_at":"2026-08-17T19:06:11+00:00","display_standard":{"editorial_review":"submitter_supplied","formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":8706,"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": "2026-09-08T03:33:06.835Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"a22716a7-b44a-461b-b16c-3588ff0ebaf2","format_issues":null,"number":6828} |
1.0.0 | problem:theoremdb-p8708 | problem | theoremdb-p8708 | Termination of the greedy algorithm for Egyptian fractions with square denominators | {
"main": "math",
"sub": "combinatorics",
"slug": "combinatorics",
"name": "Combinatorics",
"original_discipline": "mathematics"
} | [] | [
{
"code": "05",
"basis": "community-map",
"evidence": "Inferred from community field: combinatorics"
}
] | For a rational number $x$, define the greedy unit-fraction remainder sequence with respect to a set $A \subseteq \mathbb{N}$ as follows. Set $r_0 = x - 1/n_0$, where $n_0$ is the smallest element of $A$ with $n_0 \geq 1/x$. Inductively, if $r_t \leq 0$ then set $r_{t+1} = 0$; otherwise set $r_{t+1} = r_t - 1/n_{t+1}$, ... | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P8708",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P8708",
"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_282.variants.sq :
answer(sorry) ↔ ∀ x : ℚ, (x : ℝ) ∈ Set.Ico 0 (π ^ 2 / 6 - 1) ∪ Set.Ico 1 (π ^ 2 / 6) →
greedyUnitFractionRem { n | IsSquare n } x =ᶠ[atTop] 0 | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem erdos_282.variants.sq :\n answer(sorry) ↔ ∀ x : ℚ, (x : ℝ) ∈ Set.Ico 0 (π ^ 2 / 6 - 1) ∪ Set.Ico 1 (π ^ 2 / 6) →\n greedyUnitFractionRem { n | IsSquare n } x =ᶠ[atTop] 0",
"target": null,
"source_url": "https://theoremdb.org/statements/P8708",
"fo... | [] | TheoremDB P8708 · 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 P8708",
"url": "https://theoremdb.org/statements/P8708",
"note": null,
"sort_order": 0
},
{
"kind": "formalization",
"citation": "Source formal statement (not locally verified)",
"url": "https://theoremdb.org/statements/P8708",
"not... | {
"source_record": "P8708",
"imported_at": "2026-09-08",
"status_as_of": "2026-09-08",
"evidence_label": "Source candidate · not reviewed",
"note": "Full public directory statement. Source lane: backlog; source resolution: open. Source verification claims are attributed only; no local Lean verification has be... | {"created_at":"2026-08-17T19:06:25+00:00","display_standard":{"formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":8708,"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": "2026-09-08T03:33:06.291Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"15c39538-a7ca-4705-a0e4-db9d0824fad0","format_issues":null,"number":6712} |
1.0.0 | problem:theoremdb-p8710 | problem | theoremdb-p8710 | Explicit lower bound for the maximal length of a distinct consecutive-sums sequence | {
"main": "math",
"sub": "number theory",
"slug": "number-theory",
"name": "Number theory",
"original_discipline": "mathematics"
} | [] | [
{
"code": "11",
"basis": "community-map",
"evidence": "Inferred from community field: number-theory"
}
] | For each positive integer $n$, let $f(n)$ denote the largest integer $k$ for which there exist integers $1 \le a_1 < a_2 < \dots < a_k \le n$ such that all sums of the form $\sum_{i=u}^{v} a_i$ with $1 \le u \le v \le k$ are distinct. Determine an explicit function $g : \mathbb{N} \to \mathbb{R}$ such that $g(n) = O(f(... | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P8710",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P8710",
"citation": null,
"doi": null,
"arxiv_id": null,
"license": "Selected packets label their content CC0-1.0; original references retain their own terms",
"notes": "Selected problem... | theorem erdos_357.parts.ii.bigO_version :
(answer(sorry) : ℕ → ℝ) =O[atTop] (fun n ↦ (f n : ℝ)) | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem erdos_357.parts.ii.bigO_version :\n (answer(sorry) : ℕ → ℝ) =O[atTop] (fun n ↦ (f n : ℝ))",
"target": null,
"source_url": "https://theoremdb.org/statements/P8710",
"format": "declaration-or-fragment",
"sha256": "a08f86deec2898fba49978c2e47c320d7021... | [] | TheoremDB P8710 · 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 P8710",
"url": "https://theoremdb.org/statements/P8710",
"note": null,
"sort_order": 0
},
{
"kind": "formalization",
"citation": "Source formal statement (not locally verified)",
"url": "https://theoremdb.org/statements/P8710",
"not... | {
"source_record": "P8710",
"imported_at": "2026-09-08",
"status_as_of": "2026-09-08",
"evidence_label": "Source candidate · not reviewed",
"note": "Full public directory statement. Source lane: backlog; source resolution: open. Source verification claims are attributed only; no local Lean verification has be... | {"created_at":"2026-08-17T19:06:35+00:00","display_standard":{"editorial_review":"submitter_supplied","formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":8710,"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": "2026-09-08T03:32:58.897Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"5cfc3342-b8cd-4c1c-af1b-9dbb71655eae","format_issues":null,"number":5278} |
1.0.0 | problem:theoremdb-p8714 | problem | theoremdb-p8714 | Kurepa's conjecture on left 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 integer $n > 2$, define the left factorial of $n$ by
$$
!n = \sum_{m=0}^{n-1} m! = 0! + 1! + 2! + \cdots + (n-1)!.
$$
Prove that $!n \not\equiv 0 \pmod{n}$ for every integer $n > 2$. | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P8714",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P8714",
"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 kurepa_conjecture (n : ℕ) (h_n : 2 < n) : (!n : ℕ) % n ≠ 0 | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem kurepa_conjecture (n : ℕ) (h_n : 2 < n) : (!n : ℕ) % n ≠ 0",
"target": null,
"source_url": "https://theoremdb.org/statements/P8714",
"format": "declaration-or-fragment",
"sha256": "e7690b4bfc28f57ce4635017d6127b2e4f6da49209c35e1fb6db32088122f1ff",
... | [] | TheoremDB P8714 · 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 P8714",
"url": "https://theoremdb.org/statements/P8714",
"note": null,
"sort_order": 0
},
{
"kind": "formalization",
"citation": "Source formal statement (not locally verified)",
"url": "https://theoremdb.org/statements/P8714",
"not... | {
"source_record": "P8714",
"imported_at": "2026-09-08",
"status_as_of": "2026-09-08",
"evidence_label": "Source candidate · not reviewed",
"note": "Full public directory statement. Source lane: backlog; source resolution: open. Source verification claims are attributed only; no local Lean verification has be... | {"created_at":"2026-08-17T19:10:23+00:00","display_standard":{"formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":8714,"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": "2026-09-08T03:33:02.892Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"5ddebd6b-c293-494a-9410-5f80c1586db5","format_issues":null,"number":5977} |
1.0.0 | problem:theoremdb-p8716 | problem | theoremdb-p8716 | Non-existence of monochromatic quantum graphs with six vertices and five colours over the complex numbers | {
"main": "math",
"sub": "combinatorics",
"slug": "combinatorics",
"name": "Combinatorics",
"original_discipline": "mathematics"
} | [] | [
{
"code": "05",
"basis": "community-map",
"evidence": "Inferred from community field: combinatorics"
}
] | Let $N$ and $D$ be positive integers. A perfect matching on $N$ vertices is a set of $N/2$ pairwise disjoint edges whose union contains all $N$ vertices; we require $N$ to be even for such a matching to exist. An edge-coloured, edge-weighted complete graph with $N$ vertices and $D$ colours is specified by a weight func... | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P8716",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P8716",
"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 :
answer(sorry) ↔
¬ ∃ W : WeightsN 6 5 ℂ, EqSystemN 6 5 W | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem eqSystem6_no_solution_d5 :\n answer(sorry) ↔\n ¬ ∃ W : WeightsN 6 5 ℂ, EqSystemN 6 5 W",
"target": null,
"source_url": "https://theoremdb.org/statements/P8716",
"format": "declaration-or-fragment",
"sha256": "c9d44b636f2b8a333ceb20b21e1654c744... | [] | TheoremDB P8716 · 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 P8716",
"url": "https://theoremdb.org/statements/P8716",
"note": null,
"sort_order": 0
},
{
"kind": "formalization",
"citation": "Source formal statement (not locally verified)",
"url": "https://theoremdb.org/statements/P8716",
"not... | {
"source_record": "P8716",
"imported_at": "2026-09-08",
"status_as_of": "2026-09-08",
"evidence_label": "Source candidate · not reviewed",
"note": "Full public directory statement. Source lane: backlog; source resolution: open. Source verification claims are attributed only; no local Lean verification has be... | {"created_at":"2026-08-17T19:10:37+00:00","display_standard":{"editorial_review":"submitter_supplied","formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":8716,"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": "2026-09-08T03:33:04.332Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"73a71105-4ac0-4b16-87f5-883bec414232","format_issues":null,"number":6288} |
1.0.0 | problem:theoremdb-p8718 | problem | theoremdb-p8718 | The Bateman–Horn conjecture on simultaneous prime values of polynomials | {
"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 $\mathcal{S}$ be a finite nonempty set of polynomials with integer coefficients. Assume that every $f \in \mathcal{S}$ satisfies the Bunyakovsky condition: $f$ is irreducible over $\mathbb{Z}$, has positive leading coefficient, and the set of values $\{f(n) : n \in \mathbb{N}\}$ has no common divisor greater than $... | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P8718",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P8718",
"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 bateman_horn_conjecture
(polys : Finset ℤ[X])
(h_nonempty : polys.Nonempty)
(h_irreducible : ∀ f ∈ polys, BunyakovskyCondition f)
(h_compat : SchinzelCondition polys) :
(fun x : ℝ => (CountSimultaneousPrimes polys x : ℝ)) ~[atTop]
(fun x : ℝ => BatemanHornConstant polys * x / (Real.log x... | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem bateman_horn_conjecture\n (polys : Finset ℤ[X])\n (h_nonempty : polys.Nonempty)\n (h_irreducible : ∀ f ∈ polys, BunyakovskyCondition f)\n (h_compat : SchinzelCondition polys) :\n (fun x : ℝ => (CountSimultaneousPrimes polys x : ℝ)) ~[atTop]\n (fun x... | [] | TheoremDB P8718 · 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 P8718",
"url": "https://theoremdb.org/statements/P8718",
"note": null,
"sort_order": 0
},
{
"kind": "formalization",
"citation": "Source formal statement (not locally verified)",
"url": "https://theoremdb.org/statements/P8718",
"not... | {
"source_record": "P8718",
"imported_at": "2026-09-08",
"status_as_of": "2026-09-08",
"evidence_label": "Source candidate · not reviewed",
"note": "Full public directory statement. Source lane: backlog; source resolution: open. Source verification claims are attributed only; no local Lean verification has be... | {"created_at":"2026-08-17T19:10:41+00:00","display_standard":{"formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":8718,"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": "2026-09-08T03:33:06.380Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"a57c45e4-6c69-4381-83c3-3be8e9bf0d69","format_issues":null,"number":6730} |
1.0.0 | problem:theoremdb-p8720 | problem | theoremdb-p8720 | Infinitude of barriers for the number of distinct prime divisors | {
"main": "math",
"sub": "number theory",
"slug": "number-theory",
"name": "Number theory",
"original_discipline": "mathematics"
} | [] | [
{
"code": "11",
"basis": "community-map",
"evidence": "Inferred from community field: number-theory"
}
] | For a positive integer $m$, let $\omega(m)$ denote the number of distinct prime divisors of $m$. A natural number $n$ is called a *barrier* for a function $f: \mathbb{N} \to \mathbb{R}$ if $m + f(m) \leq n$ for all $m < n$. Determine whether there are infinitely many natural numbers $n$ that are barriers for the functi... | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P8720",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P8720",
"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_413.parts.i :
answer(sorry) ↔ { n | IsBarrier (fun m => ω m) n }.Infinite | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem erdos_413.parts.i :\n answer(sorry) ↔ { n | IsBarrier (fun m => ω m) n }.Infinite",
"target": null,
"source_url": "https://theoremdb.org/statements/P8720",
"format": "declaration-or-fragment",
"sha256": "675184bbeeae6a6a5a77189266d3ea51a924a2935e3f... | [] | TheoremDB P8720 · 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 P8720",
"url": "https://theoremdb.org/statements/P8720",
"note": null,
"sort_order": 0
},
{
"kind": "formalization",
"citation": "Source formal statement (not locally verified)",
"url": "https://theoremdb.org/statements/P8720",
"not... | {
"source_record": "P8720",
"imported_at": "2026-09-08",
"status_as_of": "2026-09-08",
"evidence_label": "Source candidate · not reviewed",
"note": "Full public directory statement. Source lane: backlog; source resolution: open. Source verification claims are attributed only; no local Lean verification has be... | {"created_at":"2026-08-17T19:10:53+00:00","display_standard":{"editorial_review":"submitter_supplied","formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":8720,"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": "2026-09-08T03:33:01.095Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"a2461e9c-cbae-4426-b42f-ab8f276ff838","format_issues":null,"number":5656} |
1.0.0 | problem:theoremdb-p8722 | problem | theoremdb-p8722 | Erdős's almost everywhere convergence problem for lacunary sequences with triple-logarithmic Fourier approximation | {
"main": "math",
"sub": "analysis",
"slug": "analysis",
"name": "Analysis",
"original_discipline": "mathematics"
} | [] | [] | Let $\mathbb{T} = \mathbb{R}/\mathbb{Z}$ denote the circle group with normalized Haar measure. For $f \in L^2(\mathbb{T})$ and $k \in \mathbb{N}$, define the $k$-th Fourier partial sum of $f$ by
$$
f_k(x) = \sum_{j=-k}^{k} \widehat{f}(j) e^{2\pi i j x},
$$
where $\widehat{f}(j) = \int_{\mathbb{T}} f(t) e^{-2\pi i j t... | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P8722",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P8722",
"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_996 : answer(sorry) ↔
∃ (C : ℝ), 0 < C ∧ ∀ (f : Lp ℂ 2 (haarAddCircle (T := 1))) (n : ℕ → ℕ),
IsLacunary n →
(fun k => (eLpNorm (fourierPartial f k) 2 (haarAddCircle (T := 1))).toReal) =O[atTop]
(fun k => 1 / (log (log (log k))) ^ C)
→
∀ᵐ x, Tendsto (fun N => (∑ k ∈ .range N, f (n ... | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem erdos_996 : answer(sorry) ↔\n ∃ (C : ℝ), 0 < C ∧ ∀ (f : Lp ℂ 2 (haarAddCircle (T := 1))) (n : ℕ → ℕ),\n IsLacunary n →\n (fun k => (eLpNorm (fourierPartial f k) 2 (haarAddCircle (T := 1))).toReal) =O[atTop]\n (fun k => 1 / (log (log (log k))) ^ C)\n →\... | [] | TheoremDB P8722 · 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 P8722",
"url": "https://theoremdb.org/statements/P8722",
"note": null,
"sort_order": 0
},
{
"kind": "formalization",
"citation": "Source formal statement (not locally verified)",
"url": "https://theoremdb.org/statements/P8722",
"not... | {
"source_record": "P8722",
"imported_at": "2026-09-08",
"status_as_of": "2026-09-08",
"evidence_label": "Source candidate · not reviewed",
"note": "Full public directory statement. Source lane: backlog; source resolution: open. Source verification claims are attributed only; no local Lean verification has be... | {"created_at":"2026-08-17T19:11:03+00:00","display_standard":{"editorial_review":"submitter_supplied","formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":8722,"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": "2026-09-08T03:32:56.311Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"6d8985f3-f22b-4db4-ba92-f7758102cc8b","format_issues":null,"number":4713} |
1.0.0 | problem:theoremdb-p8724 | problem | theoremdb-p8724 | Characterization of orders admitting a complete set of mutually orthogonal 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 each column. Two Latin squares of the same order are orthogonal if, when superimposed, each ordered pair of symbols occurs at most once. A family of Latin squares is mutually orthogonal if... | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P8724",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P8724",
"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 molsExistenceProblem : answer(sorry) = {n : ℕ | HasCompleteMOLS n} | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem molsExistenceProblem : answer(sorry) = {n : ℕ | HasCompleteMOLS n}",
"target": null,
"source_url": "https://theoremdb.org/statements/P8724",
"format": "declaration-or-fragment",
"sha256": "2a3f931ff0b6d7a549f94ebbbc0205d3ae360957ccf0f2453df068a0684091... | [] | TheoremDB P8724 · 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 P8724",
"url": "https://theoremdb.org/statements/P8724",
"note": null,
"sort_order": 0
},
{
"kind": "formalization",
"citation": "Source formal statement (not locally verified)",
"url": "https://theoremdb.org/statements/P8724",
"not... | {
"source_record": "P8724",
"imported_at": "2026-09-08",
"status_as_of": "2026-09-08",
"evidence_label": "Source candidate · not reviewed",
"note": "Full public directory statement. Source lane: backlog; source resolution: open. Source verification claims are attributed only; no local Lean verification has be... | {"created_at":"2026-08-17T19:11:04+00:00","display_standard":{"formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":8724,"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": "2026-09-08T03:32:51.082Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"374b51e4-162c-4fcb-a72c-0699380c9018","format_issues":null,"number":3557} |
1.0.0 | problem:theoremdb-p8726 | problem | theoremdb-p8726 | Composite Fermat Numbers Beyond Index 4 | {
"main": "math",
"sub": "number theory",
"slug": "number-theory",
"name": "Number theory",
"original_discipline": "mathematics"
} | [] | [
{
"code": "11",
"basis": "community-map",
"evidence": "Inferred from community field: number-theory"
}
] | For each non-negative integer $n$, the $n$-th Fermat number is defined as $F_n = 2^{2^n} + 1$. It is known that $F_0 = 3$, $F_1 = 5$, $F_2 = 17$, $F_3 = 257$, and $F_4 = 65537$ are all prime, while $F_5 = 4294967297 = 641 \times 6700417$ is composite. Determine whether every Fermat number $F_n$ with $n > 4$ is composit... | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P8726",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P8726",
"citation": null,
"doi": null,
"arxiv_id": null,
"license": "Selected packets label their content CC0-1.0; original references retain their own terms",
"notes": "Selected problem... | theorem fermat_number_are_composite : answer(sorry) ↔ ∀ n > 4, ¬Prime n.fermatNumber | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem fermat_number_are_composite : answer(sorry) ↔ ∀ n > 4, ¬Prime n.fermatNumber",
"target": null,
"source_url": "https://theoremdb.org/statements/P8726",
"format": "declaration-or-fragment",
"sha256": "3115ea43b320c7e5b3e863edb3ebf9b58620917a2a4074bf93fe... | [] | TheoremDB P8726 · 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 P8726",
"url": "https://theoremdb.org/statements/P8726",
"note": null,
"sort_order": 0
},
{
"kind": "formalization",
"citation": "Source formal statement (not locally verified)",
"url": "https://theoremdb.org/statements/P8726",
"not... | {
"source_record": "P8726",
"imported_at": "2026-09-08",
"status_as_of": "2026-09-08",
"evidence_label": "Source candidate · not reviewed",
"note": "Full public directory statement. Source lane: backlog; source resolution: open. Source verification claims are attributed only; no local Lean verification has be... | {"created_at":"2026-08-17T19:11:19+00:00","display_standard":{"editorial_review":"submitter_supplied","formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":8726,"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": "2026-09-08T03:32:52.533Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"b31d84b5-85cd-4cc2-8a6d-49c666395d5d","format_issues":null,"number":3881} |
1.0.0 | problem:theoremdb-p8728 | problem | theoremdb-p8728 | Erdős Problem 598 on Countable-Subset Colourings | {
"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 $m$ be an infinite set and let $\kappa = (2^{\aleph_0})^+$ be the successor cardinal of the continuum. Let $\mathcal{C}(m)$ denote the collection of all countable subsets of $m$. Does there exist a colouring $c: \mathcal{C}(m) \to \kappa$ such that for every subset $X \subseteq m$ with $|X| = \kappa$, the image of ... | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P8728",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P8728",
"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_598 : answer(sorry) ↔
∃ c : { s : Set m // s.Countable } → κ.out,
∀ X : Set m, #X = κ →
c '' { s : { sub : Set m // sub.Countable } | s.1 ⊆ X } = Set.univ | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem erdos_598 : answer(sorry) ↔\n ∃ c : { s : Set m // s.Countable } → κ.out,\n ∀ X : Set m, #X = κ →\n c '' { s : { sub : Set m // sub.Countable } | s.1 ⊆ X } = Set.univ",
"target": null,
"source_url": "https://theoremdb.org/statements/P8728",
"form... | [] | TheoremDB P8728 · 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 P8728",
"url": "https://theoremdb.org/statements/P8728",
"note": null,
"sort_order": 0
},
{
"kind": "formalization",
"citation": "Source formal statement (not locally verified)",
"url": "https://theoremdb.org/statements/P8728",
"not... | {
"source_record": "P8728",
"imported_at": "2026-09-08",
"status_as_of": "2026-09-08",
"evidence_label": "Source candidate · not reviewed",
"note": "Full public directory statement. Source lane: backlog; source resolution: open. Source verification claims are attributed only; no local Lean verification has be... | {"created_at":"2026-08-17T19:11:21+00:00","display_standard":{"formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":8728,"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": "2026-09-08T03:32:55.709Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"67e70a93-9a01-43ce-96fd-b32f04be5385","format_issues":null,"number":4579} |
1.0.0 | problem:theoremdb-p8730 | problem | theoremdb-p8730 | Non-existence of integer-weighted monochromatic quantum graphs with ten vertices and three colours | {
"main": "math",
"sub": "combinatorics",
"slug": "combinatorics",
"name": "Combinatorics",
"original_discipline": "mathematics"
} | [] | [
{
"code": "05",
"basis": "community-map",
"evidence": "Inferred from community field: combinatorics"
}
] | Let $N$ be a positive even integer and let $D$ be a positive integer. A *quantum graph* with $N$ vertices and $D$ colours consists of a complete graph on $N$ vertices together with an edge-colouring by $D$ colours and an assignment of a weight from a coefficient domain to each edge. For each vertex colouring $\iota : \... | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P8730",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P8730",
"citation": null,
"doi": null,
"arxiv_id": null,
"license": "Selected packets label their content CC0-1.0; original references retain their own terms",
"notes": "Selected problem... | theorem eqSystem10_no_solution_d3_int :
answer(sorry) ↔
¬ ∃ W : WeightsN 10 3 ℤ, EqSystemN 10 3 W | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem eqSystem10_no_solution_d3_int :\n answer(sorry) ↔\n ¬ ∃ W : WeightsN 10 3 ℤ, EqSystemN 10 3 W",
"target": null,
"source_url": "https://theoremdb.org/statements/P8730",
"format": "declaration-or-fragment",
"sha256": "34fc76e02ef3e812721a0af7010... | [] | TheoremDB P8730 · 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 P8730",
"url": "https://theoremdb.org/statements/P8730",
"note": null,
"sort_order": 0
},
{
"kind": "formalization",
"citation": "Source formal statement (not locally verified)",
"url": "https://theoremdb.org/statements/P8730",
"not... | {
"source_record": "P8730",
"imported_at": "2026-09-08",
"status_as_of": "2026-09-08",
"evidence_label": "Source candidate · not reviewed",
"note": "Full public directory statement. Source lane: backlog; source resolution: open. Source verification claims are attributed only; no local Lean verification has be... | {"created_at":"2026-08-17T19:11:33+00:00","display_standard":{"editorial_review":"submitter_supplied","formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":8730,"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": "2026-09-08T03:33:04.285Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"b47d311f-85db-459c-8b1c-faa2f2b23b8b","format_issues":null,"number":6279} |
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