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1.0.0 | problem:theoremdb-p8126 | problem | theoremdb-p8126 | Erdős's Problem on Composite Numbers Near Integers with Growing Offset | {
"main": "math",
"sub": "number theory",
"slug": "number-theory",
"name": "Number theory",
"original_discipline": "mathematics"
} | [] | [
{
"code": "11",
"basis": "community-map",
"evidence": "Inferred from community field: number-theory"
}
] | Does there exist a function $f : \mathbb{N} \to \mathbb{N}$ such that $f(n) \to \infty$ as $n \to \infty$, with the property that for all sufficiently large $n$, there exists a composite number $m$ satisfying
$$
n + f(n) < m < n + p(m),
$$
where $p(m)$ denotes the least prime factor of $m$? | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P8126",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P8126",
"citation": null,
"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_463 : answer(sorry) ↔ ∃ (f : ℕ → ℕ) (_ : Tendsto f atTop atTop),
∀ᶠ n in atTop,
∃ m, m.Composite ∧
n + f n < m ∧ m < n + m.minFac | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem erdos_463 : answer(sorry) ↔ ∃ (f : ℕ → ℕ) (_ : Tendsto f atTop atTop),\n ∀ᶠ n in atTop,\n ∃ m, m.Composite ∧\n n + f n < m ∧ m < n + m.minFac",
"target": null,
"source_url": "https://theoremdb.org/statements/P8126",
"format": "declaration-o... | [] | TheoremDB P8126 · Candidate awaiting source review | null | Source-reported assessment, retrieved 8 September 2026. Frontier Science has not independently reviewed this status.
Open. No signed Lean verification is attached to this problem or its selected resolution.
This record is in the source backlog; its novelty and unresolved status have not passed source review. | null | published | incremental | hard | [
"source-candidate",
"theoremdb"
] | null | null | null | null | [
{
"kind": "website",
"citation": "TheoremDB P8126",
"url": "https://theoremdb.org/statements/P8126",
"note": null,
"sort_order": 0
},
{
"kind": "formalization",
"citation": "Source formal statement (not locally verified)",
"url": "https://theoremdb.org/statements/P8126",
"not... | {
"source_record": "P8126",
"imported_at": "2026-09-08",
"status_as_of": "2026-09-08",
"evidence_label": "Source candidate · not reviewed",
"note": "Full public directory statement. Source lane: backlog; source resolution: open. Source verification claims are attributed only; no local Lean verification has be... | {"created_at":"2026-08-17T18:34:53+00:00","display_standard":{"editorial_review":"submitter_supplied","formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":8126,"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": "2026-09-08T03:32:56.456Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"431fa63b-19db-4fa4-8e81-4e7d746e6dd1","format_issues":null,"number":4744} |
1.0.0 | problem:theoremdb-p8128 | problem | theoremdb-p8128 | The p-adic formulation of the Hilbert–Smith conjecture | {
"main": "math",
"sub": "geometry and topology",
"slug": "geometry-and-topology",
"name": "Geometry and topology",
"original_discipline": "mathematics"
} | [] | [] | Let $p$ be a prime number. Let $\mathbb{Z}_p$ denote the ring of $p$-adic integers, equipped with its profinite topology. Let $X$ be a connected, Hausdorff, finite-dimensional topological manifold. Suppose $\mathbb{Z}_p$ acts continuously and faithfully on $X$ by homeomorphisms. Does this lead to a contradiction? | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P8128",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P8128",
"citation": null,
"doi": null,
"arxiv_id": null,
"license": "Selected packets label their content CC0-1.0; original references retain their own terms",
"notes": "Selected problem... | theorem hilbert_smith_padic_formulation (p : ℕ) [Fact p.Prime]
[AddAction (PadicInt p) X] [ContinuousVAdd (PadicInt p) X]
[FaithfulVAdd (PadicInt p) X] :
False | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem hilbert_smith_padic_formulation (p : ℕ) [Fact p.Prime]\n [AddAction (PadicInt p) X] [ContinuousVAdd (PadicInt p) X]\n [FaithfulVAdd (PadicInt p) X] :\n False",
"target": null,
"source_url": "https://theoremdb.org/statements/P8128",
"format": "dec... | [] | TheoremDB P8128 · Candidate awaiting source review | null | Source-reported assessment, retrieved 8 September 2026. Frontier Science has not independently reviewed this status.
Open. No signed Lean verification is attached to this problem or its selected resolution.
This record is in the source backlog; its novelty and unresolved status have not passed source review. | null | published | incremental | hard | [
"source-candidate",
"theoremdb"
] | null | null | null | null | [
{
"kind": "website",
"citation": "TheoremDB P8128",
"url": "https://theoremdb.org/statements/P8128",
"note": null,
"sort_order": 0
},
{
"kind": "formalization",
"citation": "Source formal statement (not locally verified)",
"url": "https://theoremdb.org/statements/P8128",
"not... | {
"source_record": "P8128",
"imported_at": "2026-09-08",
"status_as_of": "2026-09-08",
"evidence_label": "Source candidate · not reviewed",
"note": "Full public directory statement. Source lane: backlog; source resolution: open. Source verification claims are attributed only; no local Lean verification has be... | {"created_at":"2026-08-17T18:34:54+00:00","display_standard":{"formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":8128,"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": "2026-09-08T03:33:06.694Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"14bb95f1-b4e4-4a10-a592-4e585a6d3b4b","format_issues":null,"number":6798} |
1.0.0 | problem:theoremdb-p8130 | problem | theoremdb-p8130 | Non-vanishing of the fractional parts of powers of e | {
"main": "math",
"sub": "number theory",
"slug": "number-theory",
"name": "Number theory",
"original_discipline": "mathematics"
} | [] | [
{
"code": "11",
"basis": "community-map",
"evidence": "Inferred from community field: number-theory"
}
] | For a real number $x$, let $\lVert x \rVert$ denote the distance from $x$ to the nearest integer, that is, $\lVert x \rVert = \min_{m \in \mathbb{Z}} |x - m|$. Prove that the sequence $\lVert e^n \rVert$ does not converge to $0$ as $n \to \infty$, where $e$ denotes the base of the natural logarithm. | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P8130",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P8130",
"citation": null,
"doi": null,
"arxiv_id": null,
"license": "Selected packets label their content CC0-1.0; original references retain their own terms",
"notes": "Selected problem... | theorem problem_10_2 :
¬ Filter.Tendsto (fun n : ℕ ↦ distToNearestInt (Real.exp n)) Filter.atTop (nhds 0) | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem problem_10_2 :\n ¬ Filter.Tendsto (fun n : ℕ ↦ distToNearestInt (Real.exp n)) Filter.atTop (nhds 0)",
"target": null,
"source_url": "https://theoremdb.org/statements/P8130",
"format": "declaration-or-fragment",
"sha256": "a8846cfc9d78c40f315437420b... | [] | TheoremDB P8130 · Candidate awaiting source review | null | Source-reported assessment, retrieved 8 September 2026. Frontier Science has not independently reviewed this status.
Open. No signed Lean verification is attached to this problem or its selected resolution.
This record is in the source backlog; its novelty and unresolved status have not passed source review. | null | published | incremental | hard | [
"source-candidate",
"theoremdb"
] | null | null | null | null | [
{
"kind": "website",
"citation": "TheoremDB P8130",
"url": "https://theoremdb.org/statements/P8130",
"note": null,
"sort_order": 0
},
{
"kind": "formalization",
"citation": "Source formal statement (not locally verified)",
"url": "https://theoremdb.org/statements/P8130",
"not... | {
"source_record": "P8130",
"imported_at": "2026-09-08",
"status_as_of": "2026-09-08",
"evidence_label": "Source candidate · not reviewed",
"note": "Full public directory statement. Source lane: backlog; source resolution: open. Source verification claims are attributed only; no local Lean verification has be... | {"created_at":"2026-08-17T18:34:57+00:00","display_standard":{"editorial_review":"submitter_supplied","formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":8130,"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": "2026-09-08T03:33:04.459Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"cd03e782-1870-4baa-b870-28bc9292a7fe","format_issues":null,"number":6314} |
1.0.0 | problem:theoremdb-p8132 | problem | theoremdb-p8132 | Coprime reciprocal-sum pairs avoiding 0 and 1 | {
"main": "math",
"sub": "combinatorics",
"slug": "combinatorics",
"name": "Combinatorics",
"original_discipline": "mathematics"
} | [] | [
{
"code": "05",
"basis": "community-map",
"evidence": "Inferred from community field: combinatorics"
}
] | Let two finite sets of natural numbers $P$ and $Q$ be called a coprime reciprocal-sum pair if every pair of distinct elements in $P$ is coprime, every pair of distinct elements in $Q$ is coprime, and
$$
1 = \left(\sum_{p \in P} \frac{1}{p}\right)\left(\sum_{q \in Q} \frac{1}{q}\right).
$$
Does there exist a coprime r... | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P8132",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P8132",
"citation": null,
"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_307.variants.coprime_one_notMem : answer(sorry) ↔ ∃ P Q : Finset ℕ, 0 ∉ P ∩ Q ∧ 1 ∉ P ∪ Q ∧
1 < #P ∧ 1 < #Q ∧ Set.Pairwise P Nat.Coprime ∧ Set.Pairwise Q Nat.Coprime ∧
1 = (∑ p ∈ P, (p : ℚ)⁻¹) * (∑ q ∈ Q, (q : ℚ)⁻¹) | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem erdos_307.variants.coprime_one_notMem : answer(sorry) ↔ ∃ P Q : Finset ℕ, 0 ∉ P ∩ Q ∧ 1 ∉ P ∪ Q ∧\n 1 < #P ∧ 1 < #Q ∧ Set.Pairwise P Nat.Coprime ∧ Set.Pairwise Q Nat.Coprime ∧\n 1 = (∑ p ∈ P, (p : ℚ)⁻¹) * (∑ q ∈ Q, (q : ℚ)⁻¹)",
"target": null,
"source_u... | [] | TheoremDB P8132 · Candidate awaiting source review | null | Source-reported assessment, retrieved 8 September 2026. Frontier Science has not independently reviewed this status.
Open. No signed Lean verification is attached to this problem or its selected resolution.
This record is in the source backlog; its novelty and unresolved status have not passed source review. | null | published | incremental | hard | [
"source-candidate",
"theoremdb"
] | null | null | null | null | [
{
"kind": "website",
"citation": "TheoremDB P8132",
"url": "https://theoremdb.org/statements/P8132",
"note": null,
"sort_order": 0
},
{
"kind": "formalization",
"citation": "Source formal statement (not locally verified)",
"url": "https://theoremdb.org/statements/P8132",
"not... | {
"source_record": "P8132",
"imported_at": "2026-09-08",
"status_as_of": "2026-09-08",
"evidence_label": "Source candidate · not reviewed",
"note": "Full public directory statement. Source lane: backlog; source resolution: open. Source verification claims are attributed only; no local Lean verification has be... | {"created_at":"2026-08-17T18:34:58+00:00","display_standard":{"editorial_review":"submitter_supplied","formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":8132,"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": "2026-09-08T03:32:52.735Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"5c058bc7-2dea-4cd1-8a2f-c17cb30681b9","format_issues":null,"number":3924} |
1.0.0 | problem:theoremdb-p8134 | problem | theoremdb-p8134 | The limit superior of a prime reciprocal sum | {
"main": "math",
"sub": "number theory",
"slug": "number-theory",
"name": "Number theory",
"original_discipline": "mathematics"
} | [] | [
{
"code": "11",
"basis": "community-map",
"evidence": "Inferred from community field: number-theory"
}
] | For each positive integer $n$, define
$$
f(n)=\sum_{\substack{p<n\\ p\text{ prime}}}\frac{1}{n-p},
$$
where the sum ranges over all prime numbers $p$ strictly less than $n$. Does the limit superior of $f(n)$ equal $+\infty$? That is, does $\limsup_{n\to\infty}f(n)=\infty$ hold? | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P8134",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P8134",
"citation": null,
"doi": null,
"arxiv_id": null,
"license": "Selected packets label their content CC0-1.0; original references retain their own terms",
"notes": "Selected problem... | theorem erdos_950.parts.ii :
answer(sorry) ↔ atTop.limsup (fun n : ℕ ↦ (f n : EReal)) = ⊤ | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem erdos_950.parts.ii :\n answer(sorry) ↔ atTop.limsup (fun n : ℕ ↦ (f n : EReal)) = ⊤",
"target": null,
"source_url": "https://theoremdb.org/statements/P8134",
"format": "declaration-or-fragment",
"sha256": "421d42a2a5e3bc4d289a2a5666dd7358bce055efcc... | [] | TheoremDB P8134 · Candidate awaiting source review | null | Source-reported assessment, retrieved 8 September 2026. Frontier Science has not independently reviewed this status.
Open. No signed Lean verification is attached to this problem or its selected resolution.
This record is in the source backlog; its novelty and unresolved status have not passed source review. | null | published | incremental | hard | [
"source-candidate",
"theoremdb"
] | null | null | null | null | [
{
"kind": "website",
"citation": "TheoremDB P8134",
"url": "https://theoremdb.org/statements/P8134",
"note": null,
"sort_order": 0
},
{
"kind": "formalization",
"citation": "Source formal statement (not locally verified)",
"url": "https://theoremdb.org/statements/P8134",
"not... | {
"source_record": "P8134",
"imported_at": "2026-09-08",
"status_as_of": "2026-09-08",
"evidence_label": "Source candidate · not reviewed",
"note": "Full public directory statement. Source lane: backlog; source resolution: open. Source verification claims are attributed only; no local Lean verification has be... | {"created_at":"2026-08-17T18:34:59+00:00","display_standard":{"formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":8134,"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": "2026-09-08T03:33:06.627Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"1e472707-1e39-4f2a-b828-03b8bb349363","format_issues":null,"number":6783} |
1.0.0 | problem:theoremdb-p8136 | problem | theoremdb-p8136 | Irrationality of reciprocal Fibonacci subseries with superlinear index growth | {
"main": "math",
"sub": "number theory",
"slug": "number-theory",
"name": "Number theory",
"original_discipline": "mathematics"
} | [] | [
{
"code": "11",
"basis": "community-map",
"evidence": "Inferred from community field: number-theory"
}
] | Let $(F_n)_{n\ge 1}$ denote the Fibonacci sequence, defined by $F_1 = F_2 = 1$ and $F_{n+1} = F_n + F_{n-1}$ for $n \ge 2$. Let $(n_k)_{k\ge 1}$ be a strictly increasing sequence of positive integers such that $\displaystyle\lim_{k\to\infty} \frac{n_{k+1}}{k} = \infty$. Must the series $\displaystyle\sum_{k=1}^{\infty}... | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P8136",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P8136",
"citation": null,
"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_267.variants.generalisation_ratio_limit_to_infinity : answer(sorry) ↔ ∀ (n : ℕ → ℕ),
StrictMono n → Filter.Tendsto (fun k => (n (k+1) / k.succ : ℝ)) Filter.atTop Filter.atTop →
Irrational (∑' k, 1 / (Nat.fib <| n k)) | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem erdos_267.variants.generalisation_ratio_limit_to_infinity : answer(sorry) ↔ ∀ (n : ℕ → ℕ),\n StrictMono n → Filter.Tendsto (fun k => (n (k+1) / k.succ : ℝ)) Filter.atTop Filter.atTop →\n Irrational (∑' k, 1 / (Nat.fib <| n k))",
"target": null,
"source_... | [] | TheoremDB P8136 · Candidate awaiting source review | null | Source-reported assessment, retrieved 8 September 2026. Frontier Science has not independently reviewed this status.
Open. No signed Lean verification is attached to this problem or its selected resolution.
This record is in the source backlog; its novelty and unresolved status have not passed source review. | null | published | incremental | hard | [
"source-candidate",
"theoremdb"
] | null | null | null | null | [
{
"kind": "website",
"citation": "TheoremDB P8136",
"url": "https://theoremdb.org/statements/P8136",
"note": null,
"sort_order": 0
},
{
"kind": "formalization",
"citation": "Source formal statement (not locally verified)",
"url": "https://theoremdb.org/statements/P8136",
"not... | {
"source_record": "P8136",
"imported_at": "2026-09-08",
"status_as_of": "2026-09-08",
"evidence_label": "Source candidate · not reviewed",
"note": "Full public directory statement. Source lane: backlog; source resolution: open. Source verification claims are attributed only; no local Lean verification has be... | {"created_at":"2026-08-17T18:35:01+00:00","display_standard":{"editorial_review":"submitter_supplied","formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":8136,"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": "2026-09-08T03:33:01.404Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"70da81e3-5d5d-4a1b-aac9-d8ed7e3c4593","format_issues":null,"number":5723} |
1.0.0 | problem:theoremdb-p8138 | problem | theoremdb-p8138 | Asymptotic growth of the maximal length of a distinct consecutive-sums sequence | {
"main": "math",
"sub": "number theory",
"slug": "number-theory",
"name": "Number theory",
"original_discipline": "mathematics"
} | [] | [
{
"code": "11",
"basis": "community-map",
"evidence": "Inferred from community field: number-theory"
}
] | For each positive integer $n$, let $f(n)$ denote the largest integer $k$ for which there exist integers $1 \leq a_1 < a_2 < \dots < a_k \leq n$ such that all sums of the form $\sum_{i=u}^{v} a_i$ with $1 \leq u \leq v \leq k$ are distinct. Prove that $f(n) = o(n)$. | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P8138",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P8138",
"citation": null,
"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.i : (fun n ↦ (f n : ℝ)) =o[atTop] (fun n ↦ (n : ℝ)) | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem erdos_357.parts.i : (fun n ↦ (f n : ℝ)) =o[atTop] (fun n ↦ (n : ℝ))",
"target": null,
"source_url": "https://theoremdb.org/statements/P8138",
"format": "declaration-or-fragment",
"sha256": "b544397478546a14565418e0c63108300c8d656f1c15ea3ce994e5d335320... | [] | TheoremDB P8138 · Candidate awaiting source review | null | Source-reported assessment, retrieved 8 September 2026. Frontier Science has not independently reviewed this status.
Open. No signed Lean verification is attached to this problem or its selected resolution.
This record is in the source backlog; its novelty and unresolved status have not passed source review. | null | published | incremental | hard | [
"source-candidate",
"theoremdb"
] | null | null | null | null | [
{
"kind": "website",
"citation": "TheoremDB P8138",
"url": "https://theoremdb.org/statements/P8138",
"note": null,
"sort_order": 0
},
{
"kind": "formalization",
"citation": "Source formal statement (not locally verified)",
"url": "https://theoremdb.org/statements/P8138",
"not... | {
"source_record": "P8138",
"imported_at": "2026-09-08",
"status_as_of": "2026-09-08",
"evidence_label": "Source candidate · not reviewed",
"note": "Full public directory statement. Source lane: backlog; source resolution: open. Source verification claims are attributed only; no local Lean verification has be... | {"created_at":"2026-08-17T18:35:02+00:00","display_standard":{"editorial_review":"submitter_supplied","formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":8138,"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": "2026-09-08T03:32:50.203Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"4c8a1af5-d925-4640-b46c-ce1922e476b2","format_issues":null,"number":3372} |
1.0.0 | problem:theoremdb-p8140 | problem | theoremdb-p8140 | The Beck–Fiala Conjecture on Discrepancy of Bounded-Degree 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 $n$, $m$, and $t$ be natural numbers. Consider a collection of $m$ subsets $S_1, \dots, S_m \subseteq \{1, \dots, n\}$ with the property that every element $j \in \{1, \dots, n\}$ belongs to at most $t$ of these sets. Such a collection is called a set system of degree at most $t$. A colouring of the ground set is a... | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P8140",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P8140",
"citation": null,
"doi": null,
"arxiv_id": null,
"license": "Selected packets label their content CC0-1.0; original references retain their own terms",
"notes": "Selected problem... | theorem beck_fiala_conjecture :
∃ C : ℝ, 0 < C ∧ ∀ (n m t : ℕ) (S : Fin m → Finset (Fin n)),
(∀ j, (Finset.univ.filter fun i => j ∈ S i).card ≤ t) →
∃ χ : Fin n → ℝ, (∀ j, χ j = 1 ∨ χ j = -1) ∧
∀ i, |∑ j ∈ S i, χ j| ≤ C * Real.sqrt t | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem beck_fiala_conjecture :\n ∃ C : ℝ, 0 < C ∧ ∀ (n m t : ℕ) (S : Fin m → Finset (Fin n)),\n (∀ j, (Finset.univ.filter fun i => j ∈ S i).card ≤ t) →\n ∃ χ : Fin n → ℝ, (∀ j, χ j = 1 ∨ χ j = -1) ∧\n ∀ i, |∑ j ∈ S i, χ j| ≤ C * Real.sqrt t",
"target... | [] | TheoremDB P8140 · Candidate awaiting source review | null | Source-reported assessment, retrieved 8 September 2026. Frontier Science has not independently reviewed this status.
Open. No signed Lean verification is attached to this problem or its selected resolution.
This record is in the source backlog; its novelty and unresolved status have not passed source review. | null | published | incremental | hard | [
"source-candidate",
"theoremdb"
] | null | null | null | null | [
{
"kind": "website",
"citation": "TheoremDB P8140",
"url": "https://theoremdb.org/statements/P8140",
"note": null,
"sort_order": 0
},
{
"kind": "formalization",
"citation": "Source formal statement (not locally verified)",
"url": "https://theoremdb.org/statements/P8140",
"not... | {
"source_record": "P8140",
"imported_at": "2026-09-08",
"status_as_of": "2026-09-08",
"evidence_label": "Source candidate · not reviewed",
"note": "Full public directory statement. Source lane: backlog; source resolution: open. Source verification claims are attributed only; no local Lean verification has be... | {"created_at":"2026-08-17T18:35:05+00:00","display_standard":{"formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":8140,"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": "2026-09-08T03:33:06.389Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"adc69fe6-8d8e-4788-941c-6098ac94f387","format_issues":null,"number":6732} |
1.0.0 | problem:theoremdb-p8142 | problem | theoremdb-p8142 | Characterization of obligatory finite 3-uniform hypergraphs | {
"main": "math",
"sub": "combinatorics",
"slug": "combinatorics",
"name": "Combinatorics",
"original_discipline": "mathematics"
} | [] | [
{
"code": "05",
"basis": "community-map",
"evidence": "Inferred from community field: combinatorics"
}
] | A 3-uniform hypergraph on a vertex set $V$ is a collection of 3-element subsets of $V$, called hyperedges. A hypergraph is 2-colorable if its vertices can be colored with two colors such that no hyperedge is monochromatic. The chromatic cardinal of a hypergraph is the smallest cardinal $\kappa$ for which there exists a... | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P8142",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P8142",
"citation": null,
"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_593 : answer(sorry) ↔
∀ (W : Type) [Fintype W] (F : ThreeUniformHypergraph W),
IsObligatory F ↔ F.IsTwoColorable | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem erdos_593 : answer(sorry) ↔\n ∀ (W : Type) [Fintype W] (F : ThreeUniformHypergraph W),\n IsObligatory F ↔ F.IsTwoColorable",
"target": null,
"source_url": "https://theoremdb.org/statements/P8142",
"format": "declaration-or-fragment",
"sha256":... | [] | TheoremDB P8142 · Candidate awaiting source review | null | Source-reported assessment, retrieved 8 September 2026. Frontier Science has not independently reviewed this status.
Open. Not Lean-verified
This record is in the source backlog; its novelty and unresolved status have not passed source review. | null | published | incremental | hard | [
"source-candidate",
"theoremdb"
] | null | null | null | null | [
{
"kind": "website",
"citation": "TheoremDB P8142",
"url": "https://theoremdb.org/statements/P8142",
"note": null,
"sort_order": 0
},
{
"kind": "formalization",
"citation": "Source formal statement (not locally verified)",
"url": "https://theoremdb.org/statements/P8142",
"not... | {
"source_record": "P8142",
"imported_at": "2026-09-08",
"status_as_of": "2026-09-08",
"evidence_label": "Source candidate · not reviewed",
"note": "Full public directory statement. Source lane: backlog; source resolution: open. Source verification claims are attributed only; no local Lean verification has be... | {"created_at":"2026-08-17T18:35:07+00:00","display_standard":{"formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":8142,"public_status":{"evidence":{"hasFormalization":true,"hasInformalProof":false,"hasRecordedWork":true,"solutionRecord":null},"lean":{"kernelAccepted":false,"label":"Not Lean-v... | [
"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": "2026-09-08T03:32:51.077Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"599618ef-6b6f-4974-b9e8-ea3edac52b2f","format_issues":null,"number":3556} |
1.0.0 | problem:theoremdb-p8144 | problem | theoremdb-p8144 | Characterization of terminating greedy unit fraction expansions | {
"main": "math",
"sub": "combinatorics",
"slug": "combinatorics",
"name": "Combinatorics",
"original_discipline": "mathematics"
} | [] | [
{
"code": "05",
"basis": "community-map",
"evidence": "Inferred from community field: combinatorics"
}
] | Let $A \subseteq \mathbb{N}$ be an infinite set. For a rational number $x$, define the greedy unit fraction remainder as follows. Set $r_0 = x - \frac{1}{n_0}$ where $n_0$ is the minimal element of $A$ satisfying $n_0 \geq 1/x$. For each $t \geq 0$, if $r_t \leq 0$ then set $r_{t+1} = 0$; otherwise set $r_{t+1} = r_t -... | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P8144",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P8144",
"citation": null,
"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.general (x : ℚ) (A : Set ℕ) :
greedyUnitFractionRem A x =ᶠ[atTop] 0 ↔ (x, A) ∈ (answer(sorry) : Set (ℚ × Set ℕ)) | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem erdos_282.variants.general (x : ℚ) (A : Set ℕ) :\n greedyUnitFractionRem A x =ᶠ[atTop] 0 ↔ (x, A) ∈ (answer(sorry) : Set (ℚ × Set ℕ))",
"target": null,
"source_url": "https://theoremdb.org/statements/P8144",
"format": "declaration-or-fragment",
"... | [] | TheoremDB P8144 · Candidate awaiting source review | null | Source-reported assessment, retrieved 8 September 2026. Frontier Science has not independently reviewed this status.
Open. No signed Lean verification is attached to this problem or its selected resolution.
This record is in the source backlog; its novelty and unresolved status have not passed source review. | null | published | incremental | hard | [
"source-candidate",
"theoremdb"
] | null | null | null | null | [
{
"kind": "website",
"citation": "TheoremDB P8144",
"url": "https://theoremdb.org/statements/P8144",
"note": null,
"sort_order": 0
},
{
"kind": "formalization",
"citation": "Source formal statement (not locally verified)",
"url": "https://theoremdb.org/statements/P8144",
"not... | {
"source_record": "P8144",
"imported_at": "2026-09-08",
"status_as_of": "2026-09-08",
"evidence_label": "Source candidate · not reviewed",
"note": "Full public directory statement. Source lane: backlog; source resolution: open. Source verification claims are attributed only; no local Lean verification has be... | {"created_at":"2026-08-17T18:35:08+00:00","display_standard":{"formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":8144,"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": "2026-09-08T03:32:51.099Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"de33d4c5-672f-4e39-99fe-a5a94222accb","format_issues":null,"number":3560} |
1.0.0 | problem:theoremdb-p8146 | problem | theoremdb-p8146 | Existence of a Wall–Sun–Sun Prime | {
"main": "math",
"sub": "number theory",
"slug": "number-theory",
"name": "Number theory",
"original_discipline": "mathematics"
} | [] | [
{
"code": "11",
"basis": "community-map",
"evidence": "Inferred from community field: number-theory"
}
] | Let the Lucas numbers be defined by $L_0 = 2$, $L_1 = 1$, and $L_{n} = L_{n-1} + L_{n-2}$ for $n \geq 2$. A prime $p$ is called a Wall–Sun–Sun prime if $L_p \equiv 1 \pmod{p^2}$. Does there exist at least one Wall–Sun–Sun prime? | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P8146",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P8146",
"citation": null,
"doi": null,
"arxiv_id": null,
"license": "Selected packets label their content CC0-1.0; original references retain their own terms",
"notes": "Selected problem... | theorem exists_isWallSunSunPrime : ∃ p, IsWallSunSunPrime p | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem exists_isWallSunSunPrime : ∃ p, IsWallSunSunPrime p",
"target": null,
"source_url": "https://theoremdb.org/statements/P8146",
"format": "declaration-or-fragment",
"sha256": "1cdac1c39d2c052da8c610b554e924aaf64eb45f5f17431fde878b5955e191e1",
"licen... | [] | TheoremDB P8146 · Candidate awaiting source review | null | Source-reported assessment, retrieved 8 September 2026. Frontier Science has not independently reviewed this status.
Open. No signed Lean verification is attached to this problem or its selected resolution.
This record is in the source backlog; its novelty and unresolved status have not passed source review. | null | published | incremental | hard | [
"source-candidate",
"theoremdb"
] | null | null | null | null | [
{
"kind": "website",
"citation": "TheoremDB P8146",
"url": "https://theoremdb.org/statements/P8146",
"note": null,
"sort_order": 0
},
{
"kind": "formalization",
"citation": "Source formal statement (not locally verified)",
"url": "https://theoremdb.org/statements/P8146",
"not... | {
"source_record": "P8146",
"imported_at": "2026-09-08",
"status_as_of": "2026-09-08",
"evidence_label": "Source candidate · not reviewed",
"note": "Full public directory statement. Source lane: backlog; source resolution: open. Source verification claims are attributed only; no local Lean verification has be... | {"created_at":"2026-08-17T18:35:16+00:00","display_standard":{"editorial_review":"submitter_supplied","formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":8146,"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": "2026-09-08T03:32:58.589Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"e7319f17-c0f4-4b84-b00e-f0fd30d9dea8","format_issues":null,"number":5214} |
1.0.0 | problem:theoremdb-p8148 | problem | theoremdb-p8148 | Erdős Problem 699 on Common Prime Divisors 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"
}
] | For positive integers $n$, $i$, and $j$ with $1 \le i < j \le n/2$, does there always exist a prime $p$ satisfying $p \ge i$ such that $p$ divides both $\binom{n}{i}$ and $\binom{n}{j}$? In other words, does every pair of distinct binomial coefficients from the same row of Pascal's triangle, both lying in the left half... | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P8148",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P8148",
"citation": null,
"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_699 : answer(sorry) ↔
∀ n i j : ℕ,
1 ≤ i →
i < j →
j ≤ n / 2 →
∃ p : ℕ, p.Prime ∧ i ≤ p ∧ p ∣ Nat.gcd (Nat.choose n i) (Nat.choose n j) | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem erdos_699 : answer(sorry) ↔\n ∀ n i j : ℕ,\n 1 ≤ i →\n i < j →\n j ≤ n / 2 →\n ∃ p : ℕ, p.Prime ∧ i ≤ p ∧ p ∣ Nat.gcd (Nat.choose n i) (Nat.choose n j)",
"target": null,
"source_url": "https://theoremdb.org/statements/P8148",
"forma... | [] | TheoremDB P8148 · Candidate awaiting source review | null | Source-reported assessment, retrieved 8 September 2026. Frontier Science has not independently reviewed this status.
Open. No signed Lean verification is attached to this problem or its selected resolution.
This record is in the source backlog; its novelty and unresolved status have not passed source review. | null | published | incremental | hard | [
"source-candidate",
"theoremdb"
] | null | null | null | null | [
{
"kind": "website",
"citation": "TheoremDB P8148",
"url": "https://theoremdb.org/statements/P8148",
"note": null,
"sort_order": 0
},
{
"kind": "formalization",
"citation": "Source formal statement (not locally verified)",
"url": "https://theoremdb.org/statements/P8148",
"not... | {
"source_record": "P8148",
"imported_at": "2026-09-08",
"status_as_of": "2026-09-08",
"evidence_label": "Source candidate · not reviewed",
"note": "Full public directory statement. Source lane: backlog; source resolution: open. Source verification claims are attributed only; no local Lean verification has be... | {"created_at":"2026-08-17T18:35:19+00:00","display_standard":{"formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":8148,"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": "2026-09-08T03:32:55.917Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"aafa0a0e-cc8e-4ca3-b388-0839c692b15f","format_issues":null,"number":4625} |
1.0.0 | problem:theoremdb-p8150 | problem | theoremdb-p8150 | Powerful values of $2^n + 1$ | {
"main": "math",
"sub": "number theory",
"slug": "number-theory",
"name": "Number theory",
"original_discipline": "mathematics"
} | [] | [
{
"code": "11",
"basis": "community-map",
"evidence": "Inferred from community field: number-theory"
}
] | A positive integer $m$ is called powerful if for every prime $p$ dividing $m$, the square $p^2$ also divides $m$. Determine whether there exist only finitely many positive integers $n$ such that $2^n + 1$ is powerful. | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P8150",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P8150",
"citation": null,
"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_936.variants.two_pow_add_one :
answer(sorry) ↔ EventuallyNotPowerful (2 ^ · + 1) | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem erdos_936.variants.two_pow_add_one :\n answer(sorry) ↔ EventuallyNotPowerful (2 ^ · + 1)",
"target": null,
"source_url": "https://theoremdb.org/statements/P8150",
"format": "declaration-or-fragment",
"sha256": "eb63162f010e4ffc0ae45ac99175e85c467711... | [] | TheoremDB P8150 · Candidate awaiting source review | null | Source-reported assessment, retrieved 8 September 2026. Frontier Science has not independently reviewed this status.
Open. No signed Lean verification is attached to this problem or its selected resolution.
This record is in the source backlog; its novelty and unresolved status have not passed source review. | null | published | incremental | hard | [
"source-candidate",
"theoremdb"
] | null | null | null | null | [
{
"kind": "website",
"citation": "TheoremDB P8150",
"url": "https://theoremdb.org/statements/P8150",
"note": null,
"sort_order": 0
},
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"kind": "formalization",
"citation": "Source formal statement (not locally verified)",
"url": "https://theoremdb.org/statements/P8150",
"not... | {
"source_record": "P8150",
"imported_at": "2026-09-08",
"status_as_of": "2026-09-08",
"evidence_label": "Source candidate · not reviewed",
"note": "Full public directory statement. Source lane: backlog; source resolution: open. Source verification claims are attributed only; no local Lean verification has be... | {"created_at":"2026-08-17T18:35:20+00:00","display_standard":{"formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":8150,"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": "2026-09-08T03:33:05.013Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"acf5df47-dc20-40f2-94a6-8d25117cb53e","format_issues":null,"number":6434} |
1.0.0 | problem:theoremdb-p8152 | problem | theoremdb-p8152 | Almost sure convergence of the normalized root count in the unit disk for Kac polynomials with {0,1} coefficients | {
"main": "math",
"sub": "algebra",
"slug": "algebra",
"name": "Algebra",
"original_discipline": "mathematics"
} | [] | [] | Let $\Omega$ be a probability space with probability measure $\mathbb{P}$. Let $n \geq 1$ be a natural number. A Kac polynomial of degree $n$ with coefficients in $\{0, 1\}$ is a random polynomial of the form $f(z) = \sum_{k=0}^{n} \epsilon_k z^k$, where the coefficients $\epsilon_k \in \{0, 1\}$ are independent random... | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P8152",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P8152",
"citation": null,
"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.variants.zero_one :
answer(sorry) ↔ ∀ {Ω : Type*} [MeasureSpace Ω] [IsProbabilityMeasure (ℙ : Measure Ω)]
{n : ℕ} (hn : 1 ≤ n) (f : KacCoefficients ({0, 1} : Set ℂ) Ω),
ℙ {ω | atTop.Tendsto (fun n : ℕ ↦ (2 * f.numRootsInUnitDisk n ω : ℝ) / n) (𝓝 1)} = 1 | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem erdos_522.variants.zero_one :\n answer(sorry) ↔ ∀ {Ω : Type*} [MeasureSpace Ω] [IsProbabilityMeasure (ℙ : Measure Ω)]\n {n : ℕ} (hn : 1 ≤ n) (f : KacCoefficients ({0, 1} : Set ℂ) Ω),\n ℙ {ω | atTop.Tendsto (fun n : ℕ ↦ (2 * f.numRootsInUnitDisk n ω : ℝ) ... | [] | TheoremDB P8152 · Candidate awaiting source review | null | Source-reported assessment, retrieved 8 September 2026. Frontier Science has not independently reviewed this status.
Open. No signed Lean verification is attached to this problem or its selected resolution.
This record is in the source backlog; its novelty and unresolved status have not passed source review. | null | published | incremental | hard | [
"source-candidate",
"theoremdb"
] | null | null | null | null | [
{
"kind": "website",
"citation": "TheoremDB P8152",
"url": "https://theoremdb.org/statements/P8152",
"note": null,
"sort_order": 0
},
{
"kind": "formalization",
"citation": "Source formal statement (not locally verified)",
"url": "https://theoremdb.org/statements/P8152",
"not... | {
"source_record": "P8152",
"imported_at": "2026-09-08",
"status_as_of": "2026-09-08",
"evidence_label": "Source candidate · not reviewed",
"note": "Full public directory statement. Source lane: backlog; source resolution: open. Source verification claims are attributed only; no local Lean verification has be... | {"created_at":"2026-08-17T18:35:22+00:00","display_standard":{"editorial_review":"submitter_supplied","formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":8152,"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": "2026-09-08T03:32:49.551Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"16418324-80f4-484e-a3dc-69a06182022d","format_issues":null,"number":3232} |
1.0.0 | problem:theoremdb-p8154 | problem | theoremdb-p8154 | Unboundedness of the ratio of the minimal prime-generating coefficient to iterated logarithms | {
"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 $a(n)$ denote the smallest non-negative integer $k$ such that $kn + 1$ is prime. Determine whether the set of real numbers
$$
\left\{ \frac{a(n)}{\log n \cdot \log(\log n)} : n \in \mathbb{N},\ n > 1 \right\}
$$
is unbounded above. | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P8154",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P8154",
"citation": null,
"doi": null,
"arxiv_id": null,
"license": "Selected packets label their content CC0-1.0; original references retain their own terms",
"notes": "Selected problem... | theorem a_unbounded : ¬BddAbove (Set.range fun n ↦ a n / (Real.log n * Real.log (Real.log n))) | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem a_unbounded : ¬BddAbove (Set.range fun n ↦ a n / (Real.log n * Real.log (Real.log n)))",
"target": null,
"source_url": "https://theoremdb.org/statements/P8154",
"format": "declaration-or-fragment",
"sha256": "b9a6e19b375e72fd9e5b2219f1567b992dea7c3be9... | [] | TheoremDB P8154 · Candidate awaiting source review | null | Source-reported assessment, retrieved 8 September 2026. Frontier Science has not independently reviewed this status.
Open. No signed Lean verification is attached to this problem or its selected resolution.
This record is in the source backlog; its novelty and unresolved status have not passed source review. | null | published | incremental | hard | [
"source-candidate",
"theoremdb"
] | null | null | null | null | [
{
"kind": "website",
"citation": "TheoremDB P8154",
"url": "https://theoremdb.org/statements/P8154",
"note": null,
"sort_order": 0
},
{
"kind": "formalization",
"citation": "Source formal statement (not locally verified)",
"url": "https://theoremdb.org/statements/P8154",
"not... | {
"source_record": "P8154",
"imported_at": "2026-09-08",
"status_as_of": "2026-09-08",
"evidence_label": "Source candidate · not reviewed",
"note": "Full public directory statement. Source lane: backlog; source resolution: open. Source verification claims are attributed only; no local Lean verification has be... | {"created_at":"2026-08-17T18:35:22+00:00","display_standard":{"editorial_review":"submitter_supplied","formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":8154,"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": "2026-09-08T03:33:07.185Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"1ef383a2-d6d1-4166-8a15-a213f69fb105","format_issues":null,"number":6905} |
1.0.0 | problem:theoremdb-p8156 | problem | theoremdb-p8156 | Existence of Natural Densities for Level Sets of the Minimal Coprime Exponent Function | {
"main": "math",
"sub": "number theory",
"slug": "number-theory",
"name": "Number theory",
"original_discipline": "mathematics"
} | [] | [
{
"code": "11",
"basis": "community-map",
"evidence": "Inferred from community field: number-theory"
}
] | For each positive integer $n$, define $h(n)$ to be the least integer $m > 2$ such that the numbers $2^n - 1, 3^n - 1, \dots, m^n - 1$ are collectively coprime; that is, their greatest common divisor equals $1$. (If no such $m$ exists, $h(n)$ is taken to be infinite.) For a set $S \subseteq \mathbb{N}$, we say that $S$ ... | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P8156",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P8156",
"citation": null,
"doi": null,
"arxiv_id": null,
"license": "Selected packets label their content CC0-1.0; original references retain their own terms",
"notes": "Selected problem... | theorem erdos_770.parts.i : answer(sorry) ↔ ∀ p : ℕ, p.Prime → ∃ a, HasDensity {n | h n = p} a | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem erdos_770.parts.i : answer(sorry) ↔ ∀ p : ℕ, p.Prime → ∃ a, HasDensity {n | h n = p} a",
"target": null,
"source_url": "https://theoremdb.org/statements/P8156",
"format": "declaration-or-fragment",
"sha256": "0eef13e4fd10129dbb10000e56857476d3618a41bd... | [] | TheoremDB P8156 · Candidate awaiting source review | null | Source-reported assessment, retrieved 8 September 2026. Frontier Science has not independently reviewed this status.
Open. No signed Lean verification is attached to this problem or its selected resolution.
This record is in the source backlog; its novelty and unresolved status have not passed source review. | null | published | incremental | hard | [
"source-candidate",
"theoremdb"
] | null | null | null | null | [
{
"kind": "website",
"citation": "TheoremDB P8156",
"url": "https://theoremdb.org/statements/P8156",
"note": null,
"sort_order": 0
},
{
"kind": "formalization",
"citation": "Source formal statement (not locally verified)",
"url": "https://theoremdb.org/statements/P8156",
"not... | {
"source_record": "P8156",
"imported_at": "2026-09-08",
"status_as_of": "2026-09-08",
"evidence_label": "Source candidate · not reviewed",
"note": "Full public directory statement. Source lane: backlog; source resolution: open. Source verification claims are attributed only; no local Lean verification has be... | {"created_at":"2026-08-17T18:35:24+00:00","display_standard":{"editorial_review":"submitter_supplied","formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":8156,"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": "2026-09-08T03:32:58.779Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"7cb022da-41b2-46b1-acf0-e3e1e799eb55","format_issues":null,"number":5253} |
1.0.0 | problem:theoremdb-p8158 | problem | theoremdb-p8158 | Existence of Quasiperfect 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 a positive integer $n$, let $\sigma_1(n)$ denote the sum of all positive divisors of $n$. A positive integer $n$ is called quasiperfect if $\sigma_1(n) = 2n + 1$. Does there exist a quasiperfect number? | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P8158",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P8158",
"citation": null,
"doi": null,
"arxiv_id": null,
"license": "Selected packets label their content CC0-1.0; original references retain their own terms",
"notes": "Selected problem... | theorem exists_quasiperfect :
answer(sorry) ↔ ∃ n, Quasiperfect n | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem exists_quasiperfect :\n answer(sorry) ↔ ∃ n, Quasiperfect n",
"target": null,
"source_url": "https://theoremdb.org/statements/P8158",
"format": "declaration-or-fragment",
"sha256": "f09d6167fa40a50ddf3f24b06f79351bee1e88878a7807684aea71d9305b4778",... | [] | TheoremDB P8158 · Candidate awaiting source review | null | Source-reported assessment, retrieved 8 September 2026. Frontier Science has not independently reviewed this status.
Open. No signed Lean verification is attached to this problem or its selected resolution.
This record is in the source backlog; its novelty and unresolved status have not passed source review. | null | published | incremental | hard | [
"source-candidate",
"theoremdb"
] | null | null | null | null | [
{
"kind": "website",
"citation": "TheoremDB P8158",
"url": "https://theoremdb.org/statements/P8158",
"note": null,
"sort_order": 0
},
{
"kind": "formalization",
"citation": "Source formal statement (not locally verified)",
"url": "https://theoremdb.org/statements/P8158",
"not... | {
"source_record": "P8158",
"imported_at": "2026-09-08",
"status_as_of": "2026-09-08",
"evidence_label": "Source candidate · not reviewed",
"note": "Full public directory statement. Source lane: backlog; source resolution: open. Source verification claims are attributed only; no local Lean verification has be... | {"created_at":"2026-08-17T18:35:30+00:00","display_standard":{"editorial_review":"submitter_supplied","formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":8158,"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": "2026-09-08T03:32:58.802Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"f17778a8-9d4b-453f-8c73-12a27f259fde","format_issues":null,"number":5258} |
1.0.0 | problem:theoremdb-p8160 | problem | theoremdb-p8160 | Infinitely many barriers for the total prime multiplicity function | {
"main": "math",
"sub": "number theory",
"slug": "number-theory",
"name": "Number theory",
"original_discipline": "mathematics"
} | [] | [
{
"code": "11",
"basis": "community-map",
"evidence": "Inferred from community field: number-theory"
}
] | For a function $f : \mathbb{N} \to \mathbb{R}$, a natural number $n$ is called a *barrier* for $f$ if $(m : \mathbb{R}) + f(m) \leq n$ for all $m < n$. Let $\Omega(m)$ denote the total number of prime factors of $m$ counted with multiplicity (so if $m = p_1^{k_1} \cdots p_r^{k_r}$, then $\Omega(m) = k_1 + \cdots + k_r$... | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P8160",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P8160",
"citation": null,
"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.variants.bigOmega :
answer(sorry) ↔ { n | IsBarrier (fun m => Ω m) n }.Infinite | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem erdos_413.variants.bigOmega :\n answer(sorry) ↔ { n | IsBarrier (fun m => Ω m) n }.Infinite",
"target": null,
"source_url": "https://theoremdb.org/statements/P8160",
"format": "declaration-or-fragment",
"sha256": "a42c37a9ca4b40c01dc9cdce5ba6fc2054... | [] | TheoremDB P8160 · Candidate awaiting source review | null | Source-reported assessment, retrieved 8 September 2026. Frontier Science has not independently reviewed this status.
Open. No signed Lean verification is attached to this problem or its selected resolution.
This record is in the source backlog; its novelty and unresolved status have not passed source review. | null | published | incremental | hard | [
"source-candidate",
"theoremdb"
] | null | null | null | null | [
{
"kind": "website",
"citation": "TheoremDB P8160",
"url": "https://theoremdb.org/statements/P8160",
"note": null,
"sort_order": 0
},
{
"kind": "formalization",
"citation": "Source formal statement (not locally verified)",
"url": "https://theoremdb.org/statements/P8160",
"not... | {
"source_record": "P8160",
"imported_at": "2026-09-08",
"status_as_of": "2026-09-08",
"evidence_label": "Source candidate · not reviewed",
"note": "Full public directory statement. Source lane: backlog; source resolution: open. Source verification claims are attributed only; no local Lean verification has be... | {"created_at":"2026-08-17T18:35:34+00:00","display_standard":{"formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":8160,"public_status":{"evidence":{"hasFormalization":false,"hasInformalProof":false,"hasRecordedWork":false,"solutionRecord":null},"lean":{"kernelAccepted":false,"label":"","reaso... | [
"formalization-contains-placeholder",
"formalization-not-compiled-during-export",
"current-status-unassessed",
"source-full-text-not-present"
] | [] | {
"created_at": "2026-09-08T03:32:15.935Z",
"updated_at": "2026-09-08T03:32:15.935Z",
"published_at": "2026-09-08T03:33:00.825Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"f3a4bec9-1976-4b7c-bd02-d7e4147f5172","format_issues":null,"number":5598} |
1.0.0 | problem:theoremdb-p8162 | problem | theoremdb-p8162 | Perfect Powers as Products of Arithmetic Progressions | {
"main": "math",
"sub": "number theory",
"slug": "number-theory",
"name": "Number theory",
"original_discipline": "mathematics"
} | [] | [
{
"code": "11",
"basis": "community-map",
"evidence": "Inferred from community field: number-theory"
}
] | For integers $k \geq 4$ and $l > 1$, let an arithmetic progression of length $k$ with difference $d$ and starting term $n$ be a set of the form $\{n, n+d, n+2d, \ldots, n+(k-1)d\}$ where $n > 0$, $d > 0$, and $\gcd(n,d) = 1$. Is it true that the product of all terms in such an arithmetic progression can never equal $q^... | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P8162",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P8162",
"citation": null,
"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_672 :
answer(sorry) ↔ ∀ᵉ (k) (l > 1), k ≥ 4 → Erdos672With k l | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem erdos_672 :\n answer(sorry) ↔ ∀ᵉ (k) (l > 1), k ≥ 4 → Erdos672With k l",
"target": null,
"source_url": "https://theoremdb.org/statements/P8162",
"format": "declaration-or-fragment",
"sha256": "c6e2aeeaed89f5cd26cf5c5a5c5db747eb5b942927ff8d1d602f2dd... | [] | TheoremDB P8162 · Candidate awaiting source review | null | Source-reported assessment, retrieved 8 September 2026. Frontier Science has not independently reviewed this status.
Open. No signed Lean verification is attached to this problem or its selected resolution.
This record is in the source backlog; its novelty and unresolved status have not passed source review. | null | published | incremental | hard | [
"source-candidate",
"theoremdb"
] | null | null | null | null | [
{
"kind": "website",
"citation": "TheoremDB P8162",
"url": "https://theoremdb.org/statements/P8162",
"note": null,
"sort_order": 0
},
{
"kind": "formalization",
"citation": "Source formal statement (not locally verified)",
"url": "https://theoremdb.org/statements/P8162",
"not... | {
"source_record": "P8162",
"imported_at": "2026-09-08",
"status_as_of": "2026-09-08",
"evidence_label": "Source candidate · not reviewed",
"note": "Full public directory statement. Source lane: backlog; source resolution: open. Source verification claims are attributed only; no local Lean verification has be... | {"created_at":"2026-08-17T18:35:36+00:00","display_standard":{"editorial_review":"submitter_supplied","formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":8162,"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": "2026-09-08T03:33:04.791Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"0f9d4785-4751-4f67-accb-66e212892469","format_issues":null,"number":6387} |
1.0.0 | problem:theoremdb-p8164 | problem | theoremdb-p8164 | Limit of the Ratio of Distinct to Repeated Totient Values | {
"main": "math",
"sub": "number theory",
"slug": "number-theory",
"name": "Number theory",
"original_discipline": "mathematics"
} | [] | [
{
"code": "11",
"basis": "community-map",
"evidence": "Inferred from community field: number-theory"
}
] | For a positive integer $n$, let $\phi(n)$ denote Euler's totient function, which counts the number of integers from $1$ to $n$ that are coprime to $n$. Define two counting functions:
$$
V'(x) = \#\{\phi(m) : 1 \leq m \leq x\},
$$
the number of distinct values taken by $\phi$ on integers from $1$ to $x$, and
$$
V(x) ... | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P8164",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P8164",
"citation": null,
"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_417.parts.ii :
answer(sorry) ↔ ∃ L > (1 : ENNReal), Tendsto (fun x ↦
({ k | k ∈ range totient ∧ (k : ℝ) ≤ x }.ncard : ENNReal) /
((totient '' { m | 1 ≤ m ∧ (m : ℝ) ≤ x }).ncard : ENNReal))
atTop (𝓝 L) | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem erdos_417.parts.ii :\n answer(sorry) ↔ ∃ L > (1 : ENNReal), Tendsto (fun x ↦\n ({ k | k ∈ range totient ∧ (k : ℝ) ≤ x }.ncard : ENNReal) /\n ((totient '' { m | 1 ≤ m ∧ (m : ℝ) ≤ x }).ncard : ENNReal))\n atTop (𝓝 L)",
"target": null,
"source... | [] | TheoremDB P8164 · Candidate awaiting source review | null | Source-reported assessment, retrieved 8 September 2026. Frontier Science has not independently reviewed this status.
Open. No signed Lean verification is attached to this problem or its selected resolution.
This record is in the source backlog; its novelty and unresolved status have not passed source review. | null | published | incremental | hard | [
"source-candidate",
"theoremdb"
] | null | null | null | null | [
{
"kind": "website",
"citation": "TheoremDB P8164",
"url": "https://theoremdb.org/statements/P8164",
"note": null,
"sort_order": 0
},
{
"kind": "formalization",
"citation": "Source formal statement (not locally verified)",
"url": "https://theoremdb.org/statements/P8164",
"not... | {
"source_record": "P8164",
"imported_at": "2026-09-08",
"status_as_of": "2026-09-08",
"evidence_label": "Source candidate · not reviewed",
"note": "Full public directory statement. Source lane: backlog; source resolution: open. Source verification claims are attributed only; no local Lean verification has be... | {"created_at":"2026-08-17T18:35:52+00:00","display_standard":{"editorial_review":"submitter_supplied","formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":8164,"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": "2026-09-08T03:33:03.214Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"81d48ecf-202e-4bfe-84d8-70fa704f7ce1","format_issues":null,"number":6047} |
1.0.0 | problem:theoremdb-p8166 | problem | theoremdb-p8166 | 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"
}
] | For every integer $k \geq 3$, does there exist a set of $k$ consecutive primes in arithmetic progression? That is, for each $k \geq 3$, does there exist a set $s$ of natural numbers such that $s$ forms an arithmetic progression of length $k$ and consists of $k$ consecutive primes? | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P8166",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P8166",
"citation": null,
"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 : answer(sorry) ↔
∀ k ≥ 3, ∃ (s : Set ℕ), s.IsAPAndPrimeProgressionOfLength k | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem erdos_141 : answer(sorry) ↔\n ∀ k ≥ 3, ∃ (s : Set ℕ), s.IsAPAndPrimeProgressionOfLength k",
"target": null,
"source_url": "https://theoremdb.org/statements/P8166",
"format": "declaration-or-fragment",
"sha256": "db9b44db5b198064ce38d4f7cb211f5fe7b6... | [] | TheoremDB P8166 · Candidate awaiting source review | null | Source-reported assessment, retrieved 8 September 2026. Frontier Science has not independently reviewed this status.
Open. No signed Lean verification is attached to this problem or its selected resolution.
This record is in the source backlog; its novelty and unresolved status have not passed source review. | null | published | incremental | hard | [
"source-candidate",
"theoremdb"
] | null | null | null | null | [
{
"kind": "website",
"citation": "TheoremDB P8166",
"url": "https://theoremdb.org/statements/P8166",
"note": null,
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},
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"citation": "Source formal statement (not locally verified)",
"url": "https://theoremdb.org/statements/P8166",
"not... | {
"source_record": "P8166",
"imported_at": "2026-09-08",
"status_as_of": "2026-09-08",
"evidence_label": "Source candidate · not reviewed",
"note": "Full public directory statement. Source lane: backlog; source resolution: open. Source verification claims are attributed only; no local Lean verification has be... | {"created_at":"2026-08-17T18:35:57+00:00","display_standard":{"editorial_review":"submitter_supplied","formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":8166,"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": "2026-09-08T03:32:52.622Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"5d56a88e-fc1a-461e-ae27-39cc8faedc42","format_issues":null,"number":3900} |
1.0.0 | problem:theoremdb-p8168 | problem | theoremdb-p8168 | Infinitude of primes with maximal prime divisor property for shifted quadratic products | {
"main": "math",
"sub": "number theory",
"slug": "number-theory",
"name": "Number theory",
"original_discipline": "mathematics"
} | [] | [
{
"code": "11",
"basis": "community-map",
"evidence": "Inferred from community field: number-theory"
}
] | For a positive integer $n$, let $P(n)$ denote the largest prime divisor of $n$. For each nonnegative integer $k$ and each prime $p$, consider the product
$$
\prod_{i=0}^{k} (p^2 + i) = (p^2)(p^2+1)(p^2+2)\cdots(p^2+k).
$$
Is it true that for every nonnegative integer $k$, there exist infinitely many primes $p$ such... | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P8168",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P8168",
"citation": null,
"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_383 : answer(sorry) ↔
∀ k, {p : ℕ | p.Prime ∧ Nat.maxPrimeFac (∏ i ∈ Finset.Icc 0 k, (p ^ 2 + i)) = p}.Infinite | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem erdos_383 : answer(sorry) ↔\n ∀ k, {p : ℕ | p.Prime ∧ Nat.maxPrimeFac (∏ i ∈ Finset.Icc 0 k, (p ^ 2 + i)) = p}.Infinite",
"target": null,
"source_url": "https://theoremdb.org/statements/P8168",
"format": "declaration-or-fragment",
"sha256": "a52419... | [] | TheoremDB P8168 · Candidate awaiting source review | null | Source-reported assessment, retrieved 8 September 2026. Frontier Science has not independently reviewed this status.
Open. No signed Lean verification is attached to this problem or its selected resolution.
This record is in the source backlog; its novelty and unresolved status have not passed source review. | null | published | incremental | hard | [
"source-candidate",
"theoremdb"
] | null | null | null | null | [
{
"kind": "website",
"citation": "TheoremDB P8168",
"url": "https://theoremdb.org/statements/P8168",
"note": null,
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},
{
"kind": "formalization",
"citation": "Source formal statement (not locally verified)",
"url": "https://theoremdb.org/statements/P8168",
"not... | {
"source_record": "P8168",
"imported_at": "2026-09-08",
"status_as_of": "2026-09-08",
"evidence_label": "Source candidate · not reviewed",
"note": "Full public directory statement. Source lane: backlog; source resolution: open. Source verification claims are attributed only; no local Lean verification has be... | {"created_at":"2026-08-17T18:36:01+00:00","display_standard":{"editorial_review":"submitter_supplied","formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":8168,"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": "2026-09-08T03:33:01.230Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"a6bc4d2b-5c83-432e-8daa-c70eec240420","format_issues":null,"number":5686} |
1.0.0 | problem:theoremdb-p8170 | problem | theoremdb-p8170 | Solitary Status of Ten | {
"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 the positive divisors of $n$. The abundancy index of $n$ is the ratio $\sigma(n)/n$. Two positive integers $m$ and $n$ are called friendly if they have the same abundancy index, that is, $\sigma(m)/m = \sigma(n)/n$, or equivalently $\sigma(m) \cdot n = \sigm... | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P8170",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P8170",
"citation": null,
"doi": null,
"arxiv_id": null,
"license": "Selected packets label their content CC0-1.0; original references retain their own terms",
"notes": "Selected problem... | theorem is_ten_solitary : answer(sorry) ↔ IsSolitary 10 | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem is_ten_solitary : answer(sorry) ↔ IsSolitary 10",
"target": null,
"source_url": "https://theoremdb.org/statements/P8170",
"format": "declaration-or-fragment",
"sha256": "a404bac70d1e7e9a3621510291bca7e6d6a098b468ec8f3fc6db49f93f55b49e",
"license":... | [] | TheoremDB P8170 · Candidate awaiting source review | null | Source-reported assessment, retrieved 8 September 2026. Frontier Science has not independently reviewed this status.
Open. No signed Lean verification is attached to this problem or its selected resolution.
This record is in the source backlog; its novelty and unresolved status have not passed source review. | null | published | incremental | hard | [
"source-candidate",
"theoremdb"
] | null | null | null | null | [
{
"kind": "website",
"citation": "TheoremDB P8170",
"url": "https://theoremdb.org/statements/P8170",
"note": null,
"sort_order": 0
},
{
"kind": "formalization",
"citation": "Source formal statement (not locally verified)",
"url": "https://theoremdb.org/statements/P8170",
"not... | {
"source_record": "P8170",
"imported_at": "2026-09-08",
"status_as_of": "2026-09-08",
"evidence_label": "Source candidate · not reviewed",
"note": "Full public directory statement. Source lane: backlog; source resolution: open. Source verification claims are attributed only; no local Lean verification has be... | {"created_at":"2026-08-17T18:36:10+00:00","display_standard":{"editorial_review":"submitter_supplied","formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":8170,"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": "2026-09-08T03:33:05.970Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"dfc37616-3656-4d8f-ae57-91b56bc8d716","format_issues":null,"number":6642} |
1.0.0 | problem:theoremdb-p8172 | problem | theoremdb-p8172 | Infinitely many even terms in the sequence of integer parts of powers of 3/2 | {
"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 $\lfloor x \rfloor$ denote the greatest integer less than or equal to $x$. Consider the sequence $\left\lfloor \left(\frac{3}{2}\right)^n \right\rfloor$ for $n = 0, 1, 2, \dots$. Is the set of indices $n$ for which $\left\lfloor \left(\frac{3}{2}\right)^n \right\rfloor$ is even an infinite set? | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P8172",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P8172",
"citation": null,
"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_349.variants.floor_3_halves_even :
answer(sorry) ↔ {n | Even ⌊(3/2 : ℝ) ^ n⌋}.Infinite | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem erdos_349.variants.floor_3_halves_even :\n answer(sorry) ↔ {n | Even ⌊(3/2 : ℝ) ^ n⌋}.Infinite",
"target": null,
"source_url": "https://theoremdb.org/statements/P8172",
"format": "declaration-or-fragment",
"sha256": "416c290acbb0b95a736ff30484c4851... | [] | TheoremDB P8172 · Candidate awaiting source review | null | Source-reported assessment, retrieved 8 September 2026. Frontier Science has not independently reviewed this status.
Open. No signed Lean verification is attached to this problem or its selected resolution.
This record is in the source backlog; its novelty and unresolved status have not passed source review. | null | published | incremental | hard | [
"source-candidate",
"theoremdb"
] | null | null | null | null | [
{
"kind": "website",
"citation": "TheoremDB P8172",
"url": "https://theoremdb.org/statements/P8172",
"note": null,
"sort_order": 0
},
{
"kind": "formalization",
"citation": "Source formal statement (not locally verified)",
"url": "https://theoremdb.org/statements/P8172",
"not... | {
"source_record": "P8172",
"imported_at": "2026-09-08",
"status_as_of": "2026-09-08",
"evidence_label": "Source candidate · not reviewed",
"note": "Full public directory statement. Source lane: backlog; source resolution: open. Source verification claims are attributed only; no local Lean verification has be... | {"created_at":"2026-08-17T18:36:12+00:00","display_standard":{"editorial_review":"submitter_supplied","formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":8172,"public_status":{"evidence":{"hasFormalization":false,"hasInformalProof":false,"hasRecordedWork":false,"solutionRecord":null},"lean":{... | [
"formalization-contains-placeholder",
"formalization-not-compiled-during-export",
"current-status-unassessed",
"source-full-text-not-present"
] | [] | {
"created_at": "2026-09-08T03:32:15.935Z",
"updated_at": "2026-09-08T03:32:15.935Z",
"published_at": "2026-09-08T03:33:00.882Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"5dcd6e62-fcf8-4ec8-b686-596d87bad2b2","format_issues":null,"number":5610} |
1.0.0 | problem:theoremdb-p8174 | problem | theoremdb-p8174 | Square-freeness of all Fermat 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 each natural number $n$, the $n$th Fermat number is defined as $F_n = 2^{2^n} + 1$. A positive integer is called square-free if it is not divisible by the square of any prime. Does every Fermat number $F_n$ have the property that it is square-free? | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P8174",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P8174",
"citation": null,
"doi": null,
"arxiv_id": null,
"license": "Selected packets label their content CC0-1.0; original references retain their own terms",
"notes": "Selected problem... | theorem all_fermat_squarefree : answer(sorry) ↔ ∀ n : ℕ, Squarefree n.fermatNumber | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem all_fermat_squarefree : answer(sorry) ↔ ∀ n : ℕ, Squarefree n.fermatNumber",
"target": null,
"source_url": "https://theoremdb.org/statements/P8174",
"format": "declaration-or-fragment",
"sha256": "57595219e39a2ef7cc63e419d080f39ea367f0d017ae2f24144099... | [] | TheoremDB P8174 · Candidate awaiting source review | null | Source-reported assessment, retrieved 8 September 2026. Frontier Science has not independently reviewed this status.
Open. No signed Lean verification is attached to this problem or its selected resolution.
This record is in the source backlog; its novelty and unresolved status have not passed source review. | null | published | incremental | hard | [
"source-candidate",
"theoremdb"
] | null | null | null | null | [
{
"kind": "website",
"citation": "TheoremDB P8174",
"url": "https://theoremdb.org/statements/P8174",
"note": null,
"sort_order": 0
},
{
"kind": "formalization",
"citation": "Source formal statement (not locally verified)",
"url": "https://theoremdb.org/statements/P8174",
"not... | {
"source_record": "P8174",
"imported_at": "2026-09-08",
"status_as_of": "2026-09-08",
"evidence_label": "Source candidate · not reviewed",
"note": "Full public directory statement. Source lane: backlog; source resolution: open. Source verification claims are attributed only; no local Lean verification has be... | {"created_at":"2026-08-17T18:36:14+00:00","display_standard":{"editorial_review":"submitter_supplied","formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":8174,"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": "2026-09-08T03:33:06.018Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"24778af0-548f-4156-ab80-aff0fac7521c","format_issues":null,"number":6653} |
1.0.0 | problem:theoremdb-p8176 | problem | theoremdb-p8176 | Infinitude of integers with coprimality threshold 3 | {
"main": "math",
"sub": "number theory",
"slug": "number-theory",
"name": "Number theory",
"original_discipline": "mathematics"
} | [] | [
{
"code": "11",
"basis": "community-map",
"evidence": "Inferred from community field: number-theory"
}
] | For each positive integer $n$, define $h(n)$ to be the smallest integer $m > 2$ such that the numbers $2^n - 1, 3^n - 1, \ldots, m^n - 1$ are collectively coprime; that is, their greatest common divisor equals $1$. If no such $m$ exists, set $h(n) = \infty$. Determine whether the set $\{n \in \mathbb{N} : h(n) = 3\}$ i... | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P8176",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P8176",
"citation": null,
"doi": null,
"arxiv_id": null,
"license": "Selected packets label their content CC0-1.0; original references retain their own terms",
"notes": "Selected problem... | theorem erdos_770.variants.three : {n | h n = 3}.Infinite | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem erdos_770.variants.three : {n | h n = 3}.Infinite",
"target": null,
"source_url": "https://theoremdb.org/statements/P8176",
"format": "declaration-or-fragment",
"sha256": "77c7617372684599bcfe58d2905945a0312104a01b938aeadcb7eabedbbf26d9",
"license... | [] | TheoremDB P8176 · Candidate awaiting source review | null | Source-reported assessment, retrieved 8 September 2026. Frontier Science has not independently reviewed this status.
Open. No signed Lean verification is attached to this problem or its selected resolution.
This record is in the source backlog; its novelty and unresolved status have not passed source review. | null | published | incremental | hard | [
"source-candidate",
"theoremdb"
] | null | null | null | null | [
{
"kind": "website",
"citation": "TheoremDB P8176",
"url": "https://theoremdb.org/statements/P8176",
"note": null,
"sort_order": 0
},
{
"kind": "formalization",
"citation": "Source formal statement (not locally verified)",
"url": "https://theoremdb.org/statements/P8176",
"not... | {
"source_record": "P8176",
"imported_at": "2026-09-08",
"status_as_of": "2026-09-08",
"evidence_label": "Source candidate · not reviewed",
"note": "Full public directory statement. Source lane: backlog; source resolution: open. Source verification claims are attributed only; no local Lean verification has be... | {"created_at":"2026-08-17T18:36:17+00:00","display_standard":{"editorial_review":"submitter_supplied","formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":8176,"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": "2026-09-08T03:33:01.159Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"0ce9db0d-7dda-40e8-bce4-4bc14e4c48d3","format_issues":null,"number":5670} |
1.0.0 | problem:theoremdb-p8178 | problem | theoremdb-p8178 | Erdős's variant on almost-bipartite subgraphs of infinite-chromatic graphs | {
"main": "math",
"sub": "combinatorics",
"slug": "combinatorics",
"name": "Combinatorics",
"original_discipline": "mathematics"
} | [] | [
{
"code": "05",
"basis": "community-map",
"evidence": "Inferred from community field: combinatorics"
}
] | For a simple graph $G$ and a natural number $n$, let the edge distance to bipartite of a subgraph $A$ of $G$ be the minimum number of edges that must be deleted from $A$ to obtain a bipartite graph. Let the maximum subgraph edge distance to bipartite of $G$ at $n$, denoted $b_G(n)$, be the supremum of the edge distance... | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P8178",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P8178",
"citation": null,
"doi": null,
"arxiv_id": null,
"license": "Selected packets label their content CC0-1.0; original references retain their own terms",
"notes": "Selected problem... | theorem erdos_74.variants.sqrt : answer(sorry) ↔
∃ (V : Type u) (G : SimpleGraph V), G.chromaticNumber = ⊤ ∧
∀ n, G.maxSubgraphEdgeDistToBipartite n ≤ (n : ℝ).sqrt | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem erdos_74.variants.sqrt : answer(sorry) ↔\n ∃ (V : Type u) (G : SimpleGraph V), G.chromaticNumber = ⊤ ∧\n ∀ n, G.maxSubgraphEdgeDistToBipartite n ≤ (n : ℝ).sqrt",
"target": null,
"source_url": "https://theoremdb.org/statements/P8178",
"format": "decl... | [] | TheoremDB P8178 · Candidate awaiting source review | null | Source-reported assessment, retrieved 8 September 2026. Frontier Science has not independently reviewed this status.
Open. No signed Lean verification is attached to this problem or its selected resolution.
This record is in the source backlog; its novelty and unresolved status have not passed source review. | null | published | incremental | hard | [
"source-candidate",
"theoremdb"
] | null | null | null | null | [
{
"kind": "website",
"citation": "TheoremDB P8178",
"url": "https://theoremdb.org/statements/P8178",
"note": null,
"sort_order": 0
},
{
"kind": "formalization",
"citation": "Source formal statement (not locally verified)",
"url": "https://theoremdb.org/statements/P8178",
"not... | {
"source_record": "P8178",
"imported_at": "2026-09-08",
"status_as_of": "2026-09-08",
"evidence_label": "Source candidate · not reviewed",
"note": "Full public directory statement. Source lane: backlog; source resolution: open. Source verification claims are attributed only; no local Lean verification has be... | {"created_at":"2026-08-17T18:36:23+00:00","display_standard":{"editorial_review":"submitter_supplied","formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":8178,"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": "2026-09-08T03:32:56.632Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"f3b56f38-de24-49bf-b50b-919b8f66bbb6","format_issues":null,"number":4782} |
1.0.0 | problem:theoremdb-p8180 | problem | theoremdb-p8180 | Finiteness of solutions with bounded ratio for equal-prime-factor products | {
"main": "math",
"sub": "number theory",
"slug": "number-theory",
"name": "Number theory",
"original_discipline": "mathematics"
} | [] | [
{
"code": "11",
"basis": "community-map",
"evidence": "Inferred from community field: number-theory"
}
] | For integers $k_1$ and $k_2$ with $k_2 \geq 3$ and $k_2 \leq k_1$, consider pairs of natural numbers $(n_1, n_2)$ satisfying $n_1 + k_1 \leq n_2 \leq 2(n_1 + k_1)$ such that the products $\prod_{i=1}^{k_1} (n_1 + i)$ and $\prod_{j=1}^{k_2} (n_2 + j)$ have the same set of prime factors. Is the set of all such pairs $(n_... | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P8180",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P8180",
"citation": null,
"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_931.variants.additional_condition : answer(sorry) ↔ ∀ᵉ (k₁ : ℕ) (k₂ ≥ 3), k₂ ≤ k₁ →
{(n₁, n₂) | n₁ + k₁ ≤ n₂ ∧ n₂ ≤ 2 * (n₁ + k₁) ∧
(∏ i ∈ Finset.Icc 1 k₁, (n₁ + i)).primeFactors =
(∏ j ∈ Finset.Icc 1 k₂, (n₂ + j)).primeFactors}.Finite | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem erdos_931.variants.additional_condition : answer(sorry) ↔ ∀ᵉ (k₁ : ℕ) (k₂ ≥ 3), k₂ ≤ k₁ →\n {(n₁, n₂) | n₁ + k₁ ≤ n₂ ∧ n₂ ≤ 2 * (n₁ + k₁) ∧\n (∏ i ∈ Finset.Icc 1 k₁, (n₁ + i)).primeFactors =\n (∏ j ∈ Finset.Icc 1 k₂, (n₂ + j)).primeFactors}.Finite",
... | [] | TheoremDB P8180 · Candidate awaiting source review | null | Source-reported assessment, retrieved 8 September 2026. Frontier Science has not independently reviewed this status.
Open. No signed Lean verification is attached to this problem or its selected resolution.
This record is in the source backlog; its novelty and unresolved status have not passed source review. | null | published | incremental | hard | [
"source-candidate",
"theoremdb"
] | null | null | null | null | [
{
"kind": "website",
"citation": "TheoremDB P8180",
"url": "https://theoremdb.org/statements/P8180",
"note": null,
"sort_order": 0
},
{
"kind": "formalization",
"citation": "Source formal statement (not locally verified)",
"url": "https://theoremdb.org/statements/P8180",
"not... | {
"source_record": "P8180",
"imported_at": "2026-09-08",
"status_as_of": "2026-09-08",
"evidence_label": "Source candidate · not reviewed",
"note": "Full public directory statement. Source lane: backlog; source resolution: open. Source verification claims are attributed only; no local Lean verification has be... | {"created_at":"2026-08-17T18:36:27+00:00","display_standard":{"editorial_review":"submitter_supplied","formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":8180,"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": "2026-09-08T03:32:59.095Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"8add417a-cdce-4712-898d-ef7c87184139","format_issues":null,"number":5321} |
1.0.0 | problem:theoremdb-p8182 | problem | theoremdb-p8182 | Supercongruence for generalized Apéry-like numerators | {
"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 nonnegative integer $m$, define $u_m(n)$ to be the numerator of the rational number
$$
\sum_{k=1}^{n} \frac{1}{k^{2m+1}} \binom{n}{k}^2 \binom{n+k}{k}^2
$$
for $n \geq 1$, where the numerator is taken in lowest terms. Does there exist, for every $m \in \mathbb{N}$, a finite set of primes such that $u_m(p-1... | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P8182",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P8182",
"citation": null,
"doi": null,
"arxiv_id": null,
"license": "Selected packets label their content CC0-1.0; original references retain their own terms",
"notes": "Selected problem... | theorem general_supercongruence (m : ℕ) : ∃ (exceptions : Finset ℕ), ∀ p, p.Prime →
p ∉ exceptions → u m (p - 1) = (0 : ZMod (p ^ 4)) | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem general_supercongruence (m : ℕ) : ∃ (exceptions : Finset ℕ), ∀ p, p.Prime →\n p ∉ exceptions → u m (p - 1) = (0 : ZMod (p ^ 4))",
"target": null,
"source_url": "https://theoremdb.org/statements/P8182",
"format": "declaration-or-fragment",
"sha256":... | [] | TheoremDB P8182 · Candidate awaiting source review | null | Source-reported assessment, retrieved 8 September 2026. Frontier Science has not independently reviewed this status.
Open. No signed Lean verification is attached to this problem or its selected resolution.
This record is in the source backlog; its novelty and unresolved status have not passed source review. | null | published | incremental | hard | [
"source-candidate",
"theoremdb"
] | null | null | null | null | [
{
"kind": "website",
"citation": "TheoremDB P8182",
"url": "https://theoremdb.org/statements/P8182",
"note": null,
"sort_order": 0
},
{
"kind": "formalization",
"citation": "Source formal statement (not locally verified)",
"url": "https://theoremdb.org/statements/P8182",
"not... | {
"source_record": "P8182",
"imported_at": "2026-09-08",
"status_as_of": "2026-09-08",
"evidence_label": "Source candidate · not reviewed",
"note": "Full public directory statement. Source lane: backlog; source resolution: open. Source verification claims are attributed only; no local Lean verification has be... | {"created_at":"2026-08-17T18:36:32+00:00","display_standard":{"editorial_review":"submitter_supplied","formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":8182,"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": "2026-09-08T03:33:06.182Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"4b510d27-3752-46d8-9d49-9c9f38b75018","format_issues":null,"number":6688} |
1.0.0 | problem:theoremdb-p8184 | problem | theoremdb-p8184 | Almost-all asymptotic behavior of the maximal factorial quotient deficit | {
"main": "math",
"sub": "number theory",
"slug": "number-theory",
"name": "Number theory",
"original_discipline": "mathematics"
} | [] | [
{
"code": "11",
"basis": "community-map",
"evidence": "Inferred from community field: number-theory"
}
] | For any integer $k \geq 2$, define $g_k(n)$ to be the maximum value of $(a_1 + \cdots + a_k) - n$, where $a_1, \ldots, a_k$ are non-negative integers such that $a_1! \cdots a_k!$ divides $n!$. Does there exist, for each $k \geq 2$, a real constant $c_k$ such that for every $\varepsilon > 0$, the proportion of integers ... | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P8184",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P8184",
"citation": null,
"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_400.parts.ii :
answer(sorry) ↔ ∀ᵉ (k ≥ 2), ∃ c : ℝ, ∀ ε > 0,
Tendsto (fun x : ℕ ↦
(((Icc 1 x).filter (fun n ↦
|(g k n : ℝ) - c * Real.log x| ≤ ε * Real.log x)).card : ℝ) / x)
atTop (𝓝 1) | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem erdos_400.parts.ii :\n answer(sorry) ↔ ∀ᵉ (k ≥ 2), ∃ c : ℝ, ∀ ε > 0,\n Tendsto (fun x : ℕ ↦\n (((Icc 1 x).filter (fun n ↦\n |(g k n : ℝ) - c * Real.log x| ≤ ε * Real.log x)).card : ℝ) / x)\n atTop (𝓝 1)",
"target": null,
"sourc... | [] | TheoremDB P8184 · Candidate awaiting source review | null | Source-reported assessment, retrieved 8 September 2026. Frontier Science has not independently reviewed this status.
Open. No signed Lean verification is attached to this problem or its selected resolution.
This record is in the source backlog; its novelty and unresolved status have not passed source review. | null | published | incremental | hard | [
"source-candidate",
"theoremdb"
] | null | null | null | null | [
{
"kind": "website",
"citation": "TheoremDB P8184",
"url": "https://theoremdb.org/statements/P8184",
"note": null,
"sort_order": 0
},
{
"kind": "formalization",
"citation": "Source formal statement (not locally verified)",
"url": "https://theoremdb.org/statements/P8184",
"not... | {
"source_record": "P8184",
"imported_at": "2026-09-08",
"status_as_of": "2026-09-08",
"evidence_label": "Source candidate · not reviewed",
"note": "Full public directory statement. Source lane: backlog; source resolution: open. Source verification claims are attributed only; no local Lean verification has be... | {"created_at":"2026-08-17T18:36:36+00:00","display_standard":{"editorial_review":"submitter_supplied","formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":8184,"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": "2026-09-08T03:32:49.556Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"475ae409-6a8e-44c0-8733-03ffb36f13a1","format_issues":null,"number":3233} |
1.0.0 | problem:theoremdb-p8186 | problem | theoremdb-p8186 | Limit of consecutive term ratios for lacunary strongly complete sequences with hereditary incompleteness | {
"main": "math",
"sub": "number theory",
"slug": "number-theory",
"name": "Number theory",
"original_discipline": "mathematics"
} | [] | [
{
"code": "11",
"basis": "community-map",
"evidence": "Inferred from community field: number-theory"
}
] | Let $A : \mathbb{N} \to \mathbb{N}$ be a sequence of natural numbers. A sequence is called lacunary if there exists a constant $c > 1$ such that $A(n+1) > c \cdot A(n)$ for all sufficiently large $n$. A sequence is called additively complete if every sufficiently large natural number can be expressed as a sum of distin... | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P8186",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P8186",
"citation": null,
"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_346 : answer(sorry) ↔ ∀ {A : ℕ → ℕ}, IsLacunary A → IsAddStronglyCompleteNatSeq A →
(∀ B : Set ℕ, B ⊆ range A → B.Infinite → ¬ IsAddComplete (range A \ B)) →
Tendsto (fun n => A (n + 1) / (A n : ℝ)) atTop (𝓝 ((1 + √5) / 2)) | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem erdos_346 : answer(sorry) ↔ ∀ {A : ℕ → ℕ}, IsLacunary A → IsAddStronglyCompleteNatSeq A →\n (∀ B : Set ℕ, B ⊆ range A → B.Infinite → ¬ IsAddComplete (range A \\ B)) →\n Tendsto (fun n => A (n + 1) / (A n : ℝ)) atTop (𝓝 ((1 + √5) / 2))",
"target": null,
... | [] | TheoremDB P8186 · Candidate awaiting source review | null | Source-reported assessment, retrieved 8 September 2026. Frontier Science has not independently reviewed this status.
Open. No signed Lean verification is attached to this problem or its selected resolution.
This record is in the source backlog; its novelty and unresolved status have not passed source review. | null | published | incremental | hard | [
"source-candidate",
"theoremdb"
] | null | null | null | null | [
{
"kind": "website",
"citation": "TheoremDB P8186",
"url": "https://theoremdb.org/statements/P8186",
"note": null,
"sort_order": 0
},
{
"kind": "formalization",
"citation": "Source formal statement (not locally verified)",
"url": "https://theoremdb.org/statements/P8186",
"not... | {
"source_record": "P8186",
"imported_at": "2026-09-08",
"status_as_of": "2026-09-08",
"evidence_label": "Source candidate · not reviewed",
"note": "Full public directory statement. Source lane: backlog; source resolution: open. Source verification claims are attributed only; no local Lean verification has be... | {"created_at":"2026-08-17T18:36:37+00:00","display_standard":{"editorial_review":"submitter_supplied","formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":8186,"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": "2026-09-08T03:33:03.194Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"effd17ef-2fc9-42f3-9846-64a5b0b859ff","format_issues":null,"number":6043} |
1.0.0 | problem:theoremdb-p8188 | problem | theoremdb-p8188 | Infinitude of Mersenne primes | {
"main": "math",
"sub": "number theory",
"slug": "number-theory",
"name": "Number theory",
"original_discipline": "mathematics"
} | [] | [
{
"code": "11",
"basis": "community-map",
"evidence": "Inferred from community field: number-theory"
}
] | A Mersenne number is a natural number of the form $M_p = 2^p - 1$, where $p$ is a natural number. A Mersenne prime is a Mersenne number that is prime. Determine whether the set $\{p \in \mathbb{N} : 2^p - 1 \text{ is prime}\}$ is infinite. That is, are there infinitely many Mersenne primes? | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P8188",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P8188",
"citation": null,
"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_mersenne_primes :
answer(sorry) ↔ Set.Infinite { p : ℕ | (mersenne p).Prime } | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem infinitely_many_mersenne_primes :\n answer(sorry) ↔ Set.Infinite { p : ℕ | (mersenne p).Prime }",
"target": null,
"source_url": "https://theoremdb.org/statements/P8188",
"format": "declaration-or-fragment",
"sha256": "8bb8d96acc1d5594b27df9bdf5618ad7... | [] | TheoremDB P8188 · Candidate awaiting source review | null | Source-reported assessment, retrieved 8 September 2026. Frontier Science has not independently reviewed this status.
Open. No signed Lean verification is attached to this problem or its selected resolution.
This record is in the source backlog; its novelty and unresolved status have not passed source review. | null | published | incremental | hard | [
"source-candidate",
"theoremdb"
] | null | null | null | null | [
{
"kind": "website",
"citation": "TheoremDB P8188",
"url": "https://theoremdb.org/statements/P8188",
"note": null,
"sort_order": 0
},
{
"kind": "formalization",
"citation": "Source formal statement (not locally verified)",
"url": "https://theoremdb.org/statements/P8188",
"not... | {
"source_record": "P8188",
"imported_at": "2026-09-08",
"status_as_of": "2026-09-08",
"evidence_label": "Source candidate · not reviewed",
"note": "Full public directory statement. Source lane: backlog; source resolution: open. Source verification claims are attributed only; no local Lean verification has be... | {"created_at":"2026-08-17T18:36:40+00:00","display_standard":{"editorial_review":"submitter_supplied","formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":8188,"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": "2026-09-08T03:33:01.172Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"3fe84cf7-18c5-4364-91c8-84c5d50ae66b","format_issues":null,"number":5673} |
1.0.0 | problem:theoremdb-p8190 | problem | theoremdb-p8190 | Existence of an absolutely maximally entangled state for ten parties of local dimension ten | {
"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. 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 obtained by tracing out the complementary parties is maximally mixed. Does there exist ... | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P8190",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P8190",
"citation": null,
"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_10_10_open :
answer(sorry) ↔ ExistsAME 10 10 | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem ame_10_10_open :\n answer(sorry) ↔ ExistsAME 10 10",
"target": null,
"source_url": "https://theoremdb.org/statements/P8190",
"format": "declaration-or-fragment",
"sha256": "b429dd3ff0d8afa53dd862b5f5b19b18d328703d7f80150dfd6ff28a0fbe1ee7",
"lic... | [] | TheoremDB P8190 · Candidate awaiting source review | null | Source-reported assessment, retrieved 8 September 2026. Frontier Science has not independently reviewed this status.
Open. No signed Lean verification is attached to this problem or its selected resolution.
This record is in the source backlog; its novelty and unresolved status have not passed source review. | null | published | incremental | hard | [
"source-candidate",
"theoremdb"
] | null | null | null | null | [
{
"kind": "website",
"citation": "TheoremDB P8190",
"url": "https://theoremdb.org/statements/P8190",
"note": null,
"sort_order": 0
},
{
"kind": "formalization",
"citation": "Source formal statement (not locally verified)",
"url": "https://theoremdb.org/statements/P8190",
"not... | {
"source_record": "P8190",
"imported_at": "2026-09-08",
"status_as_of": "2026-09-08",
"evidence_label": "Source candidate · not reviewed",
"note": "Full public directory statement. Source lane: backlog; source resolution: open. Source verification claims are attributed only; no local Lean verification has be... | {"created_at":"2026-08-17T18:36:43+00:00","display_standard":{"editorial_review":"submitter_supplied","formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":8190,"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": "2026-09-08T03:32:58.659Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"fc4502dc-5656-4d17-bcfc-86b4716521ce","format_issues":null,"number":5228} |
1.0.0 | problem:theoremdb-p8192 | problem | theoremdb-p8192 | Unbounded growth of the smallest missing even prime gap | {
"main": "math",
"sub": "number theory",
"slug": "number-theory",
"name": "Number theory",
"original_discipline": "mathematics"
} | [] | [
{
"code": "11",
"basis": "community-map",
"evidence": "Inferred from community field: number-theory"
}
] | Let $p_n$ denote the $n$th prime number, and let $d_n = p_{n+1} - p_n$ denote the $n$th prime gap. For each natural number $x$, let $r(x)$ be the smallest even positive integer $t$ such that there exists no index $n \leq x$ with $d_n = t$. Does $r(x) \to \infty$ as $x \to \infty$? | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P8192",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P8192",
"citation": null,
"doi": null,
"arxiv_id": null,
"license": "Selected packets label their content CC0-1.0; original references retain their own terms",
"notes": "Selected problem... | theorem erdos_853.parts.i : atTop.Tendsto r atTop | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem erdos_853.parts.i : atTop.Tendsto r atTop",
"target": null,
"source_url": "https://theoremdb.org/statements/P8192",
"format": "declaration-or-fragment",
"sha256": "e804c54e4aaeca082666eeca77aab97ecb62b4fdcf16c9cfc476efa1df3de4e3",
"license": "Sele... | [] | TheoremDB P8192 · Candidate awaiting source review | null | Source-reported assessment, retrieved 8 September 2026. Frontier Science has not independently reviewed this status.
Open. No signed Lean verification is attached to this problem or its selected resolution.
This record is in the source backlog; its novelty and unresolved status have not passed source review. | null | published | incremental | hard | [
"source-candidate",
"theoremdb"
] | null | null | null | null | [
{
"kind": "website",
"citation": "TheoremDB P8192",
"url": "https://theoremdb.org/statements/P8192",
"note": null,
"sort_order": 0
},
{
"kind": "formalization",
"citation": "Source formal statement (not locally verified)",
"url": "https://theoremdb.org/statements/P8192",
"not... | {
"source_record": "P8192",
"imported_at": "2026-09-08",
"status_as_of": "2026-09-08",
"evidence_label": "Source candidate · not reviewed",
"note": "Full public directory statement. Source lane: backlog; source resolution: open. Source verification claims are attributed only; no local Lean verification has be... | {"created_at":"2026-08-17T18:36:47+00:00","display_standard":{"editorial_review":"submitter_supplied","formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":8192,"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": "2026-09-08T03:33:07.110Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"9f960d30-bcc1-44a0-8085-6e5d31b53e2c","format_issues":null,"number":6888} |
1.0.0 | problem:theoremdb-p8194 | problem | theoremdb-p8194 | Improved Upper Bound for the Chromatic Number of Four-Term Arithmetic Progressions | {
"main": "math",
"sub": "combinatorics",
"slug": "combinatorics",
"name": "Combinatorics",
"original_discipline": "mathematics"
} | [] | [
{
"code": "05",
"basis": "community-map",
"evidence": "Inferred from community field: combinatorics"
}
] | For each positive integer $n$, let $h(n)$ denote the smallest number of colours $k$ such that the set $\{1, 2, \ldots, n\}$ can be coloured with $k$ colours in such a way that every four-term arithmetic progression contained in $\{1, 2, \ldots, n\}$ contains at least three distinct colours. It is known that $h(n) = O\l... | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P8194",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P8194",
"citation": null,
"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_160.better_upper :
let upper_bound : ℕ → ℝ := answer(sorry)
(fun n => (erdos_160.h n : ℝ)) =O[atTop] upper_bound ∧
upper_bound =o[atTop] fun n => (n : ℝ) ^ ((2 : ℝ) / 3) | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem erdos_160.better_upper :\n let upper_bound : ℕ → ℝ := answer(sorry)\n (fun n => (erdos_160.h n : ℝ)) =O[atTop] upper_bound ∧\n upper_bound =o[atTop] fun n => (n : ℝ) ^ ((2 : ℝ) / 3)",
"target": null,
"source_url": "https://theoremdb.org/statements/P8... | [] | TheoremDB P8194 · Candidate awaiting source review | null | Source-reported assessment, retrieved 8 September 2026. Frontier Science has not independently reviewed this status.
Open. No signed Lean verification is attached to this problem or its selected resolution.
This record is in the source backlog; its novelty and unresolved status have not passed source review. | null | published | incremental | hard | [
"source-candidate",
"theoremdb"
] | null | null | null | null | [
{
"kind": "website",
"citation": "TheoremDB P8194",
"url": "https://theoremdb.org/statements/P8194",
"note": null,
"sort_order": 0
},
{
"kind": "formalization",
"citation": "Source formal statement (not locally verified)",
"url": "https://theoremdb.org/statements/P8194",
"not... | {
"source_record": "P8194",
"imported_at": "2026-09-08",
"status_as_of": "2026-09-08",
"evidence_label": "Source candidate · not reviewed",
"note": "Full public directory statement. Source lane: backlog; source resolution: open. Source verification claims are attributed only; no local Lean verification has be... | {"created_at":"2026-08-17T18:36:49+00:00","display_standard":{"editorial_review":"submitter_supplied","formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":8194,"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": "2026-09-08T03:32:59.666Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"e63e27e9-a254-4828-8e26-cc87a1f5e5e8","format_issues":null,"number":5444} |
1.0.0 | problem:theoremdb-p8196 | problem | theoremdb-p8196 | Asymptotic order of coprimality count for harmonic number denominators | {
"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 $L_n$ denote the least common multiple of $\{1, 2, \ldots, n\}$, and let $a_n$ be the unique positive integer defined by the identity
$$
\sum_{k=1}^{n} \frac{1}{k} = \frac{a_n}{L_n}.
$$
For a real number $x \geq 1$, let $N(x)$ denote the number of integers $n$ with $1 \leq n \leq ... | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P8196",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P8196",
"citation": null,
"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_291.variants.shiu_heuristic_asymptotic :
Asymptotics.IsTheta atTop
(fun x : ℕ ↦ ((((Finset.Icc 1 x).filter (fun n ↦ Nat.gcd (a n) (L n) = 1)).card : ℝ)))
(fun x : ℕ ↦ (x : ℝ) / Real.log x) | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem erdos_291.variants.shiu_heuristic_asymptotic :\n Asymptotics.IsTheta atTop\n (fun x : ℕ ↦ ((((Finset.Icc 1 x).filter (fun n ↦ Nat.gcd (a n) (L n) = 1)).card : ℝ)))\n (fun x : ℕ ↦ (x : ℝ) / Real.log x)",
"target": null,
"source_url": "https://theo... | [] | TheoremDB P8196 · Candidate awaiting source review | null | Source-reported assessment, retrieved 8 September 2026. Frontier Science has not independently reviewed this status.
Open. No signed Lean verification is attached to this problem or its selected resolution.
This record is in the source backlog; its novelty and unresolved status have not passed source review. | null | published | incremental | hard | [
"source-candidate",
"theoremdb"
] | null | null | null | null | [
{
"kind": "website",
"citation": "TheoremDB P8196",
"url": "https://theoremdb.org/statements/P8196",
"note": null,
"sort_order": 0
},
{
"kind": "formalization",
"citation": "Source formal statement (not locally verified)",
"url": "https://theoremdb.org/statements/P8196",
"not... | {
"source_record": "P8196",
"imported_at": "2026-09-08",
"status_as_of": "2026-09-08",
"evidence_label": "Source candidate · not reviewed",
"note": "Full public directory statement. Source lane: backlog; source resolution: open. Source verification claims are attributed only; no local Lean verification has be... | {"created_at":"2026-08-17T18:36:52+00:00","display_standard":{"editorial_review":"submitter_supplied","formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":8196,"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": "2026-09-08T03:32:50.350Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"f8b4ed3d-4e2f-4541-9724-92684ba58de9","format_issues":null,"number":3401} |
1.0.0 | problem:theoremdb-p8198 | problem | theoremdb-p8198 | Positive Upper Density of Odd Numbers Not Representable as a Prime Plus Two Powers of 2 | {
"main": "math",
"sub": "combinatorics",
"slug": "combinatorics",
"name": "Combinatorics",
"original_discipline": "mathematics"
} | [] | [
{
"code": "05",
"basis": "community-map",
"evidence": "Inferred from community field: combinatorics"
}
] | Let $\mathbb{N}$ denote the set of natural numbers. Define a set $A \subseteq \mathbb{N}$ by
$$
A = \{\, n \in \mathbb{N} \mid n \text{ is odd and there do not exist } p, k, l \in \mathbb{N} \text{ such that } p \text{ is prime and } n = p + 2^k + 2^l \,\}.
$$
The set $A$ is known to be infinite. For a subset $S \s... | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P8198",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P8198",
"citation": null,
"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_9 : answer(sorry) ↔ 0 < Erdos9A.upperDensity | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem erdos_9 : answer(sorry) ↔ 0 < Erdos9A.upperDensity",
"target": null,
"source_url": "https://theoremdb.org/statements/P8198",
"format": "declaration-or-fragment",
"sha256": "b93cc3efcabf77e92c531284025518d7d89b416462a0ec868fc4fce4bed46968",
"licens... | [] | TheoremDB P8198 · Candidate awaiting source review | null | Source-reported assessment, retrieved 8 September 2026. Frontier Science has not independently reviewed this status.
Open. No signed Lean verification is attached to this problem or its selected resolution.
This record is in the source backlog; its novelty and unresolved status have not passed source review. | null | published | incremental | hard | [
"source-candidate",
"theoremdb"
] | null | null | null | null | [
{
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"citation": "TheoremDB P8198",
"url": "https://theoremdb.org/statements/P8198",
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"citation": "Source formal statement (not locally verified)",
"url": "https://theoremdb.org/statements/P8198",
"not... | {
"source_record": "P8198",
"imported_at": "2026-09-08",
"status_as_of": "2026-09-08",
"evidence_label": "Source candidate · not reviewed",
"note": "Full public directory statement. Source lane: backlog; source resolution: open. Source verification claims are attributed only; no local Lean verification has be... | {"created_at":"2026-08-17T18:36:55+00:00","display_standard":{"formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":8198,"public_status":{"evidence":{"hasFormalization":false,"hasInformalProof":false,"hasRecordedWork":false,"solutionRecord":null},"lean":{"kernelAccepted":false,"label":"","reaso... | [
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] | [] | {
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"updated_at": "2026-09-08T03:32:15.935Z",
"published_at": "2026-09-08T03:33:04.970Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"796f090f-f755-4da5-9b97-2f00af88a4e8","format_issues":null,"number":6425} |
1.0.0 | problem:theoremdb-p8200 | problem | theoremdb-p8200 | The Zariski Cancellation Problem for Multivariate Polynomial Rings over Fields of Characteristic Zero | {
"main": "math",
"sub": "algebra",
"slug": "algebra",
"name": "Algebra",
"original_discipline": "mathematics"
} | [] | [] | Let $k$ be a field of characteristic zero and let $\iota$ be a finite type. The multivariate polynomial ring $k[(X_i)_{i \in \iota}]$ is defined to be cancellative as follows: a finitely generated $k$-algebra $A$ is called cancellative if for every finitely generated $k$-algebra $B$, an isomorphism of polynomial rings ... | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P8200",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P8200",
"citation": null,
"doi": null,
"arxiv_id": null,
"license": "Selected packets label their content CC0-1.0; original references retain their own terms",
"notes": "Selected problem... | theorem zariski_cancellation_problem {k : Type*} [Field k]
[CharZero k] {ι : Type*} [Fintype ι] : IsCancellative k (MvPolynomial ι k) | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem zariski_cancellation_problem {k : Type*} [Field k]\n [CharZero k] {ι : Type*} [Fintype ι] : IsCancellative k (MvPolynomial ι k)",
"target": null,
"source_url": "https://theoremdb.org/statements/P8200",
"format": "declaration-or-fragment",
"sha256":... | [] | TheoremDB P8200 · Candidate awaiting source review | null | Source-reported assessment, retrieved 8 September 2026. Frontier Science has not independently reviewed this status.
Open. No signed Lean verification is attached to this problem or its selected resolution.
This record is in the source backlog; its novelty and unresolved status have not passed source review. | null | published | incremental | hard | [
"source-candidate",
"theoremdb"
] | null | null | null | null | [
{
"kind": "website",
"citation": "TheoremDB P8200",
"url": "https://theoremdb.org/statements/P8200",
"note": null,
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},
{
"kind": "formalization",
"citation": "Source formal statement (not locally verified)",
"url": "https://theoremdb.org/statements/P8200",
"not... | {
"source_record": "P8200",
"imported_at": "2026-09-08",
"status_as_of": "2026-09-08",
"evidence_label": "Source candidate · not reviewed",
"note": "Full public directory statement. Source lane: backlog; source resolution: open. Source verification claims are attributed only; no local Lean verification has be... | {"created_at":"2026-08-17T18:36:58+00:00","display_standard":{"editorial_review":"submitter_supplied","formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":8200,"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": "2026-09-08T03:33:06.866Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"976378bb-9c0f-4189-953c-0a387c15eed9","format_issues":null,"number":6834} |
1.0.0 | problem:theoremdb-p8202 | problem | theoremdb-p8202 | Existence of a SIC-POVM in dimension 69 | {
"main": "math",
"sub": "algebra",
"slug": "algebra",
"name": "Algebra",
"original_discipline": "mathematics"
} | [] | [] | A state vector in dimension $d$ is an element of the complex Hilbert space $\mathbb{C}^d$ with the standard Euclidean inner product. A state vector $\psi$ is normalized if its $L^2$ norm equals $1$. For two state vectors $\varphi$ and $\psi$ in the same dimension, their squared overlap is the squared magnitude of their... | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P8202",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P8202",
"citation": null,
"doi": null,
"arxiv_id": null,
"license": "Selected packets label their content CC0-1.0; original references retain their own terms",
"notes": "Selected problem... | theorem hasSICPOVM_69 : answer(sorry) ↔ HasSICPOVM 69 | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem hasSICPOVM_69 : answer(sorry) ↔ HasSICPOVM 69",
"target": null,
"source_url": "https://theoremdb.org/statements/P8202",
"format": "declaration-or-fragment",
"sha256": "ddb32a0c6f6cb826dd0520faeb3e013143f531656002a031f7f74916bc502e66",
"license": "... | [] | TheoremDB P8202 · Candidate awaiting source review | null | Source-reported assessment, retrieved 8 September 2026. Frontier Science has not independently reviewed this status.
Open. No signed Lean verification is attached to this problem or its selected resolution.
This record is in the source backlog; its novelty and unresolved status have not passed source review. | null | published | incremental | hard | [
"source-candidate",
"theoremdb"
] | null | null | null | null | [
{
"kind": "website",
"citation": "TheoremDB P8202",
"url": "https://theoremdb.org/statements/P8202",
"note": null,
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},
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"kind": "formalization",
"citation": "Source formal statement (not locally verified)",
"url": "https://theoremdb.org/statements/P8202",
"not... | {
"source_record": "P8202",
"imported_at": "2026-09-08",
"status_as_of": "2026-09-08",
"evidence_label": "Source candidate · not reviewed",
"note": "Full public directory statement. Source lane: backlog; source resolution: open. Source verification claims are attributed only; no local Lean verification has be... | {"created_at":"2026-08-17T18:37:01+00:00","display_standard":{"editorial_review":"submitter_supplied","formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":8202,"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": "2026-09-08T03:32:58.518Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"067e8e90-bea5-4aae-b92d-337d582866da","format_issues":null,"number":5199} |
1.0.0 | problem:theoremdb-p8204 | problem | theoremdb-p8204 | Distinct consecutive totient values in long runs | {
"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 $\varphi(n)$ denote Euler's totient function, which counts the number of integers from $1$ to $n$ that are coprime to $n$. For natural numbers $n$ and $K$, we say that $\textit{$n$ starts a distinct totient run of length $K$}$ if the values $\varphi(n+1), \varphi(n+2), \ldots, \varphi(n+... | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P8204",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P8204",
"citation": null,
"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_1004 :
answer(sorry) ↔ ∀ c > (0 : ℝ), ∀ᶠ x in atTop, ∃ n ≤ x,
IsDistinctTotientRun n ⌊(Real.log (x : ℝ)) ^ c⌋₊ | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem erdos_1004 :\n answer(sorry) ↔ ∀ c > (0 : ℝ), ∀ᶠ x in atTop, ∃ n ≤ x,\n IsDistinctTotientRun n ⌊(Real.log (x : ℝ)) ^ c⌋₊",
"target": null,
"source_url": "https://theoremdb.org/statements/P8204",
"format": "declaration-or-fragment",
"sha256": "... | [] | TheoremDB P8204 · Candidate awaiting source review | null | Source-reported assessment, retrieved 8 September 2026. Frontier Science has not independently reviewed this status.
Open. No signed Lean verification is attached to this problem or its selected resolution.
This record is in the source backlog; its novelty and unresolved status have not passed source review. | null | published | incremental | hard | [
"source-candidate",
"theoremdb"
] | null | null | null | null | [
{
"kind": "website",
"citation": "TheoremDB P8204",
"url": "https://theoremdb.org/statements/P8204",
"note": null,
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},
{
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"citation": "Source formal statement (not locally verified)",
"url": "https://theoremdb.org/statements/P8204",
"not... | {
"source_record": "P8204",
"imported_at": "2026-09-08",
"status_as_of": "2026-09-08",
"evidence_label": "Source candidate · not reviewed",
"note": "Full public directory statement. Source lane: backlog; source resolution: open. Source verification claims are attributed only; no local Lean verification has be... | {"created_at":"2026-08-17T18:37:04+00:00","display_standard":{"editorial_review":"submitter_supplied","formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":8204,"public_status":{"evidence":{"hasFormalization":false,"hasInformalProof":false,"hasRecordedWork":false,"solutionRecord":null},"lean":{... | [
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"updated_at": "2026-09-08T03:32:15.935Z",
"published_at": "2026-09-08T03:32:53.217Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"f4c82764-bdcf-4d40-b3ae-5e84e12fa2db","format_issues":{"errors":2,"fields":["statement"]},"number":4028} |
1.0.0 | problem:theoremdb-p8206 | problem | theoremdb-p8206 | Asymptotic formula for the maximum size of a 3-term-AP-free subset of \(\{1,\dots,N\}\) | {
"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 a positive integer $N$, let $r_3(N)$ denote the largest possible cardinality of a subset of $\{1, 2, \dots, N\}$ that contains no non-trivial 3-term arithmetic progression. A 3-term arithmetic progression is called non-trivial if its terms are distinct. Determine whether there exists a function $f: \mathbb{N} \to \... | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P8206",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P8206",
"citation": null,
"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_142.variants.three : (fun N => (r 3 N : ℝ)) =Θ[atTop] (answer(sorry) : ℕ → ℝ) | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem erdos_142.variants.three : (fun N => (r 3 N : ℝ)) =Θ[atTop] (answer(sorry) : ℕ → ℝ)",
"target": null,
"source_url": "https://theoremdb.org/statements/P8206",
"format": "declaration-or-fragment",
"sha256": "a2fad92c55c88a1c6553d9c30089cce17aa98b0aa7e83... | [] | TheoremDB P8206 · Candidate awaiting source review | null | Source-reported assessment, retrieved 8 September 2026. Frontier Science has not independently reviewed this status.
Open. No signed Lean verification is attached to this problem or its selected resolution.
This record is in the source backlog; its novelty and unresolved status have not passed source review. | null | published | incremental | hard | [
"source-candidate",
"theoremdb"
] | null | null | null | null | [
{
"kind": "website",
"citation": "TheoremDB P8206",
"url": "https://theoremdb.org/statements/P8206",
"note": null,
"sort_order": 0
},
{
"kind": "formalization",
"citation": "Source formal statement (not locally verified)",
"url": "https://theoremdb.org/statements/P8206",
"not... | {
"source_record": "P8206",
"imported_at": "2026-09-08",
"status_as_of": "2026-09-08",
"evidence_label": "Source candidate · not reviewed",
"note": "Full public directory statement. Source lane: backlog; source resolution: open. Source verification claims are attributed only; no local Lean verification has be... | {"created_at":"2026-08-17T18:37:05+00:00","display_standard":{"editorial_review":"submitter_supplied","formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":8206,"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": "2026-09-08T03:32:50.044Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"4f0cf349-76d9-43e2-b7ac-ad50043301a4","format_issues":null,"number":3339} |
1.0.0 | problem:theoremdb-p8208 | problem | theoremdb-p8208 | Consecutive $2$-full and $3$-full integers | {
"main": "math",
"sub": "number theory",
"slug": "number-theory",
"name": "Number theory",
"original_discipline": "mathematics"
} | [] | [
{
"code": "11",
"basis": "community-map",
"evidence": "Inferred from community field: number-theory"
}
] | A positive integer $n$ is called $k$-*full* if for every prime $p$ dividing $n$, the exponent of $p$ in the prime factorization of $n$ is at least $k$. Does there exist a positive integer $n$ such that $n$ is $2$-full and $n+1$ is $3$-full? | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P8208",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P8208",
"citation": null,
"doi": null,
"arxiv_id": null,
"license": "Selected packets label their content CC0-1.0; original references retain their own terms",
"notes": "Selected problem... | theorem erdos_366 : answer(sorry) ↔ ∃ n > 0, (2).Full n ∧ (3).Full (n + 1) | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem erdos_366 : answer(sorry) ↔ ∃ n > 0, (2).Full n ∧ (3).Full (n + 1)",
"target": null,
"source_url": "https://theoremdb.org/statements/P8208",
"format": "declaration-or-fragment",
"sha256": "7a864b3e3612fe87f9b22f66bfe71806aec8e2e0ae3c293dbf513d2b1ee2b7... | [] | TheoremDB P8208 · Candidate awaiting source review | null | Source-reported assessment, retrieved 8 September 2026. Frontier Science has not independently reviewed this status.
Open. No signed Lean verification is attached to this problem or its selected resolution.
This record is in the source backlog; its novelty and unresolved status have not passed source review. | null | published | incremental | hard | [
"source-candidate",
"theoremdb"
] | null | null | null | null | [
{
"kind": "website",
"citation": "TheoremDB P8208",
"url": "https://theoremdb.org/statements/P8208",
"note": null,
"sort_order": 0
},
{
"kind": "formalization",
"citation": "Source formal statement (not locally verified)",
"url": "https://theoremdb.org/statements/P8208",
"not... | {
"source_record": "P8208",
"imported_at": "2026-09-08",
"status_as_of": "2026-09-08",
"evidence_label": "Source candidate · not reviewed",
"note": "Full public directory statement. Source lane: backlog; source resolution: open. Source verification claims are attributed only; no local Lean verification has be... | {"created_at":"2026-08-17T18:37:08+00:00","display_standard":{"editorial_review":"submitter_supplied","formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":8208,"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": "2026-09-08T03:32:52.591Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"14a4cae1-356d-4160-8515-b808fac9072a","format_issues":null,"number":3893} |
1.0.0 | problem:theoremdb-p8210 | problem | theoremdb-p8210 | Infinitude of central binomial coefficients coprime to 105 | {
"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 $\binom{2n}{n}$ denote the central binomial coefficient. Determine whether the set of natural numbers $n$ such that $\binom{2n}{n}$ is coprime to $105$ is infinite. | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P8210",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P8210",
"citation": null,
"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_376 : answer(sorry) ↔ { (n : ℕ) | n.centralBinom.Coprime 105 }.Infinite | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem erdos_376 : answer(sorry) ↔ { (n : ℕ) | n.centralBinom.Coprime 105 }.Infinite",
"target": null,
"source_url": "https://theoremdb.org/statements/P8210",
"format": "declaration-or-fragment",
"sha256": "7fc7045140304685297670fb1983dab846eb28abef2373e6e52... | [] | TheoremDB P8210 · Candidate awaiting source review | null | Source-reported assessment, retrieved 8 September 2026. Frontier Science has not independently reviewed this status.
Open. No signed Lean verification is attached to this problem or its selected resolution.
This record is in the source backlog; its novelty and unresolved status have not passed source review. | null | published | incremental | hard | [
"source-candidate",
"theoremdb"
] | null | null | null | null | [
{
"kind": "website",
"citation": "TheoremDB P8210",
"url": "https://theoremdb.org/statements/P8210",
"note": null,
"sort_order": 0
},
{
"kind": "formalization",
"citation": "Source formal statement (not locally verified)",
"url": "https://theoremdb.org/statements/P8210",
"not... | {
"source_record": "P8210",
"imported_at": "2026-09-08",
"status_as_of": "2026-09-08",
"evidence_label": "Source candidate · not reviewed",
"note": "Full public directory statement. Source lane: backlog; source resolution: open. Source verification claims are attributed only; no local Lean verification has be... | {"created_at":"2026-08-17T18:37:09+00:00","display_standard":{"editorial_review":"submitter_supplied","formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":8210,"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": "2026-09-08T03:33:01.104Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"258fe676-ccd1-454b-991d-929cf5e490cc","format_issues":null,"number":5658} |
1.0.0 | problem:theoremdb-p8212 | problem | theoremdb-p8212 | Grimm's Conjecture on Distinct Prime Divisors of Consecutive Composite Numbers | {
"main": "math",
"sub": "number theory",
"slug": "number-theory",
"name": "Number theory",
"original_discipline": "mathematics"
} | [] | [
{
"code": "11",
"basis": "community-map",
"evidence": "Inferred from community field: number-theory"
}
] | Let $n$ and $k$ be positive integers. Suppose that each of the $k$ consecutive integers $n, n+1, \ldots, n+k-1$ is composite. Does there exist a set of $k$ distinct prime numbers $p_0, p_1, \ldots, p_{k-1}$ such that $p_i$ divides $n+i$ for each $i \in \{0, 1, \ldots, k-1\}$? | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P8212",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P8212",
"citation": null,
"doi": null,
"arxiv_id": null,
"license": "Selected packets label their content CC0-1.0; original references retain their own terms",
"notes": "Selected problem... | theorem grimm_conjecture (n k : ℕ) (hn : 1 ≤ n) (hk : 1 ≤ k)
(h : ∀ i : Fin k, (n + i).Composite) :
∃ ps : Fin k ↪ ℕ, ∀ i : Fin k, (ps i).Prime ∧ ps i ∣ (n + i) | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem grimm_conjecture (n k : ℕ) (hn : 1 ≤ n) (hk : 1 ≤ k)\n (h : ∀ i : Fin k, (n + i).Composite) :\n ∃ ps : Fin k ↪ ℕ, ∀ i : Fin k, (ps i).Prime ∧ ps i ∣ (n + i)",
"target": null,
"source_url": "https://theoremdb.org/statements/P8212",
"format": "declar... | [] | TheoremDB P8212 · Candidate awaiting source review | null | Source-reported assessment, retrieved 8 September 2026. Frontier Science has not independently reviewed this status.
Open. No signed Lean verification is attached to this problem or its selected resolution.
This record is in the source backlog; its novelty and unresolved status have not passed source review. | null | published | incremental | hard | [
"source-candidate",
"theoremdb"
] | null | null | null | null | [
{
"kind": "website",
"citation": "TheoremDB P8212",
"url": "https://theoremdb.org/statements/P8212",
"note": null,
"sort_order": 0
},
{
"kind": "formalization",
"citation": "Source formal statement (not locally verified)",
"url": "https://theoremdb.org/statements/P8212",
"not... | {
"source_record": "P8212",
"imported_at": "2026-09-08",
"status_as_of": "2026-09-08",
"evidence_label": "Source candidate · not reviewed",
"note": "Full public directory statement. Source lane: backlog; source resolution: open. Source verification claims are attributed only; no local Lean verification has be... | {"created_at":"2026-08-17T18:37:11+00:00","display_standard":{"editorial_review":"submitter_supplied","formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":8212,"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": "2026-09-08T03:32:59.484Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"87aae858-50d3-4ea1-b69d-7138217658cf","format_issues":null,"number":5405} |
1.0.0 | problem:theoremdb-p8214 | problem | theoremdb-p8214 | Equality of finite suprema for a function and its gradient-shifted variant | {
"main": "math",
"sub": "analysis",
"slug": "analysis",
"name": "Analysis",
"original_discipline": "mathematics"
} | [] | [] | Let $n \geq 2$ be an integer. Let $f : \mathbb{R}^n \to \mathbb{R}$ be a continuously differentiable function. Suppose that $f$ is bounded above and that the function $x \mapsto f(x + \nabla f(x))$ is also bounded above. Does the equality
$$
\sup_{x \in \mathbb{R}^n} f(x) = \sup_{x \in \mathbb{R}^n} f(x + \nabla f(x)... | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P8214",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P8214",
"citation": null,
"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_347178.variants.bounded_only :
answer(sorry) ↔ ∀ᵉ (n ≥ 2) (f : ℝ^n → ℝ) (hf : ContDiff ℝ 1 f)
(h : BddAbove (range f)) (h' : BddAbove (range (fun x ↦ f (x + gradient f x)))),
(⨆ x, f x) = ⨆ x, f (x + gradient f x) | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem mathoverflow_347178.variants.bounded_only :\n answer(sorry) ↔ ∀ᵉ (n ≥ 2) (f : ℝ^n → ℝ) (hf : ContDiff ℝ 1 f)\n (h : BddAbove (range f)) (h' : BddAbove (range (fun x ↦ f (x + gradient f x)))),\n (⨆ x, f x) = ⨆ x, f (x + gradient f x)",
"target": n... | [] | TheoremDB P8214 · Candidate awaiting source review | null | Source-reported assessment, retrieved 8 September 2026. Frontier Science has not independently reviewed this status.
Open. No signed Lean verification is attached to this problem or its selected resolution.
This record is in the source backlog; its novelty and unresolved status have not passed source review. | null | published | incremental | hard | [
"source-candidate",
"theoremdb"
] | null | null | null | null | [
{
"kind": "website",
"citation": "TheoremDB P8214",
"url": "https://theoremdb.org/statements/P8214",
"note": null,
"sort_order": 0
},
{
"kind": "formalization",
"citation": "Source formal statement (not locally verified)",
"url": "https://theoremdb.org/statements/P8214",
"not... | {
"source_record": "P8214",
"imported_at": "2026-09-08",
"status_as_of": "2026-09-08",
"evidence_label": "Source candidate · not reviewed",
"note": "Full public directory statement. Source lane: backlog; source resolution: open. Source verification claims are attributed only; no local Lean verification has be... | {"created_at":"2026-08-17T18:37:12+00:00","display_standard":{"formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":8214,"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": "2026-09-08T03:32:54.755Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"b8cdb8fb-5228-4ebe-b1ed-32084491f2d7","format_issues":null,"number":4371} |
1.0.0 | problem:theoremdb-p8216 | problem | theoremdb-p8216 | Exact value of the Landau constant | {
"main": "math",
"sub": "analysis",
"slug": "analysis",
"name": "Analysis",
"original_discipline": "mathematics"
} | [] | [] | Let the Landau constant $L$ be defined as the supremum of all real numbers $B$ such that for every holomorphic function $f : \mathbb{C} \to \mathbb{C}$ on the unit disk $\{z \in \mathbb{C} : |z| < 1\}$ with $f'(0) = 1$, there exists a point $x \in \mathbb{C}$ for which the open disk centered at $x$ with radius $B$ is c... | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P8216",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P8216",
"citation": null,
"doi": null,
"arxiv_id": null,
"license": "Selected packets label their content CC0-1.0; original references retain their own terms",
"notes": "Selected problem... | theorem landauConstant_exact_value :
landauConstant = Real.Gamma (1 / 3) * Real.Gamma (5 / 6) / Real.Gamma (1 / 6) | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem landauConstant_exact_value :\n landauConstant = Real.Gamma (1 / 3) * Real.Gamma (5 / 6) / Real.Gamma (1 / 6)",
"target": null,
"source_url": "https://theoremdb.org/statements/P8216",
"format": "declaration-or-fragment",
"sha256": "25d699f56e8a71006... | [] | TheoremDB P8216 · Candidate awaiting source review | null | Source-reported assessment, retrieved 8 September 2026. Frontier Science has not independently reviewed this status.
Open. No signed Lean verification is attached to this problem or its selected resolution.
This record is in the source backlog; its novelty and unresolved status have not passed source review. | null | published | incremental | hard | [
"source-candidate",
"theoremdb"
] | null | null | null | null | [
{
"kind": "website",
"citation": "TheoremDB P8216",
"url": "https://theoremdb.org/statements/P8216",
"note": null,
"sort_order": 0
},
{
"kind": "formalization",
"citation": "Source formal statement (not locally verified)",
"url": "https://theoremdb.org/statements/P8216",
"not... | {
"source_record": "P8216",
"imported_at": "2026-09-08",
"status_as_of": "2026-09-08",
"evidence_label": "Source candidate · not reviewed",
"note": "Full public directory statement. Source lane: backlog; source resolution: open. Source verification claims are attributed only; no local Lean verification has be... | {"created_at":"2026-08-17T18:37:13+00:00","display_standard":{"editorial_review":"submitter_supplied","formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":8216,"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": "2026-09-08T03:32:58.345Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"1e40dce0-cb54-4c81-8bc3-b1fb75b6457e","format_issues":null,"number":5164} |
1.0.0 | problem:theoremdb-p8218 | problem | theoremdb-p8218 | Lower bound for the subset sum separation threshold | {
"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 $h(n)$ be the largest integer $m$ such that for every set $A$ of $n$ integers, there exists a subset $B \subseteq A$ with $|B| \geq m$ having the following property: if two nonempty subsets of $B$ have equal sums, then they have equal cardinalities. Prove that $\sqrt{n} = O(h(n))$ as ... | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P8218",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P8218",
"citation": null,
"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_789.variants.sq_isBigO :
(fun n : ℕ ↦ √n) =O[atTop] fun n ↦ (subsetSumThreshold n : ℝ) | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem erdos_789.variants.sq_isBigO :\n (fun n : ℕ ↦ √n) =O[atTop] fun n ↦ (subsetSumThreshold n : ℝ)",
"target": null,
"source_url": "https://theoremdb.org/statements/P8218",
"format": "declaration-or-fragment",
"sha256": "057406b1362e6f5d8946cb58e2053ca... | [] | TheoremDB P8218 · Candidate awaiting source review | null | Source-reported assessment, retrieved 8 September 2026. Frontier Science has not independently reviewed this status.
Open. No signed Lean verification is attached to this problem or its selected resolution.
This record is in the source backlog; its novelty and unresolved status have not passed source review. | null | published | incremental | hard | [
"source-candidate",
"theoremdb"
] | null | null | null | null | [
{
"kind": "website",
"citation": "TheoremDB P8218",
"url": "https://theoremdb.org/statements/P8218",
"note": null,
"sort_order": 0
},
{
"kind": "formalization",
"citation": "Source formal statement (not locally verified)",
"url": "https://theoremdb.org/statements/P8218",
"not... | {
"source_record": "P8218",
"imported_at": "2026-09-08",
"status_as_of": "2026-09-08",
"evidence_label": "Source candidate · not reviewed",
"note": "Full public directory statement. Source lane: backlog; source resolution: open. Source verification claims are attributed only; no local Lean verification has be... | {"created_at":"2026-08-17T18:37:15+00:00","display_standard":{"formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":8218,"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": "2026-09-08T03:33:03.509Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"cfe055f7-7fce-4106-8579-e6611e4b8e09","format_issues":null,"number":6110} |
1.0.0 | problem:theoremdb-p8220 | problem | theoremdb-p8220 | Asymptotic bounds for transversals in cyclic Latin squares | {
"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 $z_n$ denote the number of transversals in the Cayley table of the cyclic group $\mathbb{Z}_n$ (that is, the $n \times n$ Latin square whose $(i,j)$-entry is $i + j \pmod{n}$). It is known that $z_n = 0$ for all even $n$. Do there exist real constants $c_1$ and $c_2$ such that $0 < c_... | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P8220",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P8220",
"citation": null,
"doi": null,
"arxiv_id": null,
"license": "Selected packets label their content CC0-1.0; original references retain their own terms",
"notes": "Selected problem... | theorem numTransversalsZn : answer(sorry) ↔
∃ᵉ (c₁ > (0 : ℝ)) (c₂ < (1 : ℝ)) (_ : c₁ < c₂),
∀ n ≥ 3, Odd n →
(z n : ℝ) ∈ Set.Icc (c₁ ^ n * n.factorial) (c₂ ^ n * n.factorial) | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem numTransversalsZn : answer(sorry) ↔\n ∃ᵉ (c₁ > (0 : ℝ)) (c₂ < (1 : ℝ)) (_ : c₁ < c₂),\n ∀ n ≥ 3, Odd n →\n (z n : ℝ) ∈ Set.Icc (c₁ ^ n * n.factorial) (c₂ ^ n * n.factorial)",
"target": null,
"source_url": "https://theoremdb.org/statements/P82... | [] | TheoremDB P8220 · Candidate awaiting source review | null | Source-reported assessment, retrieved 8 September 2026. Frontier Science has not independently reviewed this status.
Open. No signed Lean verification is attached to this problem or its selected resolution.
This record is in the source backlog; its novelty and unresolved status have not passed source review. | null | published | incremental | hard | [
"source-candidate",
"theoremdb"
] | null | null | null | null | [
{
"kind": "website",
"citation": "TheoremDB P8220",
"url": "https://theoremdb.org/statements/P8220",
"note": null,
"sort_order": 0
},
{
"kind": "formalization",
"citation": "Source formal statement (not locally verified)",
"url": "https://theoremdb.org/statements/P8220",
"not... | {
"source_record": "P8220",
"imported_at": "2026-09-08",
"status_as_of": "2026-09-08",
"evidence_label": "Source candidate · not reviewed",
"note": "Full public directory statement. Source lane: backlog; source resolution: open. Source verification claims are attributed only; no local Lean verification has be... | {"created_at":"2026-08-17T18:37:16+00:00","display_standard":{"editorial_review":"submitter_supplied","formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":8220,"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": "2026-09-08T03:32:49.937Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"7ac2c0d4-3c85-4c52-ae86-780a9da626c5","format_issues":null,"number":3317} |
1.0.0 | problem:theoremdb-p8222 | problem | theoremdb-p8222 | Existence of an absolutely maximally entangled state for 12 parties of local dimension 5 | {
"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 integers. Consider a quantum system of $n$ parties, each with a $d$-dimensional Hilbert space $\mathbb{C}^d$. A pure state $|\psi\rangle$ in the tensor product space $(\mathbb{C}^d)^{\otimes n}$ is called absolutely maximally entangled (AME) if, for every subset of at most $\lfloor n/2 ... | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P8222",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P8222",
"citation": null,
"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_5_open :
answer(sorry) ↔ ExistsAME 12 5 | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem ame_12_5_open :\n answer(sorry) ↔ ExistsAME 12 5",
"target": null,
"source_url": "https://theoremdb.org/statements/P8222",
"format": "declaration-or-fragment",
"sha256": "d3e75380c2bf2aeeca993144b18b9cbab48b50a9c2732fe9650648e5d1a4138c",
"licen... | [] | TheoremDB P8222 · Candidate awaiting source review | null | Source-reported assessment, retrieved 8 September 2026. Frontier Science has not independently reviewed this status.
Open. No signed Lean verification is attached to this problem or its selected resolution.
This record is in the source backlog; its novelty and unresolved status have not passed source review. | null | published | incremental | hard | [
"source-candidate",
"theoremdb"
] | null | null | null | null | [
{
"kind": "website",
"citation": "TheoremDB P8222",
"url": "https://theoremdb.org/statements/P8222",
"note": null,
"sort_order": 0
},
{
"kind": "formalization",
"citation": "Source formal statement (not locally verified)",
"url": "https://theoremdb.org/statements/P8222",
"not... | {
"source_record": "P8222",
"imported_at": "2026-09-08",
"status_as_of": "2026-09-08",
"evidence_label": "Source candidate · not reviewed",
"note": "Full public directory statement. Source lane: backlog; source resolution: open. Source verification claims are attributed only; no local Lean verification has be... | {"created_at":"2026-08-17T18:37:17+00:00","display_standard":{"editorial_review":"submitter_supplied","formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":8222,"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": "2026-09-08T03:32:58.612Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"3a264048-2dd9-47d1-ade1-1fda98b38f9d","format_issues":null,"number":5219} |
1.0.0 | problem:theoremdb-p8224 | problem | theoremdb-p8224 | Infinitude of Cluster Primes | {
"main": "math",
"sub": "number theory",
"slug": "number-theory",
"name": "Number theory",
"original_discipline": "mathematics"
} | [] | [
{
"code": "11",
"basis": "community-map",
"evidence": "Inferred from community field: number-theory"
}
] | A prime number $p$ is called a cluster prime if every even natural number $n \le p - 3$ can be written as a difference of two primes $q_1 - q_2$ with $q_1, q_2 \le p$. Are there infinitely many cluster primes? | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P8224",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P8224",
"citation": null,
"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_17 : answer(sorry) ↔ {p : ℕ | IsClusterPrime p}.Infinite | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem erdos_17 : answer(sorry) ↔ {p : ℕ | IsClusterPrime p}.Infinite",
"target": null,
"source_url": "https://theoremdb.org/statements/P8224",
"format": "declaration-or-fragment",
"sha256": "35a069ceb62c0ddd8983a5cee41444f473415194893bb8f1781822cef7eab4ab",... | [] | TheoremDB P8224 · Candidate awaiting source review | null | Source-reported assessment, retrieved 8 September 2026. Frontier Science has not independently reviewed this status.
Open. No signed Lean verification is attached to this problem or its selected resolution.
This record is in the source backlog; its novelty and unresolved status have not passed source review. | null | published | incremental | hard | [
"source-candidate",
"theoremdb"
] | null | null | null | null | [
{
"kind": "website",
"citation": "TheoremDB P8224",
"url": "https://theoremdb.org/statements/P8224",
"note": null,
"sort_order": 0
},
{
"kind": "formalization",
"citation": "Source formal statement (not locally verified)",
"url": "https://theoremdb.org/statements/P8224",
"not... | {
"source_record": "P8224",
"imported_at": "2026-09-08",
"status_as_of": "2026-09-08",
"evidence_label": "Source candidate · not reviewed",
"note": "Full public directory statement. Source lane: backlog; source resolution: open. Source verification claims are attributed only; no local Lean verification has be... | {"created_at":"2026-08-17T18:37:21+00:00","display_standard":{"editorial_review":"submitter_supplied","formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":8224,"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": "2026-09-08T03:33:01.112Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"ebe62fb3-5b92-4b14-978b-bfd9d8734ad6","format_issues":null,"number":5660} |
1.0.0 | problem:theoremdb-p8226 | problem | theoremdb-p8226 | Existence of a universal bound for equidistant points in convex polygons | {
"main": "math",
"sub": "geometry and topology",
"slug": "geometry-and-topology",
"name": "Geometry and topology",
"original_discipline": "mathematics"
} | [] | [] | For a finite set of points $A$ in the plane, we say that $A$ has the $n$-equidistant property if for every point $p \in A$, there exist at least $n$ other points in $A$ that are all at the same positive distance from $p$ (where this distance may depend on $p$). A set $A$ is convex independent if its points are the vert... | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P8226",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P8226",
"citation": null,
"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_97.variants.k_equidistant : answer(sorry) ↔
∃ k : ℕ, ∀ A : Finset ℝ², A.Nonempty → ConvexIndep A → ¬HasNEquidistantProperty k A | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem erdos_97.variants.k_equidistant : answer(sorry) ↔\n ∃ k : ℕ, ∀ A : Finset ℝ², A.Nonempty → ConvexIndep A → ¬HasNEquidistantProperty k A",
"target": null,
"source_url": "https://theoremdb.org/statements/P8226",
"format": "declaration-or-fragment",
"... | [] | TheoremDB P8226 · Candidate awaiting source review | null | Source-reported assessment, retrieved 8 September 2026. Frontier Science has not independently reviewed this status.
Open. No signed Lean verification is attached to this problem or its selected resolution.
This record is in the source backlog; its novelty and unresolved status have not passed source review. | null | published | incremental | hard | [
"source-candidate",
"theoremdb"
] | null | null | null | null | [
{
"kind": "website",
"citation": "TheoremDB P8226",
"url": "https://theoremdb.org/statements/P8226",
"note": null,
"sort_order": 0
},
{
"kind": "formalization",
"citation": "Source formal statement (not locally verified)",
"url": "https://theoremdb.org/statements/P8226",
"not... | {
"source_record": "P8226",
"imported_at": "2026-09-08",
"status_as_of": "2026-09-08",
"evidence_label": "Source candidate · not reviewed",
"note": "Full public directory statement. Source lane: backlog; source resolution: open. Source verification claims are attributed only; no local Lean verification has be... | {"created_at":"2026-08-17T18:37:21+00:00","display_standard":{"editorial_review":"submitter_supplied","formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":8226,"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": "2026-09-08T03:32:58.564Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"9a74104f-64ba-4b22-aab7-2c8d915df03a","format_issues":null,"number":5209} |
1.0.0 | problem:theoremdb-p8228 | problem | theoremdb-p8228 | Erdős's Egyptian fraction gap problem | {
"main": "math",
"sub": "number theory",
"slug": "number-theory",
"name": "Number theory",
"original_discipline": "mathematics"
} | [] | [
{
"code": "11",
"basis": "community-map",
"evidence": "Inferred from community field: number-theory"
}
] | Let $k \geq 2$ be a natural number and let $s : \{0, 1, \ldots, k-1\} \to \mathbb{N}$ be a strictly increasing sequence of natural numbers with $s(0) > 1$. Suppose that $\sum_{i=0}^{k-1} \frac{1}{s(i)} = 1$. The maximum gap of $s$ is defined as $\max_{0 \leq i < k-1} (s(i+1) - s(i))$. Is it true that the maximum gap mu... | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P8228",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P8228",
"citation": null,
"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 : answer(sorry) ↔
∀ (k : ℕ) (hk : 2 ≤ k) (s : Fin k → ℕ),
StrictMono s → 1 < s ⟨0, by omega⟩ →
∑ i : Fin k, 1/ (s i : ℝ) = 1 →
3 ≤ max_gap k s | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem erdos_287 : answer(sorry) ↔\n ∀ (k : ℕ) (hk : 2 ≤ k) (s : Fin k → ℕ),\n StrictMono s → 1 < s ⟨0, by omega⟩ →\n ∑ i : Fin k, 1/ (s i : ℝ) = 1 →\n 3 ≤ max_gap k s",
"target": null,
"source_url": "https://theoremdb.org/statements/P8228",
"format"... | [] | TheoremDB P8228 · Candidate awaiting source review | null | Source-reported assessment, retrieved 8 September 2026. Frontier Science has not independently reviewed this status.
Open. No signed Lean verification is attached to this problem or its selected resolution.
This record is in the source backlog; its novelty and unresolved status have not passed source review. | null | published | incremental | hard | [
"source-candidate",
"theoremdb"
] | null | null | null | null | [
{
"kind": "website",
"citation": "TheoremDB P8228",
"url": "https://theoremdb.org/statements/P8228",
"note": null,
"sort_order": 0
},
{
"kind": "formalization",
"citation": "Source formal statement (not locally verified)",
"url": "https://theoremdb.org/statements/P8228",
"not... | {
"source_record": "P8228",
"imported_at": "2026-09-08",
"status_as_of": "2026-09-08",
"evidence_label": "Source candidate · not reviewed",
"note": "Full public directory statement. Source lane: backlog; source resolution: open. Source verification claims are attributed only; no local Lean verification has be... | {"created_at":"2026-08-17T18:37:24+00:00","display_standard":{"editorial_review":"submitter_supplied","formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":8228,"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": "2026-09-08T03:32:56.398Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"d6420cbf-df0d-47f3-9601-69f62b4da490","format_issues":null,"number":4732} |
1.0.0 | problem:theoremdb-p8230 | problem | theoremdb-p8230 | Existence of 4-critical graphs without critical edges | {
"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 $G$ be a simple graph with vertex set $V$. The chromatic number of $G$, denoted $\chi(G)$, is the smallest number of colors needed to color the vertices of $G$ so that no two adjacent vertices receive the same color. A graph $G$ is $k$-critical if $\chi(G) = k$ and $\chi(G - v) < k$ for every vertex $v \in V$, wher... | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P8230",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P8230",
"citation": null,
"doi": null,
"arxiv_id": null,
"license": "Selected packets label their content CC0-1.0; original references retain their own terms",
"notes": "Selected problem... | theorem erdos_944.variants.dirac_conjecture.k_eq_four :
answer(sorry) ↔ ∃ (V : Type u) (G : SimpleGraph V), G.IsErdos944 4 1 | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem erdos_944.variants.dirac_conjecture.k_eq_four :\n answer(sorry) ↔ ∃ (V : Type u) (G : SimpleGraph V), G.IsErdos944 4 1",
"target": null,
"source_url": "https://theoremdb.org/statements/P8230",
"format": "declaration-or-fragment",
"sha256": "5a39022... | [] | TheoremDB P8230 · Candidate awaiting source review | null | Source-reported assessment, retrieved 8 September 2026. Frontier Science has not independently reviewed this status.
Open. No signed Lean verification is attached to this problem or its selected resolution.
This record is in the source backlog; its novelty and unresolved status have not passed source review. | null | published | incremental | hard | [
"source-candidate",
"theoremdb"
] | null | null | null | null | [
{
"kind": "website",
"citation": "TheoremDB P8230",
"url": "https://theoremdb.org/statements/P8230",
"note": null,
"sort_order": 0
},
{
"kind": "formalization",
"citation": "Source formal statement (not locally verified)",
"url": "https://theoremdb.org/statements/P8230",
"not... | {
"source_record": "P8230",
"imported_at": "2026-09-08",
"status_as_of": "2026-09-08",
"evidence_label": "Source candidate · not reviewed",
"note": "Full public directory statement. Source lane: backlog; source resolution: open. Source verification claims are attributed only; no local Lean verification has be... | {"created_at":"2026-08-17T18:37:24+00:00","display_standard":{"editorial_review":"submitter_supplied","formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":8230,"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": "2026-09-08T03:32:58.397Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"c82bac5c-416d-4e62-86f1-937e74cec35f","format_issues":null,"number":5174} |
1.0.0 | problem:theoremdb-p8232 | problem | theoremdb-p8232 | Equality of Iterated Sum-of-Divisors 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 $\sigma(n)$ denote the sum of all positive divisors of a positive integer $n$. For integers $k \geq 1$, define the iterated function $\sigma_k(n)$ recursively by $\sigma_1(n) = \sigma(n)$ and $\sigma_k(n) = \sigma(\sigma_{k-1}(n))$ for $k \geq 2$. Is it true that for every pair of integers $m \geq 2$ and $n \geq 2$... | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P8232",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P8232",
"citation": null,
"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_412 : answer(sorry) ↔ ∀ᵉ (m ≥ 2) (n ≥ 2), ∃ i j, (σ 1)^[i] m = (σ 1)^[j] n | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem erdos_412 : answer(sorry) ↔ ∀ᵉ (m ≥ 2) (n ≥ 2), ∃ i j, (σ 1)^[i] m = (σ 1)^[j] n",
"target": null,
"source_url": "https://theoremdb.org/statements/P8232",
"format": "declaration-or-fragment",
"sha256": "5080e9f267ad85f394f0b7516c5a5c1e917e1fb5d94103f8... | [] | TheoremDB P8232 · Candidate awaiting source review | null | Source-reported assessment, retrieved 8 September 2026. Frontier Science has not independently reviewed this status.
Open. No signed Lean verification is attached to this problem or its selected resolution.
This record is in the source backlog; its novelty and unresolved status have not passed source review. | null | published | incremental | hard | [
"source-candidate",
"theoremdb"
] | null | null | null | null | [
{
"kind": "website",
"citation": "TheoremDB P8232",
"url": "https://theoremdb.org/statements/P8232",
"note": null,
"sort_order": 0
},
{
"kind": "formalization",
"citation": "Source formal statement (not locally verified)",
"url": "https://theoremdb.org/statements/P8232",
"not... | {
"source_record": "P8232",
"imported_at": "2026-09-08",
"status_as_of": "2026-09-08",
"evidence_label": "Source candidate · not reviewed",
"note": "Full public directory statement. Source lane: backlog; source resolution: open. Source verification claims are attributed only; no local Lean verification has be... | {"created_at":"2026-08-17T18:37:26+00:00","display_standard":{"formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":8232,"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": "2026-09-08T03:32:54.763Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"59a8e8ac-4b5d-49e4-b5e8-4ce217523bf5","format_issues":null,"number":4373} |
1.0.0 | problem:theoremdb-p8234 | problem | theoremdb-p8234 | Erdős Problem 1041 on Short Paths Connecting Roots of Polynomials | {
"main": "math",
"sub": "mathematics general",
"slug": "mathematics-general",
"name": "Mathematics · general",
"original_discipline": "mathematics"
} | [] | [] | Let $n \geq 2$ be a natural number and let $f \in \mathbb{C}[X]$ be a monic polynomial of degree $n$ whose roots all lie in the open unit disk $\{z \in \mathbb{C} : |z| < 1\}$. For a subset $s \subseteq \mathbb{C}$, define the \emph{length} of $s$ to be its $1$-dimensional Hausdorff measure $\mathcal{H}^1(s)$. Must the... | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P8234",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P8234",
"citation": null,
"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_1041 :
∃ (z₁ z₂ : ℂ) (h : ({z₁, z₂} : Multiset ℂ) ≤ f.roots) (γ : Path z₁ z₂),
Set.range γ ⊆ { z : ℂ | ‖f.eval z‖ < 1 } ∧ length (Set.range γ) < 2 | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem erdos_1041 :\n ∃ (z₁ z₂ : ℂ) (h : ({z₁, z₂} : Multiset ℂ) ≤ f.roots) (γ : Path z₁ z₂),\n Set.range γ ⊆ { z : ℂ | ‖f.eval z‖ < 1 } ∧ length (Set.range γ) < 2",
"target": null,
"source_url": "https://theoremdb.org/statements/P8234",
"format": "decla... | [] | TheoremDB P8234 · Candidate awaiting source review | null | Source-reported assessment, retrieved 8 September 2026. Frontier Science has not independently reviewed this status.
Open. No signed Lean verification is attached to this problem or its selected resolution.
This record is in the source backlog; its novelty and unresolved status have not passed source review. | null | published | incremental | hard | [
"source-candidate",
"theoremdb"
] | null | null | null | null | [
{
"kind": "website",
"citation": "TheoremDB P8234",
"url": "https://theoremdb.org/statements/P8234",
"note": null,
"sort_order": 0
},
{
"kind": "formalization",
"citation": "Source formal statement (not locally verified)",
"url": "https://theoremdb.org/statements/P8234",
"not... | {
"source_record": "P8234",
"imported_at": "2026-09-08",
"status_as_of": "2026-09-08",
"evidence_label": "Source candidate · not reviewed",
"note": "Full public directory statement. Source lane: backlog; source resolution: open. Source verification claims are attributed only; no local Lean verification has be... | {"created_at":"2026-08-17T18:37:29+00:00","display_standard":{"formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":8234,"public_status":{"evidence":{"hasFormalization":false,"hasInformalProof":false,"hasRecordedWork":false,"solutionRecord":null},"lean":{"kernelAccepted":false,"label":"","reaso... | [
"formalization-not-compiled-during-export",
"current-status-unassessed",
"source-full-text-not-present"
] | [] | {
"created_at": "2026-09-08T03:32:15.935Z",
"updated_at": "2026-09-08T03:32:15.935Z",
"published_at": "2026-09-08T03:32:54.881Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"b2462cb7-dd44-4e7f-aa7b-8c4bee6682e9","format_issues":null,"number":4399} |
1.0.0 | problem:theoremdb-p8236 | problem | theoremdb-p8236 | Fuglede's conjecture in two dimensions | {
"main": "math",
"sub": "analysis",
"slug": "analysis",
"name": "Analysis",
"original_discipline": "mathematics"
} | [] | [] | Let $\Omega$ be a bounded subset of $\mathbb{R}^2$ with positive Lebesgue measure. We say that $\Omega$ is spectral if there exists a subset $\Lambda \subset \mathbb{R}^2$ such that the set of exponential functions $\{e^{2\pi i \langle \lambda, \cdot \rangle} : \lambda \in \Lambda\}$ forms an orthogonal basis for $L^2(... | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P8236",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P8236",
"citation": null,
"doi": null,
"arxiv_id": null,
"license": "Selected packets label their content CC0-1.0; original references retain their own terms",
"notes": "Selected problem... | theorem FugledeConjecture.variants.dim_2 :
answer(sorry) ↔ FugledeConjectureFor 2 | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem FugledeConjecture.variants.dim_2 :\n answer(sorry) ↔ FugledeConjectureFor 2",
"target": null,
"source_url": "https://theoremdb.org/statements/P8236",
"format": "declaration-or-fragment",
"sha256": "73dbe7039421b3df1d1541e8734fa5ccbf8a5d88e56164f138... | [] | TheoremDB P8236 · Candidate awaiting source review | null | Source-reported assessment, retrieved 8 September 2026. Frontier Science has not independently reviewed this status.
Open. No signed Lean verification is attached to this problem or its selected resolution.
This record is in the source backlog; its novelty and unresolved status have not passed source review. | null | published | incremental | hard | [
"source-candidate",
"theoremdb"
] | null | null | null | null | [
{
"kind": "website",
"citation": "TheoremDB P8236",
"url": "https://theoremdb.org/statements/P8236",
"note": null,
"sort_order": 0
},
{
"kind": "formalization",
"citation": "Source formal statement (not locally verified)",
"url": "https://theoremdb.org/statements/P8236",
"not... | {
"source_record": "P8236",
"imported_at": "2026-09-08",
"status_as_of": "2026-09-08",
"evidence_label": "Source candidate · not reviewed",
"note": "Full public directory statement. Source lane: backlog; source resolution: open. Source verification claims are attributed only; no local Lean verification has be... | {"created_at":"2026-08-17T18:37:29+00:00","display_standard":{"editorial_review":"submitter_supplied","formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":8236,"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": "2026-09-08T03:32:59.229Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"190d0422-a2ed-4e34-bcf0-3172b5b161ec","format_issues":null,"number":5351} |
1.0.0 | problem:theoremdb-p8238 | problem | theoremdb-p8238 | The Littlewood Conjecture on Simultaneous Diophantine Approximation | {
"main": "math",
"sub": "number theory",
"slug": "number-theory",
"name": "Number theory",
"original_discipline": "mathematics"
} | [] | [
{
"code": "11",
"basis": "community-map",
"evidence": "Inferred from community field: number-theory"
}
] | For a real number $x$, let $\|x\|$ denote the distance from $x$ to the nearest integer, that is, $\|x\| = \min_{m \in \mathbb{Z}} |x - m|$. Does there exist, for every pair of real numbers $\alpha$ and $\beta$, a sequence of natural numbers $n$ tending to infinity such that $n \|n\alpha\| \|n\beta\|$ tends to $0$? Equi... | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P8238",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P8238",
"citation": null,
"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_495 : answer(sorry) ↔ ∀ α β : ℝ, liminf (fun n : ℕ ↦ (n : ℝ) * distToNearestInt (n * α)
* distToNearestInt (n * β)) atTop = 0 | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem erdos_495 : answer(sorry) ↔ ∀ α β : ℝ, liminf (fun n : ℕ ↦ (n : ℝ) * distToNearestInt (n * α)\n * distToNearestInt (n * β)) atTop = 0",
"target": null,
"source_url": "https://theoremdb.org/statements/P8238",
"format": "declaration-or-fragment",
"sha2... | [] | TheoremDB P8238 · Candidate awaiting source review | null | Source-reported assessment, retrieved 8 September 2026. Frontier Science has not independently reviewed this status.
Open. No signed Lean verification is attached to this problem or its selected resolution.
This record is in the source backlog; its novelty and unresolved status have not passed source review. | null | published | incremental | hard | [
"source-candidate",
"theoremdb"
] | null | null | null | null | [
{
"kind": "website",
"citation": "TheoremDB P8238",
"url": "https://theoremdb.org/statements/P8238",
"note": null,
"sort_order": 0
},
{
"kind": "formalization",
"citation": "Source formal statement (not locally verified)",
"url": "https://theoremdb.org/statements/P8238",
"not... | {
"source_record": "P8238",
"imported_at": "2026-09-08",
"status_as_of": "2026-09-08",
"evidence_label": "Source candidate · not reviewed",
"note": "Full public directory statement. Source lane: backlog; source resolution: open. Source verification claims are attributed only; no local Lean verification has be... | {"created_at":"2026-08-17T18:37:34+00:00","display_standard":{"formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":8238,"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": "2026-09-08T03:33:06.631Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"c3e70c86-c344-455a-82a6-376d2239eec4","format_issues":null,"number":6784} |
1.0.0 | problem:theoremdb-p8240 | problem | theoremdb-p8240 | Erdős Problem 680, Part (i) | {
"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(m)$ denote the least prime factor of a positive integer $m$. Is it true that for all sufficiently large positive integers $n$, there exists a nonzero integer $k$ such that $p(n+k) > k^2 + 1$? | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P8240",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P8240",
"citation": null,
"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_680.parts.i :
answer(sorry) ↔ ∀ᶠ (n : ℕ) in .atTop, ∃ k ≠ 0, (n + k).minFac > k^2 + 1 | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem erdos_680.parts.i :\n answer(sorry) ↔ ∀ᶠ (n : ℕ) in .atTop, ∃ k ≠ 0, (n + k).minFac > k^2 + 1",
"target": null,
"source_url": "https://theoremdb.org/statements/P8240",
"format": "declaration-or-fragment",
"sha256": "3a75e2fa0bce985e6466d42c5e1aa9e0... | [] | TheoremDB P8240 · Candidate awaiting source review | null | Source-reported assessment, retrieved 8 September 2026. Frontier Science has not independently reviewed this status.
Open. No signed Lean verification is attached to this problem or its selected resolution.
This record is in the source backlog; its novelty and unresolved status have not passed source review. | null | published | incremental | hard | [
"source-candidate",
"theoremdb"
] | null | null | null | null | [
{
"kind": "website",
"citation": "TheoremDB P8240",
"url": "https://theoremdb.org/statements/P8240",
"note": null,
"sort_order": 0
},
{
"kind": "formalization",
"citation": "Source formal statement (not locally verified)",
"url": "https://theoremdb.org/statements/P8240",
"not... | {
"source_record": "P8240",
"imported_at": "2026-09-08",
"status_as_of": "2026-09-08",
"evidence_label": "Source candidate · not reviewed",
"note": "Full public directory statement. Source lane: backlog; source resolution: open. Source verification claims are attributed only; no local Lean verification has be... | {"created_at":"2026-08-17T18:37:36+00:00","display_standard":{"formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":8240,"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": "2026-09-08T03:32:55.883Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"5b530925-232d-456d-864c-1335adb8ff9b","format_issues":null,"number":4618} |
1.0.0 | problem:theoremdb-p8242 | problem | theoremdb-p8242 | Cofinite 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, constructed by beginning 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$ with $a, b, c, d \in A$). Consider the differ... | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P8242",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P8242",
"citation": null,
"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.cofinite_sub : answer(sorry) ↔
∀ᶠ n in cofinite, n ∈ Set.range greedySidon - Set.range greedySidon | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem erdos_340.variants.cofinite_sub : answer(sorry) ↔\n ∀ᶠ n in cofinite, n ∈ Set.range greedySidon - Set.range greedySidon",
"target": null,
"source_url": "https://theoremdb.org/statements/P8242",
"format": "declaration-or-fragment",
"sha256": "57ed2c... | [] | TheoremDB P8242 · Candidate awaiting source review | null | Source-reported assessment, retrieved 8 September 2026. Frontier Science has not independently reviewed this status.
Open. No signed Lean verification is attached to this problem or its selected resolution.
This record is in the source backlog; its novelty and unresolved status have not passed source review. | null | published | incremental | hard | [
"source-candidate",
"theoremdb"
] | null | null | null | null | [
{
"kind": "website",
"citation": "TheoremDB P8242",
"url": "https://theoremdb.org/statements/P8242",
"note": null,
"sort_order": 0
},
{
"kind": "formalization",
"citation": "Source formal statement (not locally verified)",
"url": "https://theoremdb.org/statements/P8242",
"not... | {
"source_record": "P8242",
"imported_at": "2026-09-08",
"status_as_of": "2026-09-08",
"evidence_label": "Source candidate · not reviewed",
"note": "Full public directory statement. Source lane: backlog; source resolution: open. Source verification claims are attributed only; no local Lean verification has be... | {"created_at":"2026-08-17T18:37:37+00:00","display_standard":{"editorial_review":"submitter_supplied","formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":8242,"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": "2026-09-08T03:32:52.406Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"e96449cb-b50b-4c73-a8e4-4c19f75d2814","format_issues":null,"number":3854} |
1.0.0 | problem:theoremdb-p8244 | problem | theoremdb-p8244 | Tao's optimization constant 1a | {
"main": "math",
"sub": "combinatorics",
"slug": "combinatorics",
"name": "Combinatorics",
"original_discipline": "mathematics"
} | [] | [
{
"code": "05",
"basis": "community-map",
"evidence": "Inferred from community field: combinatorics"
}
] | Let $C_{1a}$ be the supremum of all real numbers $C$ such that for every nonnegative function $f: \mathbb{R} \to \mathbb{R}$, the inequality
$$
C \cdot \left(\int_{-1/4}^{1/4} f(x)\, dx\right)^2 \leq \sup_{t \in [1/2, 1]} \int_{-\infty}^{\infty} f(t-x) \cdot f(x)\, dx
$$
holds. It is known that $1.2748 \leq C_{1a} \l... | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P8244",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P8244",
"citation": null,
"doi": null,
"arxiv_id": null,
"license": "Selected packets label their content CC0-1.0; original references retain their own terms",
"notes": "Selected problem... | theorem c1a_eq : C1a = answer(sorry) | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem c1a_eq : C1a = answer(sorry)",
"target": null,
"source_url": "https://theoremdb.org/statements/P8244",
"format": "declaration-or-fragment",
"sha256": "5d54177c5522ec34c8088e769f3172daac2ac17af58247ab9e76859ad6e8655a",
"license": "Selected packets ... | [] | TheoremDB P8244 · Candidate awaiting source review | null | Source-reported assessment, retrieved 8 September 2026. Frontier Science has not independently reviewed this status.
Open. No signed Lean verification is attached to this problem or its selected resolution.
This record is in the source backlog; its novelty and unresolved status have not passed source review. | null | published | incremental | hard | [
"source-candidate",
"theoremdb"
] | null | null | null | null | [
{
"kind": "website",
"citation": "TheoremDB P8244",
"url": "https://theoremdb.org/statements/P8244",
"note": null,
"sort_order": 0
},
{
"kind": "formalization",
"citation": "Source formal statement (not locally verified)",
"url": "https://theoremdb.org/statements/P8244",
"not... | {
"source_record": "P8244",
"imported_at": "2026-09-08",
"status_as_of": "2026-09-08",
"evidence_label": "Source candidate · not reviewed",
"note": "Full public directory statement. Source lane: backlog; source resolution: open. Source verification claims are attributed only; no local Lean verification has be... | {"created_at":"2026-08-17T18:37:39+00:00","display_standard":{"formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":8244,"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": "2026-09-08T03:33:06.273Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"ea108df3-6d12-44a6-842b-a4c81d55aacc","format_issues":null,"number":6708} |
1.0.0 | problem:theoremdb-p8246 | problem | theoremdb-p8246 | Sufficiency of two-colorability for obligatory 3-uniform hypergraphs | {
"main": "math",
"sub": "combinatorics",
"slug": "combinatorics",
"name": "Combinatorics",
"original_discipline": "mathematics"
} | [] | [
{
"code": "05",
"basis": "community-map",
"evidence": "Inferred from community field: combinatorics"
}
] | A 3-uniform hypergraph on a vertex set $V$ is a collection of hyperedges, each containing exactly 3 vertices. A hypergraph is 2-colorable (or has Property B) if its vertices can be colored with two colors such that no hyperedge is monochromatic. The chromatic cardinal of a hypergraph is the smallest cardinal $\kappa$ s... | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P8246",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P8246",
"citation": null,
"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_593.variants.two_colorable_implies_obligatory : answer(sorry) ↔
∀ (W : Type) [Fintype W] (F : ThreeUniformHypergraph W),
F.IsTwoColorable → IsObligatory F | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem erdos_593.variants.two_colorable_implies_obligatory : answer(sorry) ↔\n ∀ (W : Type) [Fintype W] (F : ThreeUniformHypergraph W),\n F.IsTwoColorable → IsObligatory F",
"target": null,
"source_url": "https://theoremdb.org/statements/P8246",
"format"... | [] | TheoremDB P8246 · Candidate awaiting source review | null | Source-reported assessment, retrieved 8 September 2026. Frontier Science has not independently reviewed this status.
Open. No signed Lean verification is attached to this problem or its selected resolution.
This record is in the source backlog; its novelty and unresolved status have not passed source review. | null | published | incremental | hard | [
"source-candidate",
"theoremdb"
] | null | null | null | null | [
{
"kind": "website",
"citation": "TheoremDB P8246",
"url": "https://theoremdb.org/statements/P8246",
"note": null,
"sort_order": 0
},
{
"kind": "formalization",
"citation": "Source formal statement (not locally verified)",
"url": "https://theoremdb.org/statements/P8246",
"not... | {
"source_record": "P8246",
"imported_at": "2026-09-08",
"status_as_of": "2026-09-08",
"evidence_label": "Source candidate · not reviewed",
"note": "Full public directory statement. Source lane: backlog; source resolution: open. Source verification claims are attributed only; no local Lean verification has be... | {"created_at":"2026-08-17T18:37:40+00:00","display_standard":{"editorial_review":"submitter_supplied","formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":8246,"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": "2026-09-08T03:33:06.159Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"2f919a54-4dda-43b2-8167-610eb0e5e248","format_issues":null,"number":6683} |
1.0.0 | problem:theoremdb-p8248 | problem | theoremdb-p8248 | Upper bound on the independence number of a connected graph | {
"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 on at least two vertices such that its complement $\overline{G}$ is also connected. For each vertex $v$ of $G$, let $l(v)$ denote the independence number of the subgraph induced by the neighborhood of $v$ in $G$. Let $\lambda = \max_{v \in V(G)} l(v)$. Let $\mathrm{diam}(\over... | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P8248",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P8248",
"citation": null,
"doi": null,
"arxiv_id": null,
"license": "Selected packets label their content CC0-1.0; original references retain their own terms",
"notes": "Selected problem... | theorem conjecture100 (G : SimpleGraph α) [DecidableRel G.Adj] (h : G.Connected) :
let maxL := (Finset.univ.image (indepNeighborsCard G)).max' (by simp)
(G.indepNum : ℝ) ≤ ⌈((maxL : ℝ) + (1 / 2) * (degreeL2Norm Gᶜ : ℝ)) / 2⌉ | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem conjecture100 (G : SimpleGraph α) [DecidableRel G.Adj] (h : G.Connected) :\n let maxL := (Finset.univ.image (indepNeighborsCard G)).max' (by simp)\n (G.indepNum : ℝ) ≤ ⌈((maxL : ℝ) + (1 / 2) * (degreeL2Norm Gᶜ : ℝ)) / 2⌉",
"target": null,
"source_url": ... | [] | TheoremDB P8248 · Candidate awaiting source review | null | Source-reported assessment, retrieved 8 September 2026. Frontier Science has not independently reviewed this status.
Open. No signed Lean verification is attached to this problem or its selected resolution.
This record is in the source backlog; its novelty and unresolved status have not passed source review. | null | published | incremental | hard | [
"source-candidate",
"theoremdb"
] | null | null | null | null | [
{
"kind": "website",
"citation": "TheoremDB P8248",
"url": "https://theoremdb.org/statements/P8248",
"note": null,
"sort_order": 0
},
{
"kind": "formalization",
"citation": "Source formal statement (not locally verified)",
"url": "https://theoremdb.org/statements/P8248",
"not... | {
"source_record": "P8248",
"imported_at": "2026-09-08",
"status_as_of": "2026-09-08",
"evidence_label": "Source candidate · not reviewed",
"note": "Full public directory statement. Source lane: backlog; source resolution: open. Source verification claims are attributed only; no local Lean verification has be... | {"created_at":"2026-08-17T18:37:42+00:00","display_standard":{"editorial_review":"submitter_supplied","formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":8248,"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": "2026-09-08T03:33:07.380Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"3269e242-5816-4da1-ab99-3a973b8a0edc","format_issues":null,"number":6947} |
1.0.0 | problem:theoremdb-p8250 | problem | theoremdb-p8250 | Erdős's conjecture on non-squarefull prime divisors of consecutive integer products | {
"main": "math",
"sub": "number theory",
"slug": "number-theory",
"name": "Number theory",
"original_discipline": "mathematics"
} | [] | [
{
"code": "11",
"basis": "community-map",
"evidence": "Inferred from community field: number-theory"
}
] | For each fixed positive integer $k$, determine whether the following holds: for all sufficiently large positive integers $n$, and for every positive integer $m$, if $N$ denotes the product $m(m+1)(m+2)\cdots(m+n)$, then there exist at least $k$ distinct prime numbers $p$ such that $p$ divides $N$ but $p^2$ does not div... | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P8250",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P8250",
"citation": null,
"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_137.variants.multiple_powerful_factors (k : ℕ) : ∀ᶠ n in Filter.atTop,
∀ (m : ℕ) (hm : 0 < m),
letI N := ∏ x ∈ Finset.Ioc m (m + n), x
∃ P : Finset ℕ, P.card = k ∧ ∀ p ∈ P, p.Prime ∧
p ∣ N ∧ ¬ p ^ 2 ∣ N | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem erdos_137.variants.multiple_powerful_factors (k : ℕ) : ∀ᶠ n in Filter.atTop,\n ∀ (m : ℕ) (hm : 0 < m),\n letI N := ∏ x ∈ Finset.Ioc m (m + n), x\n ∃ P : Finset ℕ, P.card = k ∧ ∀ p ∈ P, p.Prime ∧\n p ∣ N ∧ ¬ p ^ 2 ∣ N",
"target": null,
"source_url"... | [] | TheoremDB P8250 · Candidate awaiting source review | null | Source-reported assessment, retrieved 8 September 2026. Frontier Science has not independently reviewed this status.
Open. No signed Lean verification is attached to this problem or its selected resolution.
This record is in the source backlog; its novelty and unresolved status have not passed source review. | null | published | incremental | hard | [
"source-candidate",
"theoremdb"
] | null | null | null | null | [
{
"kind": "website",
"citation": "TheoremDB P8250",
"url": "https://theoremdb.org/statements/P8250",
"note": null,
"sort_order": 0
},
{
"kind": "formalization",
"citation": "Source formal statement (not locally verified)",
"url": "https://theoremdb.org/statements/P8250",
"not... | {
"source_record": "P8250",
"imported_at": "2026-09-08",
"status_as_of": "2026-09-08",
"evidence_label": "Source candidate · not reviewed",
"note": "Full public directory statement. Source lane: backlog; source resolution: open. Source verification claims are attributed only; no local Lean verification has be... | {"created_at":"2026-08-17T18:37:43+00:00","display_standard":{"editorial_review":"submitter_supplied","formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":8250,"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": "2026-09-08T03:32:56.343Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"db5f470b-e94c-4de4-b07d-5b9d750f0494","format_issues":null,"number":4720} |
1.0.0 | problem:theoremdb-p8252 | problem | theoremdb-p8252 | Non-existence of real 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 integer. An edge of the complete graph $K_N$ is an unordered pair of distinct vertices, identified with a two-element subset of $\{1,\dots,N\}$. A weight function on $K_N$ with coefficients in a ring $\alpha$ and $D$ colours is a function $W$ that assigns to every edge $e$ and every colour $c\in\{... | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P8252",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P8252",
"citation": null,
"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_real :
answer(sorry) ↔
¬ ∃ W : WeightsN 10 3 ℝ, EqSystemN 10 3 W | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem eqSystem10_no_solution_d3_real :\n answer(sorry) ↔\n ¬ ∃ W : WeightsN 10 3 ℝ, EqSystemN 10 3 W",
"target": null,
"source_url": "https://theoremdb.org/statements/P8252",
"format": "declaration-or-fragment",
"sha256": "cb13c079c52157e7b42ce7b0ab... | [] | TheoremDB P8252 · Candidate awaiting source review | null | Source-reported assessment, retrieved 8 September 2026. Frontier Science has not independently reviewed this status.
Open. No signed Lean verification is attached to this problem or its selected resolution.
This record is in the source backlog; its novelty and unresolved status have not passed source review. | null | published | incremental | hard | [
"source-candidate",
"theoremdb"
] | null | null | null | null | [
{
"kind": "website",
"citation": "TheoremDB P8252",
"url": "https://theoremdb.org/statements/P8252",
"note": null,
"sort_order": 0
},
{
"kind": "formalization",
"citation": "Source formal statement (not locally verified)",
"url": "https://theoremdb.org/statements/P8252",
"not... | {
"source_record": "P8252",
"imported_at": "2026-09-08",
"status_as_of": "2026-09-08",
"evidence_label": "Source candidate · not reviewed",
"note": "Full public directory statement. Source lane: backlog; source resolution: open. Source verification claims are attributed only; no local Lean verification has be... | {"created_at":"2026-08-17T18:37:45+00:00","display_standard":{"editorial_review":"submitter_supplied","formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":8252,"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": "2026-09-08T03:33:04.424Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"cc636dca-9b7b-4e9d-b458-7f9924aadb63","format_issues":null,"number":6307} |
1.0.0 | problem:theoremdb-p8254 | problem | theoremdb-p8254 | Exponential lower bound for the largest prime factor of a binomial coefficient | {
"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,k) $ denote the largest prime factor of the binomial coefficient $ \binom{n}{k} $. Does there exist a constant $ c > 0 $ such that $ P(n,k) > e^{c\sqrt{k}} $ for all natural numbers $ n $ and $ k $ with $ 0 < k \leq n/2 $? | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P8254",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P8254",
"citation": null,
"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_683.variant.exp_sqrt :
∃ c > 0, ∀ n k : ℕ, 0 < k ∧ k ≤ n / 2 → P n k > Real.exp (c * Real.sqrt k) | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem erdos_683.variant.exp_sqrt :\n ∃ c > 0, ∀ n k : ℕ, 0 < k ∧ k ≤ n / 2 → P n k > Real.exp (c * Real.sqrt k)",
"target": null,
"source_url": "https://theoremdb.org/statements/P8254",
"format": "declaration-or-fragment",
"sha256": "ea59c2a42b311bfa9736... | [] | TheoremDB P8254 · Candidate awaiting source review | null | Source-reported assessment, retrieved 8 September 2026. Frontier Science has not independently reviewed this status.
Open. No signed Lean verification is attached to this problem or its selected resolution.
This record is in the source backlog; its novelty and unresolved status have not passed source review. | null | published | incremental | hard | [
"source-candidate",
"theoremdb"
] | null | null | null | null | [
{
"kind": "website",
"citation": "TheoremDB P8254",
"url": "https://theoremdb.org/statements/P8254",
"note": null,
"sort_order": 0
},
{
"kind": "formalization",
"citation": "Source formal statement (not locally verified)",
"url": "https://theoremdb.org/statements/P8254",
"not... | {
"source_record": "P8254",
"imported_at": "2026-09-08",
"status_as_of": "2026-09-08",
"evidence_label": "Source candidate · not reviewed",
"note": "Full public directory statement. Source lane: backlog; source resolution: open. Source verification claims are attributed only; no local Lean verification has be... | {"created_at":"2026-08-17T18:37:46+00:00","display_standard":{"editorial_review":"submitter_supplied","formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":8254,"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": "2026-09-08T03:32:58.917Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"a9f84de4-a397-4480-a7cc-b5ac00857ba6","format_issues":null,"number":5282} |
1.0.0 | problem:theoremdb-p8256 | problem | theoremdb-p8256 | Discreteness of Hausdorff Toronto Spaces | {
"main": "math",
"sub": "geometry and topology",
"slug": "geometry-and-topology",
"name": "Geometry and topology",
"original_discipline": "mathematics"
} | [] | [] | A topological space $X$ is called a *Toronto space* if for every subset $Y \subseteq X$ with the same cardinality as $X$, the subspace $Y$ is homeomorphic to $X$. A topological space is called *Hausdorff* (or $T_2$) if for any two distinct points, there exist disjoint open sets containing each point. A topological spac... | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P8256",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P8256",
"citation": null,
"doi": null,
"arxiv_id": null,
"license": "Selected packets label their content CC0-1.0; original references retain their own terms",
"notes": "Selected problem... | theorem DiscreteTopology.of_t2_of_torontoSpace [T2Space X] [TorontoSpace X] :
DiscreteTopology X | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem DiscreteTopology.of_t2_of_torontoSpace [T2Space X] [TorontoSpace X] :\n DiscreteTopology X",
"target": null,
"source_url": "https://theoremdb.org/statements/P8256",
"format": "declaration-or-fragment",
"sha256": "c2caa4c583fa969432c91bb8a687518ba13... | [] | TheoremDB P8256 · Candidate awaiting source review | null | Source-reported assessment, retrieved 8 September 2026. Frontier Science has not independently reviewed this status.
Open. No signed Lean verification is attached to this problem or its selected resolution.
This record is in the source backlog; its novelty and unresolved status have not passed source review. | null | published | incremental | hard | [
"source-candidate",
"theoremdb"
] | null | null | null | null | [
{
"kind": "website",
"citation": "TheoremDB P8256",
"url": "https://theoremdb.org/statements/P8256",
"note": null,
"sort_order": 0
},
{
"kind": "formalization",
"citation": "Source formal statement (not locally verified)",
"url": "https://theoremdb.org/statements/P8256",
"not... | {
"source_record": "P8256",
"imported_at": "2026-09-08",
"status_as_of": "2026-09-08",
"evidence_label": "Source candidate · not reviewed",
"note": "Full public directory statement. Source lane: backlog; source resolution: open. Source verification claims are attributed only; no local Lean verification has be... | {"created_at":"2026-08-17T18:37:48+00:00","display_standard":{"editorial_review":"submitter_supplied","formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":8256,"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": "2026-09-08T03:32:53.186Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"36831f72-93a5-4d44-80f1-9846b396a4d5","format_issues":null,"number":4022} |
1.0.0 | problem:theoremdb-p8258 | problem | theoremdb-p8258 | CDS-Colorability of Young Diagram Graphs | {
"main": "math",
"sub": "combinatorics",
"slug": "combinatorics",
"name": "Combinatorics",
"original_discipline": "mathematics"
} | [] | [
{
"code": "05",
"basis": "community-map",
"evidence": "Inferred from community field: combinatorics"
}
] | Let $\mu$ be a finite Young diagram. The graph $G_\mu$ associated to $\mu$ has as its vertex set the cells of $\mu$, with two cells adjacent if and only if they lie in the same row or the same column. A graph is said to be CDS-colorable if there exists a proper vertex coloring such that for every positive integer $k$, ... | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P8258",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P8258",
"citation": null,
"doi": null,
"arxiv_id": null,
"license": "Selected packets label their content CC0-1.0; original references retain their own terms",
"notes": "Selected problem... | theorem LatinTableauConjecture (μ : YoungDiagram) :
μ.toSimpleGraph.CDSColorable | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem LatinTableauConjecture (μ : YoungDiagram) :\n μ.toSimpleGraph.CDSColorable",
"target": null,
"source_url": "https://theoremdb.org/statements/P8258",
"format": "declaration-or-fragment",
"sha256": "0928102533816f35ab4f6b929dee2e16efe1a100dc0f20bc9f... | [] | TheoremDB P8258 · Candidate awaiting source review | null | Source-reported assessment, retrieved 8 September 2026. Frontier Science has not independently reviewed this status.
Open. No signed Lean verification is attached to this problem or its selected resolution.
This record is in the source backlog; its novelty and unresolved status have not passed source review. | null | published | incremental | hard | [
"source-candidate",
"theoremdb"
] | null | null | null | null | [
{
"kind": "website",
"citation": "TheoremDB P8258",
"url": "https://theoremdb.org/statements/P8258",
"note": null,
"sort_order": 0
},
{
"kind": "formalization",
"citation": "Source formal statement (not locally verified)",
"url": "https://theoremdb.org/statements/P8258",
"not... | {
"source_record": "P8258",
"imported_at": "2026-09-08",
"status_as_of": "2026-09-08",
"evidence_label": "Source candidate · not reviewed",
"note": "Full public directory statement. Source lane: backlog; source resolution: open. Source verification claims are attributed only; no local Lean verification has be... | {"created_at":"2026-08-17T18:37:49+00:00","display_standard":{"formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":8258,"public_status":{"evidence":{"hasFormalization":false,"hasInformalProof":false,"hasRecordedWork":false,"solutionRecord":null},"lean":{"kernelAccepted":false,"label":"","reaso... | [
"formalization-not-compiled-during-export",
"current-status-unassessed",
"source-full-text-not-present"
] | [] | {
"created_at": "2026-09-08T03:32:15.935Z",
"updated_at": "2026-09-08T03:32:15.935Z",
"published_at": "2026-09-08T03:32:51.022Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"1a588266-0927-4625-b269-f2ea78793de3","format_issues":null,"number":3544} |
1.0.0 | problem:theoremdb-p8260 | problem | theoremdb-p8260 | Finiteness of solutions to the modified prime-factor gap equation $V_1(n)=1$ | {
"main": "math",
"sub": "number theory",
"slug": "number-theory",
"name": "Number theory",
"original_discipline": "mathematics"
} | [] | [
{
"code": "11",
"basis": "community-map",
"evidence": "Inferred from community field: number-theory"
}
] | For positive integers $n$ and $k$, define $V(n,k)$ to be the number of primes $p$ such that if $p^\alpha$ is the exact power of $p$ dividing $n+k$, then $p^\alpha$ does not divide $n+i$ for any integer $i$ with $0 \le i < k$. For each nonnegative integer $l$ and each positive integer $n$, define $V_l(n) = \sup_{k \ge l... | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P8260",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P8260",
"citation": null,
"doi": null,
"arxiv_id": null,
"license": "Selected packets label their content CC0-1.0; original references retain their own terms",
"notes": "Selected problem... | theorem erdos_889.variants.V1_eq_1_finite :
answer(sorry) ↔ {n | V_l 1 n = 1}.Finite | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem erdos_889.variants.V1_eq_1_finite :\n answer(sorry) ↔ {n | V_l 1 n = 1}.Finite",
"target": null,
"source_url": "https://theoremdb.org/statements/P8260",
"format": "declaration-or-fragment",
"sha256": "bb008121a4f909eb360e09048c995c2c397803f99c08ee8... | [] | TheoremDB P8260 · Candidate awaiting source review | null | Source-reported assessment, retrieved 8 September 2026. Frontier Science has not independently reviewed this status.
Open. No signed Lean verification is attached to this problem or its selected resolution.
This record is in the source backlog; its novelty and unresolved status have not passed source review. | null | published | incremental | hard | [
"source-candidate",
"theoremdb"
] | null | null | null | null | [
{
"kind": "website",
"citation": "TheoremDB P8260",
"url": "https://theoremdb.org/statements/P8260",
"note": null,
"sort_order": 0
},
{
"kind": "formalization",
"citation": "Source formal statement (not locally verified)",
"url": "https://theoremdb.org/statements/P8260",
"not... | {
"source_record": "P8260",
"imported_at": "2026-09-08",
"status_as_of": "2026-09-08",
"evidence_label": "Source candidate · not reviewed",
"note": "Full public directory statement. Source lane: backlog; source resolution: open. Source verification claims are attributed only; no local Lean verification has be... | {"created_at":"2026-08-17T18:37:51+00:00","display_standard":{"editorial_review":"submitter_supplied","formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":8260,"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": "2026-09-08T03:32:59.090Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"48963d7c-40fa-48a7-91c1-9fab671373fe","format_issues":null,"number":5320} |
1.0.0 | problem:theoremdb-p8262 | problem | theoremdb-p8262 | Asymptotic Bound for the Gauss Circle Error Term | {
"main": "math",
"sub": "number theory",
"slug": "number-theory",
"name": "Number theory",
"original_discipline": "mathematics"
} | [] | [
{
"code": "11",
"basis": "community-map",
"evidence": "Inferred from community field: number-theory"
}
] | For a real number $r \geq 0$, let $N(r)$ denote the number of integer lattice points $(m,n) \in \mathbb{Z}^2$ satisfying $m^2 + n^2 \leq r^2$, and let $E(r) = N(r) - \pi r^2$ denote the error term in the Gauss circle problem. Does there exist a function $o: \mathbb{R} \to \mathbb{R}$ such that $\lim_{r \to \infty} o(r)... | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P8262",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P8262",
"citation": null,
"doi": null,
"arxiv_id": null,
"license": "Selected packets label their content CC0-1.0; original references retain their own terms",
"notes": "Selected problem... | theorem error_isBigO : ∃ (o : ℝ → ℝ) (_ : Tendsto o atTop (𝓝 0)),
E =O[atTop] fun r => r ^ (1/2 + o r) | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem error_isBigO : ∃ (o : ℝ → ℝ) (_ : Tendsto o atTop (𝓝 0)),\n E =O[atTop] fun r => r ^ (1/2 + o r)",
"target": null,
"source_url": "https://theoremdb.org/statements/P8262",
"format": "declaration-or-fragment",
"sha256": "9512fc1dce3702ab82549f645401... | [] | TheoremDB P8262 · Candidate awaiting source review | null | Source-reported assessment, retrieved 8 September 2026. Frontier Science has not independently reviewed this status.
Open. No signed Lean verification is attached to this problem or its selected resolution.
This record is in the source backlog; its novelty and unresolved status have not passed source review. | null | published | incremental | hard | [
"source-candidate",
"theoremdb"
] | null | null | null | null | [
{
"kind": "website",
"citation": "TheoremDB P8262",
"url": "https://theoremdb.org/statements/P8262",
"note": null,
"sort_order": 0
},
{
"kind": "formalization",
"citation": "Source formal statement (not locally verified)",
"url": "https://theoremdb.org/statements/P8262",
"not... | {
"source_record": "P8262",
"imported_at": "2026-09-08",
"status_as_of": "2026-09-08",
"evidence_label": "Source candidate · not reviewed",
"note": "Full public directory statement. Source lane: backlog; source resolution: open. Source verification claims are attributed only; no local Lean verification has be... | {"created_at":"2026-08-17T18:37:52+00:00","display_standard":{"editorial_review":"submitter_supplied","formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":8262,"public_status":{"evidence":{"hasFormalization":false,"hasInformalProof":false,"hasRecordedWork":false,"solutionRecord":null},"lean":{... | [
"formalization-not-compiled-during-export",
"current-status-unassessed",
"source-full-text-not-present"
] | [] | {
"created_at": "2026-09-08T03:32:15.935Z",
"updated_at": "2026-09-08T03:32:15.935Z",
"published_at": "2026-09-08T03:32:49.933Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"2a7cb26b-2c3f-4227-a2bd-3c777cc2136c","format_issues":null,"number":3316} |
1.0.0 | problem:theoremdb-p8264 | problem | theoremdb-p8264 | Asymptotic lower bound for the Heilbronn triangle problem | {
"main": "math",
"sub": "geometry and topology",
"slug": "geometry-and-topology",
"name": "Geometry and topology",
"original_discipline": "mathematics"
} | [] | [] | For each positive integer $n$, let $\alpha(n)$ denote the supremum of the minimum triangle area over all sets of $n$ non-collinear points in the closed unit disk in $\mathbb{R}^2$, where the minimum is taken over all triangles formed by three distinct points of the set. Let $L(n) = \frac{\log n}{n^2}$ denote the lower ... | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P8264",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P8264",
"citation": null,
"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_507.lower:
let ans := (answer(sorry) : ℕ → ℝ)
(lowerBest =o[atTop] ans) ∧ (ans ≪ α) | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem erdos_507.lower:\n let ans := (answer(sorry) : ℕ → ℝ)\n (lowerBest =o[atTop] ans) ∧ (ans ≪ α)",
"target": null,
"source_url": "https://theoremdb.org/statements/P8264",
"format": "declaration-or-fragment",
"sha256": "90b177ba67a9170f4b120e48a55bd... | [] | TheoremDB P8264 · Candidate awaiting source review | null | Source-reported assessment, retrieved 8 September 2026. Frontier Science has not independently reviewed this status.
Open. No signed Lean verification is attached to this problem or its selected resolution.
This record is in the source backlog; its novelty and unresolved status have not passed source review. | null | published | incremental | hard | [
"source-candidate",
"theoremdb"
] | null | null | null | null | [
{
"kind": "website",
"citation": "TheoremDB P8264",
"url": "https://theoremdb.org/statements/P8264",
"note": null,
"sort_order": 0
},
{
"kind": "formalization",
"citation": "Source formal statement (not locally verified)",
"url": "https://theoremdb.org/statements/P8264",
"not... | {
"source_record": "P8264",
"imported_at": "2026-09-08",
"status_as_of": "2026-09-08",
"evidence_label": "Source candidate · not reviewed",
"note": "Full public directory statement. Source lane: backlog; source resolution: open. Source verification claims are attributed only; no local Lean verification has be... | {"created_at":"2026-08-17T18:37:54+00:00","display_standard":{"editorial_review":"submitter_supplied","formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":8264,"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": "2026-09-08T03:32:50.323Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"562f0c5d-06b1-45cf-bf70-4ce05659ff53","format_issues":null,"number":3396} |
1.0.0 | problem:theoremdb-p8266 | problem | theoremdb-p8266 | Infinitude of primes $p$ with $8p^2-1$ prime | {
"main": "math",
"sub": "number theory",
"slug": "number-theory",
"name": "Number theory",
"original_discipline": "mathematics"
} | [] | [
{
"code": "11",
"basis": "community-map",
"evidence": "Inferred from community field: number-theory"
}
] | Let $p$ denote a prime number. Does there exist infinitely many primes $p$ such that $8p^2 - 1$ is also prime? | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P8266",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P8266",
"citation": null,
"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_913.variants.infinite_many_8p_sq_add_one_primes :
{ p | p.Prime ∧ (8 * p ^ 2 - 1).Prime }.Infinite | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem erdos_913.variants.infinite_many_8p_sq_add_one_primes :\n { p | p.Prime ∧ (8 * p ^ 2 - 1).Prime }.Infinite",
"target": null,
"source_url": "https://theoremdb.org/statements/P8266",
"format": "declaration-or-fragment",
"sha256": "441cb43eb8252f0fd2e... | [] | TheoremDB P8266 · Candidate awaiting source review | null | Source-reported assessment, retrieved 8 September 2026. Frontier Science has not independently reviewed this status.
Open. No signed Lean verification is attached to this problem or its selected resolution.
This record is in the source backlog; its novelty and unresolved status have not passed source review. | null | published | incremental | hard | [
"source-candidate",
"theoremdb"
] | null | null | null | null | [
{
"kind": "website",
"citation": "TheoremDB P8266",
"url": "https://theoremdb.org/statements/P8266",
"note": null,
"sort_order": 0
},
{
"kind": "formalization",
"citation": "Source formal statement (not locally verified)",
"url": "https://theoremdb.org/statements/P8266",
"not... | {
"source_record": "P8266",
"imported_at": "2026-09-08",
"status_as_of": "2026-09-08",
"evidence_label": "Source candidate · not reviewed",
"note": "Full public directory statement. Source lane: backlog; source resolution: open. Source verification claims are attributed only; no local Lean verification has be... | {"created_at":"2026-08-17T18:37:57+00:00","display_standard":{"editorial_review":"submitter_supplied","formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":8266,"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": "2026-09-08T03:33:01.208Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"bb6dfbc6-66c6-48d1-8b09-1fdcc5322fc0","format_issues":null,"number":5681} |
1.0.0 | problem:theoremdb-p8268 | problem | theoremdb-p8268 | Asymptotic growth of the totient-prime iteration count | {
"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 $c(n)$ to be the minimum nonnegative integer $i$ such that the $i$-fold iteration of the map $f(m) = \varphi(m) + 1$ applied to $n$ yields a prime number, where $\varphi$ denotes Euler's totient function. That is, $c(n)$ is the smallest $i \geq 0$ for which $f^{(i)}(n)$ is prime. D... | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P8268",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P8268",
"citation": null,
"doi": null,
"arxiv_id": null,
"license": "Selected packets label their content CC0-1.0; original references retain their own terms",
"notes": "Selected problem... | theorem erdos_409.parts.i.isLittleO (c : ℕ → ℕ)
(h : ∀ n > 0, IsLeast { i | (φ · + 1)^[i] n |>.Prime } (c n)) :
(fun n => (c n : ℝ)) =o[atTop] (answer(sorry) : ℕ → ℝ) | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem erdos_409.parts.i.isLittleO (c : ℕ → ℕ)\n (h : ∀ n > 0, IsLeast { i | (φ · + 1)^[i] n |>.Prime } (c n)) :\n (fun n => (c n : ℝ)) =o[atTop] (answer(sorry) : ℕ → ℝ)",
"target": null,
"source_url": "https://theoremdb.org/statements/P8268",
"format": "d... | [] | TheoremDB P8268 · Candidate awaiting source review | null | Source-reported assessment, retrieved 8 September 2026. Frontier Science has not independently reviewed this status.
Open. No signed Lean verification is attached to this problem or its selected resolution.
This record is in the source backlog; its novelty and unresolved status have not passed source review. | null | published | incremental | hard | [
"source-candidate",
"theoremdb"
] | null | null | null | null | [
{
"kind": "website",
"citation": "TheoremDB P8268",
"url": "https://theoremdb.org/statements/P8268",
"note": null,
"sort_order": 0
},
{
"kind": "formalization",
"citation": "Source formal statement (not locally verified)",
"url": "https://theoremdb.org/statements/P8268",
"not... | {
"source_record": "P8268",
"imported_at": "2026-09-08",
"status_as_of": "2026-09-08",
"evidence_label": "Source candidate · not reviewed",
"note": "Full public directory statement. Source lane: backlog; source resolution: open. Source verification claims are attributed only; no local Lean verification has be... | {"created_at":"2026-08-17T18:38:02+00:00","display_standard":{"editorial_review":"submitter_supplied","formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":8268,"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": "2026-09-08T03:32:50.268Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"bd282aac-9e36-42c3-af06-4de7c61986a1","format_issues":null,"number":3385} |
1.0.0 | problem:theoremdb-p8270 | problem | theoremdb-p8270 | Erdős's problem on sum-free complements of cardinality continuum | {
"main": "math",
"sub": "combinatorics",
"slug": "combinatorics",
"name": "Combinatorics",
"original_discipline": "mathematics"
} | [] | [
{
"code": "05",
"basis": "community-map",
"evidence": "Inferred from community field: combinatorics"
}
] | Let $\mathfrak{c}$ denote the cardinality of the continuum. A set $S \subseteq \mathbb{R}$ is called sum-free if for all $a \in S$ and $b \in S$, we have $a + b \notin S$. Must every sum-free set $S \subseteq \mathbb{R}$ contain a subset $A \subseteq \mathbb{R} \setminus S$ of cardinality $\mathfrak{c}$ such that $A + ... | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P8270",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P8270",
"citation": null,
"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_949 : answer(sorry) ↔
∀ S : Set ℝ, (∀ a ∈ S, ∀ b ∈ S, a + b ∉ S) → ∃ A ⊆ Sᶜ, #A = 𝔠 ∧ A + A ⊆ Sᶜ | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem erdos_949 : answer(sorry) ↔\n ∀ S : Set ℝ, (∀ a ∈ S, ∀ b ∈ S, a + b ∉ S) → ∃ A ⊆ Sᶜ, #A = 𝔠 ∧ A + A ⊆ Sᶜ",
"target": null,
"source_url": "https://theoremdb.org/statements/P8270",
"format": "declaration-or-fragment",
"sha256": "5dc9175f9985812a3db2... | [] | TheoremDB P8270 · Candidate awaiting source review | null | Source-reported assessment, retrieved 8 September 2026. Frontier Science has not independently reviewed this status.
Open. No signed Lean verification is attached to this problem or its selected resolution.
This record is in the source backlog; its novelty and unresolved status have not passed source review. | null | published | incremental | hard | [
"source-candidate",
"theoremdb"
] | null | null | null | null | [
{
"kind": "website",
"citation": "TheoremDB P8270",
"url": "https://theoremdb.org/statements/P8270",
"note": null,
"sort_order": 0
},
{
"kind": "formalization",
"citation": "Source formal statement (not locally verified)",
"url": "https://theoremdb.org/statements/P8270",
"not... | {
"source_record": "P8270",
"imported_at": "2026-09-08",
"status_as_of": "2026-09-08",
"evidence_label": "Source candidate · not reviewed",
"note": "Full public directory statement. Source lane: backlog; source resolution: open. Source verification claims are attributed only; no local Lean verification has be... | {"created_at":"2026-08-17T18:38:02+00:00","display_standard":{"editorial_review":"submitter_supplied","formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":8270,"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": "2026-09-08T03:32:56.549Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"e886bdff-d383-4fae-a5bd-535865c4f3d7","format_issues":null,"number":4764} |
1.0.0 | problem:theoremdb-p8272 | problem | theoremdb-p8272 | Convergence of a weighted sum over well-separated sets | {
"main": "math",
"sub": "number theory",
"slug": "number-theory",
"name": "Number theory",
"original_discipline": "mathematics"
} | [] | [
{
"code": "11",
"basis": "community-map",
"evidence": "Inferred from community field: number-theory"
}
] | A set $A \subseteq (1, \infty)$ of real numbers is called well-separated if it is countably infinite and satisfies the separation condition: for all distinct $x, y \in A$ and every integer $k \geq 1$, we have $|kx - y| \geq 1$. Let $A$ be a well-separated set. Is the series
$$
\sum_{x \in A} \frac{1}{x \log x}
$$
c... | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P8272",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P8272",
"citation": null,
"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_143.parts.ii (A : Set ℝ) (h : WellSeparatedSet A) :
Summable fun (x : A) ↦ 1 / (x * Real.log x) | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem erdos_143.parts.ii (A : Set ℝ) (h : WellSeparatedSet A) :\n Summable fun (x : A) ↦ 1 / (x * Real.log x)",
"target": null,
"source_url": "https://theoremdb.org/statements/P8272",
"format": "declaration-or-fragment",
"sha256": "18f6bc5ee2e6d114e4f661... | [] | TheoremDB P8272 · Candidate awaiting source review | null | Source-reported assessment, retrieved 8 September 2026. Frontier Science has not independently reviewed this status.
Open. No signed Lean verification is attached to this problem or its selected resolution.
This record is in the source backlog; its novelty and unresolved status have not passed source review. | null | published | incremental | hard | [
"source-candidate",
"theoremdb"
] | null | null | null | null | [
{
"kind": "website",
"citation": "TheoremDB P8272",
"url": "https://theoremdb.org/statements/P8272",
"note": null,
"sort_order": 0
},
{
"kind": "formalization",
"citation": "Source formal statement (not locally verified)",
"url": "https://theoremdb.org/statements/P8272",
"not... | {
"source_record": "P8272",
"imported_at": "2026-09-08",
"status_as_of": "2026-09-08",
"evidence_label": "Source candidate · not reviewed",
"note": "Full public directory statement. Source lane: backlog; source resolution: open. Source verification claims are attributed only; no local Lean verification has be... | {"created_at":"2026-08-17T18:38:06+00:00","display_standard":{"editorial_review":"submitter_supplied","formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":8272,"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": "2026-09-08T03:32:52.650Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"476b99e8-13c1-4979-bf04-9ec4a5d2d153","format_issues":null,"number":3906} |
1.0.0 | problem:theoremdb-p8274 | problem | theoremdb-p8274 | Asymptotic growth of consecutive van der Waerden number quotients | {
"main": "math",
"sub": "number theory",
"slug": "number-theory",
"name": "Number theory",
"original_discipline": "mathematics"
} | [] | [
{
"code": "11",
"basis": "community-map",
"evidence": "Inferred from community field: number-theory"
}
] | For each positive integer $k$, let $W(k)$ denote the van der Waerden number for $2$ colors, defined as the smallest integer $N$ such that every $2$-coloring of $\{1, \ldots, N\}$ contains a monochromatic arithmetic progression of length $k$. Does the sequence of quotients $\frac{W(k+1)}{W(k)}$ tend to infinity as $k \t... | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P8274",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P8274",
"citation": null,
"doi": null,
"arxiv_id": null,
"license": "Selected packets label their content CC0-1.0; original references retain their own terms",
"notes": "Selected problem... | theorem erdos_138.variants.quotient :
answer(sorry) ↔ atTop.Tendsto (fun k => ((W (k + 1) : ℚ)/(W k))) atTop | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem erdos_138.variants.quotient :\n answer(sorry) ↔ atTop.Tendsto (fun k => ((W (k + 1) : ℚ)/(W k))) atTop",
"target": null,
"source_url": "https://theoremdb.org/statements/P8274",
"format": "declaration-or-fragment",
"sha256": "0330fa139526aba4f33259d... | [] | TheoremDB P8274 · Candidate awaiting source review | null | Source-reported assessment, retrieved 8 September 2026. Frontier Science has not independently reviewed this status.
Open. No signed Lean verification is attached to this problem or its selected resolution.
This record is in the source backlog; its novelty and unresolved status have not passed source review. | null | published | incremental | hard | [
"source-candidate",
"theoremdb"
] | null | null | null | null | [
{
"kind": "website",
"citation": "TheoremDB P8274",
"url": "https://theoremdb.org/statements/P8274",
"note": null,
"sort_order": 0
},
{
"kind": "formalization",
"citation": "Source formal statement (not locally verified)",
"url": "https://theoremdb.org/statements/P8274",
"not... | {
"source_record": "P8274",
"imported_at": "2026-09-08",
"status_as_of": "2026-09-08",
"evidence_label": "Source candidate · not reviewed",
"note": "Full public directory statement. Source lane: backlog; source resolution: open. Source verification claims are attributed only; no local Lean verification has be... | {"created_at":"2026-08-17T18:38:06+00:00","display_standard":{"editorial_review":"submitter_supplied","formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":8274,"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": "2026-09-08T03:32:50.070Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"68518d6d-aa2a-40f2-8263-7943124687c2","format_issues":null,"number":3344} |
1.0.0 | problem:theoremdb-p8276 | problem | theoremdb-p8276 | Improving the upper bound on Tao's optimization constant 1a | {
"main": "math",
"sub": "combinatorics",
"slug": "combinatorics",
"name": "Combinatorics",
"original_discipline": "mathematics"
} | [] | [
{
"code": "05",
"basis": "community-map",
"evidence": "Inferred from community field: combinatorics"
}
] | Let $C_{1a}$ denote the supremum of all real numbers $C$ such that for every nonnegative function $f: \mathbb{R} \to \mathbb{R}$, the inequality
$$
C \left(\int_{-1/4}^{1/4} f(x)\, dx\right)^2 \leq \sup_{t \in [1/2, 1]} \int_{-\infty}^{\infty} f(t-x)\, f(x)\, dx
$$
holds. It is known that $1.2748 \leq C_{1a} \leq 1... | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P8276",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P8276",
"citation": null,
"doi": null,
"arxiv_id": null,
"license": "Selected packets label their content CC0-1.0; original references retain their own terms",
"notes": "Selected problem... | theorem mem_Ico_c1a : answer(sorry) ∈ Set.Ico C1a 1.5029 | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem mem_Ico_c1a : answer(sorry) ∈ Set.Ico C1a 1.5029",
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"license"... | [] | TheoremDB P8276 · Candidate awaiting source review | null | Source-reported assessment, retrieved 8 September 2026. Frontier Science has not independently reviewed this status.
Open. No signed Lean verification is attached to this problem or its selected resolution.
This record is in the source backlog; its novelty and unresolved status have not passed source review. | null | published | incremental | hard | [
"source-candidate",
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] | null | null | null | null | [
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"url": "https://theoremdb.org/statements/P8276",
"not... | {
"source_record": "P8276",
"imported_at": "2026-09-08",
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"evidence_label": "Source candidate · not reviewed",
"note": "Full public directory statement. Source lane: backlog; source resolution: open. Source verification claims are attributed only; no local Lean verification has be... | {"created_at":"2026-08-17T18:38:09+00:00","display_standard":{"editorial_review":"submitter_supplied","formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":8276,"public_status":{"evidence":{"hasFormalization":false,"hasInformalProof":false,"hasRecordedWork":false,"solutionRecord":null},"lean":{... | [
"formalization-contains-placeholder",
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"current-status-unassessed",
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] | [] | {
"created_at": "2026-09-08T03:32:15.935Z",
"updated_at": "2026-09-08T03:32:15.935Z",
"published_at": "2026-09-08T03:32:59.690Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"3cedebf5-59ca-4eba-b3d9-71e39fd6de30","format_issues":null,"number":5449} |
1.0.0 | problem:theoremdb-p8278 | problem | theoremdb-p8278 | Asymptotic growth of maximal arithmetic intersection families | {
"main": "math",
"sub": "combinatorics",
"slug": "combinatorics",
"name": "Combinatorics",
"original_discipline": "mathematics"
} | [] | [
{
"code": "05",
"basis": "community-map",
"evidence": "Inferred from community field: combinatorics"
}
] | For each positive integer $N$, define an arithmetic intersection set to be a family $\{A_1, \ldots, A_t\}$ of subsets of $\{1, 2, \ldots, N\}$ such that for every pair of distinct indices $i \neq j$, the intersection $A_i \cap A_j$ is a non-empty arithmetic progression. Let $t(N)$ denote the maximum possible size $t$ o... | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P8278",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P8278",
"citation": null,
"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_272 :
(fun N ↦ (maxArithInterCard N : ℝ)) ~[atTop] (answer(sorry) : ℕ → ℝ) | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem erdos_272 :\n (fun N ↦ (maxArithInterCard N : ℝ)) ~[atTop] (answer(sorry) : ℕ → ℝ)",
"target": null,
"source_url": "https://theoremdb.org/statements/P8278",
"format": "declaration-or-fragment",
"sha256": "f5dde07e940e09dc48caf408e19a6c5ab70b0d95255... | [] | TheoremDB P8278 · Candidate awaiting source review | null | Source-reported assessment, retrieved 8 September 2026. Frontier Science has not independently reviewed this status.
Open. No signed Lean verification is attached to this problem or its selected resolution.
This record is in the source backlog; its novelty and unresolved status have not passed source review. | null | published | incremental | hard | [
"source-candidate",
"theoremdb"
] | null | null | null | null | [
{
"kind": "website",
"citation": "TheoremDB P8278",
"url": "https://theoremdb.org/statements/P8278",
"note": null,
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{
"kind": "formalization",
"citation": "Source formal statement (not locally verified)",
"url": "https://theoremdb.org/statements/P8278",
"not... | {
"source_record": "P8278",
"imported_at": "2026-09-08",
"status_as_of": "2026-09-08",
"evidence_label": "Source candidate · not reviewed",
"note": "Full public directory statement. Source lane: backlog; source resolution: open. Source verification claims are attributed only; no local Lean verification has be... | {"created_at":"2026-08-17T18:38:11+00:00","display_standard":{"editorial_review":"submitter_supplied","formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":8278,"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": "2026-09-08T03:32:50.103Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"a561f94f-6068-4e13-85c9-991e02370e15","format_issues":null,"number":3351} |
1.0.0 | problem:theoremdb-p8280 | problem | theoremdb-p8280 | Additive completeness of floor sequences with parameters in the interval $(1, \frac{1+\sqrt{5}}{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 $t$ and $\alpha$ be real numbers with $t > 0$ and $1 < \alpha < \frac{1+\sqrt{5}}{2}$. A sequence of integers is called additively complete if every sufficiently large integer can be expressed as a sum of distinct terms from the sequence. For fixed $t, \alpha > 0$, a pair $(t, \alpha)$ is called a good pair if the ... | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P8280",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P8280",
"citation": null,
"doi": null,
"arxiv_id": null,
"license": "Selected packets label their content CC0-1.0; original references retain their own terms",
"notes": "Selected problem... | theorem complete_for_alpha_in_Ioo_one_to_goldenRatio (t α : ℝ) (ht : 0 < t)
(hα : α ∈ Set.Ioo 1 ((1 + √5) / 2)) : IsGoodPair t α | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem complete_for_alpha_in_Ioo_one_to_goldenRatio (t α : ℝ) (ht : 0 < t)\n (hα : α ∈ Set.Ioo 1 ((1 + √5) / 2)) : IsGoodPair t α",
"target": null,
"source_url": "https://theoremdb.org/statements/P8280",
"format": "declaration-or-fragment",
"sha256": "d0f... | [] | TheoremDB P8280 · Candidate awaiting source review | null | Source-reported assessment, retrieved 8 September 2026. Frontier Science has not independently reviewed this status.
Open. No signed Lean verification is attached to this problem or its selected resolution.
This record is in the source backlog; its novelty and unresolved status have not passed source review. | null | published | incremental | hard | [
"source-candidate",
"theoremdb"
] | null | null | null | null | [
{
"kind": "website",
"citation": "TheoremDB P8280",
"url": "https://theoremdb.org/statements/P8280",
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},
{
"kind": "formalization",
"citation": "Source formal statement (not locally verified)",
"url": "https://theoremdb.org/statements/P8280",
"not... | {
"source_record": "P8280",
"imported_at": "2026-09-08",
"status_as_of": "2026-09-08",
"evidence_label": "Source candidate · not reviewed",
"note": "Full public directory statement. Source lane: backlog; source resolution: open. Source verification claims are attributed only; no local Lean verification has be... | {"created_at":"2026-08-17T18:38:13+00:00","display_standard":{"editorial_review":"submitter_supplied","formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":8280,"public_status":{"evidence":{"hasFormalization":false,"hasInformalProof":false,"hasRecordedWork":false,"solutionRecord":null},"lean":{... | [
"formalization-not-compiled-during-export",
"current-status-unassessed",
"source-full-text-not-present"
] | [] | {
"created_at": "2026-09-08T03:32:15.935Z",
"updated_at": "2026-09-08T03:32:15.935Z",
"published_at": "2026-09-08T03:32:49.478Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"a628c045-5bb5-49a3-805c-2c06db0d59e3","format_issues":null,"number":3216} |
1.0.0 | problem:theoremdb-p8282 | problem | theoremdb-p8282 | Existence of a SIC-POVM in dimension 59 | {
"main": "math",
"sub": "algebra",
"slug": "algebra",
"name": "Algebra",
"original_discipline": "mathematics"
} | [] | [] | Let $d$ be a positive integer. A state vector in $\mathbb{C}^d$ is a vector in the $d$-dimensional complex Hilbert space with the standard Euclidean inner product. A state vector is normalized if its $L^2$ norm equals $1$. For two state vectors $\varphi$ and $\psi$ in $\mathbb{C}^d$, the squared overlap is defined as t... | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P8282",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P8282",
"citation": null,
"doi": null,
"arxiv_id": null,
"license": "Selected packets label their content CC0-1.0; original references retain their own terms",
"notes": "Selected problem... | theorem hasSICPOVM_59 : answer(sorry) ↔ HasSICPOVM 59 | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem hasSICPOVM_59 : answer(sorry) ↔ HasSICPOVM 59",
"target": null,
"source_url": "https://theoremdb.org/statements/P8282",
"format": "declaration-or-fragment",
"sha256": "c6feba9a38d673960b45feda0ac36af6129fd6c3f4e0eb19ed9d07af17a9c2d0",
"license": "... | [] | TheoremDB P8282 · Candidate awaiting source review | null | Source-reported assessment, retrieved 8 September 2026. Frontier Science has not independently reviewed this status.
Open. No signed Lean verification is attached to this problem or its selected resolution.
This record is in the source backlog; its novelty and unresolved status have not passed source review. | null | published | incremental | hard | [
"source-candidate",
"theoremdb"
] | null | null | null | null | [
{
"kind": "website",
"citation": "TheoremDB P8282",
"url": "https://theoremdb.org/statements/P8282",
"note": null,
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},
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"citation": "Source formal statement (not locally verified)",
"url": "https://theoremdb.org/statements/P8282",
"not... | {
"source_record": "P8282",
"imported_at": "2026-09-08",
"status_as_of": "2026-09-08",
"evidence_label": "Source candidate · not reviewed",
"note": "Full public directory statement. Source lane: backlog; source resolution: open. Source verification claims are attributed only; no local Lean verification has be... | {"created_at":"2026-08-17T18:38:14+00:00","display_standard":{"editorial_review":"submitter_supplied","formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":8282,"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": "2026-09-08T03:32:58.496Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"44f3df28-5f57-4f87-ac5a-983570322e8b","format_issues":null,"number":5195} |
1.0.0 | problem:theoremdb-p8284 | problem | theoremdb-p8284 | Covering the unit square with rectangles of dimensions $1/(n+1) \times 1/(n+2)$ | {
"main": "math",
"sub": "geometry and topology",
"slug": "geometry-and-topology",
"name": "Geometry and topology",
"original_discipline": "mathematics"
} | [] | [] | For each natural number $n \geq 0$, let $R_n$ denote a rectangle with width $\frac{1}{n+1}$ and height $\frac{1}{n+2}$. A rectangle is placed in the plane by specifying a starting point (its lower-left corner before rotation), a rotation angle, and then applying the corresponding rigid motion (rotation about the origin... | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P8284",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P8284",
"citation": null,
"doi": null,
"arxiv_id": null,
"license": "Selected packets label their content CC0-1.0; original references retain their own terms",
"notes": "Selected problem... | theorem rectangles_cover_unit_square :
answer(sorry) ↔ ∃ c : Configuration, ∀ p ∈ unitSquare, ∃ n, p ∈ (c.rect n).toSet | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem rectangles_cover_unit_square :\n answer(sorry) ↔ ∃ c : Configuration, ∀ p ∈ unitSquare, ∃ n, p ∈ (c.rect n).toSet",
"target": null,
"source_url": "https://theoremdb.org/statements/P8284",
"format": "declaration-or-fragment",
"sha256": "9fb0892a66b9... | [] | TheoremDB P8284 · Candidate awaiting source review | null | Source-reported assessment, retrieved 8 September 2026. Frontier Science has not independently reviewed this status.
Open. No signed Lean verification is attached to this problem or its selected resolution.
This record is in the source backlog; its novelty and unresolved status have not passed source review. | null | published | incremental | hard | [
"source-candidate",
"theoremdb"
] | null | null | null | null | [
{
"kind": "website",
"citation": "TheoremDB P8284",
"url": "https://theoremdb.org/statements/P8284",
"note": null,
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},
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"kind": "formalization",
"citation": "Source formal statement (not locally verified)",
"url": "https://theoremdb.org/statements/P8284",
"not... | {
"source_record": "P8284",
"imported_at": "2026-09-08",
"status_as_of": "2026-09-08",
"evidence_label": "Source candidate · not reviewed",
"note": "Full public directory statement. Source lane: backlog; source resolution: open. Source verification claims are attributed only; no local Lean verification has be... | {"created_at":"2026-08-17T18:38:16+00:00","display_standard":{"editorial_review":"submitter_supplied","formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":8284,"public_status":{"evidence":{"hasFormalization":false,"hasInformalProof":false,"hasRecordedWork":false,"solutionRecord":null},"lean":{... | [
"formalization-contains-placeholder",
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"current-status-unassessed",
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] | [] | {
"created_at": "2026-09-08T03:32:15.935Z",
"updated_at": "2026-09-08T03:32:15.935Z",
"published_at": "2026-09-08T03:32:52.860Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"ead41a14-eb5b-4978-8390-bdb0b7c2ec34","format_issues":null,"number":3951} |
1.0.0 | problem:theoremdb-p8286 | problem | theoremdb-p8286 | Existence of an absolutely maximally entangled state for eleven parties of local dimension three | {
"main": "math",
"sub": "combinatorics",
"slug": "combinatorics",
"name": "Combinatorics",
"original_discipline": "mathematics"
} | [] | [
{
"code": "05",
"basis": "community-map",
"evidence": "Inferred from community field: combinatorics"
}
] | Let $n \geq 2$ be a positive integer and let $d \geq 2$ be a positive integer. Consider a multipartite quantum system consisting of $n$ parties, each with a local Hilbert space of dimension $d$. A pure state $\psi$ of this system is called absolutely maximally entangled (AME) if, for every subset of at most $\lfloor n/... | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P8286",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P8286",
"citation": null,
"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_3_open :
answer(sorry) ↔ ExistsAME 11 3 | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem ame_11_3_open :\n answer(sorry) ↔ ExistsAME 11 3",
"target": null,
"source_url": "https://theoremdb.org/statements/P8286",
"format": "declaration-or-fragment",
"sha256": "b431ec969ecf73ec42f1d5728d49875c307bb4a3c3a01fd99ce9aa784def48c5",
"licen... | [] | TheoremDB P8286 · Candidate awaiting source review | null | Source-reported assessment, retrieved 8 September 2026. Frontier Science has not independently reviewed this status.
Open. No signed Lean verification is attached to this problem or its selected resolution.
This record is in the source backlog; its novelty and unresolved status have not passed source review. | null | published | incremental | hard | [
"source-candidate",
"theoremdb"
] | null | null | null | null | [
{
"kind": "website",
"citation": "TheoremDB P8286",
"url": "https://theoremdb.org/statements/P8286",
"note": null,
"sort_order": 0
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"citation": "Source formal statement (not locally verified)",
"url": "https://theoremdb.org/statements/P8286",
"not... | {
"source_record": "P8286",
"imported_at": "2026-09-08",
"status_as_of": "2026-09-08",
"evidence_label": "Source candidate · not reviewed",
"note": "Full public directory statement. Source lane: backlog; source resolution: open. Source verification claims are attributed only; no local Lean verification has be... | {"created_at":"2026-08-17T18:38:17+00:00","display_standard":{"editorial_review":"submitter_supplied","formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":8286,"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": "2026-09-08T03:32:58.640Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"1568f735-45bf-4440-93ee-d5e10f30f6fd","format_issues":null,"number":5224} |
1.0.0 | problem:theoremdb-p8288 | problem | theoremdb-p8288 | The Bing–Borsuk Conjecture on Homogeneous Absolute Neighborhood Retracts | {
"main": "math",
"sub": "geometry and topology",
"slug": "geometry-and-topology",
"name": "Geometry and topology",
"original_discipline": "mathematics"
} | [] | [] | Let $n$ be a natural number and let $X$ be a metrizable topological space that is homogeneous and an absolute neighborhood retract. Suppose that $X$ has Lebesgue covering dimension equal to $n$. Does there exist an atlas of charts making $X$ a topological $n$-manifold? | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P8288",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P8288",
"citation": null,
"doi": null,
"arxiv_id": null,
"license": "Selected packets label their content CC0-1.0; original references retain their own terms",
"notes": "Selected problem... | theorem bing_borsuk_conjecture : ∀ n : ℕ, ∀ (X : Type) [TopologicalSpace X] [MetrizableSpace X] [HomogeneousSpace X] [IsAbsoluteNeighborhoodRetract X],
HasLebesgueCoveringDimensionEq X n → Nonempty (ChartedSpace (Fin n → ℝ) X) | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem bing_borsuk_conjecture : ∀ n : ℕ, ∀ (X : Type) [TopologicalSpace X] [MetrizableSpace X] [HomogeneousSpace X] [IsAbsoluteNeighborhoodRetract X],\n HasLebesgueCoveringDimensionEq X n → Nonempty (ChartedSpace (Fin n → ℝ) X)",
"target": null,
"source_url": "ht... | [] | TheoremDB P8288 · Candidate awaiting source review | null | Source-reported assessment, retrieved 8 September 2026. Frontier Science has not independently reviewed this status.
Open. No signed Lean verification is attached to this problem or its selected resolution.
This record is in the source backlog; its novelty and unresolved status have not passed source review. | null | published | incremental | hard | [
"source-candidate",
"theoremdb"
] | null | null | null | null | [
{
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"citation": "TheoremDB P8288",
"url": "https://theoremdb.org/statements/P8288",
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},
{
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"citation": "Source formal statement (not locally verified)",
"url": "https://theoremdb.org/statements/P8288",
"not... | {
"source_record": "P8288",
"imported_at": "2026-09-08",
"status_as_of": "2026-09-08",
"evidence_label": "Source candidate · not reviewed",
"note": "Full public directory statement. Source lane: backlog; source resolution: open. Source verification claims are attributed only; no local Lean verification has be... | {"created_at":"2026-08-17T18:38:18+00:00","display_standard":{"formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":8288,"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": "2026-09-08T03:33:06.394Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"28606eb0-dbf3-409c-855d-471ab646f3eb","format_issues":null,"number":6733} |
1.0.0 | problem:theoremdb-p8290 | problem | theoremdb-p8290 | Erdős Problem 307 on Reciprocal Sums of 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"
}
] | Do there exist two finite sets of prime numbers $P$ and $Q$ such that
$$
1 = \left( \sum_{p \in P} \frac{1}{p} \right) \left( \sum_{q \in Q} \frac{1}{q} \right)?
$$
Equivalently, can the number $1$ be expressed as a product of two rational numbers, each of which is the sum of reciprocals of a finite set of primes? | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P8290",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P8290",
"citation": null,
"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_307 : answer(sorry) ↔ ∃ P Q : Finset ℕ, (∀ p ∈ P, p.Prime) ∧ (∀ q ∈ Q, q.Prime) ∧
1 = (∑ p ∈ P, (p : ℚ)⁻¹) * (∑ q ∈ Q, (q : ℚ)⁻¹) | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem erdos_307 : answer(sorry) ↔ ∃ P Q : Finset ℕ, (∀ p ∈ P, p.Prime) ∧ (∀ q ∈ Q, q.Prime) ∧\n 1 = (∑ p ∈ P, (p : ℚ)⁻¹) * (∑ q ∈ Q, (q : ℚ)⁻¹)",
"target": null,
"source_url": "https://theoremdb.org/statements/P8290",
"format": "declaration-or-fragment",
... | [] | TheoremDB P8290 · Candidate awaiting source review | null | Source-reported assessment, retrieved 8 September 2026. Frontier Science has not independently reviewed this status.
Open. No signed Lean verification is attached to this problem or its selected resolution.
This record is in 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/P8290",
"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":"2026-08-17T18:38:19+00:00","display_standard":{"formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":8290,"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": "2026-09-08T03:32:55.315Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"0fdd620f-efec-4bc6-8093-67cb6511da84","format_issues":null,"number":4493} |
1.0.0 | problem:theoremdb-p8292 | problem | theoremdb-p8292 | An accumulation point of the fractional parts of powers of 3/2 | {
"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"
}
] | For each positive integer $n$, define $s_n = \left(\frac{3}{2}\right)^n - \left\lfloor\left(\frac{3}{2}\right)^n\right\rfloor$, the fractional part of $\left(\frac{3}{2}\right)^n$. A real number $x$ is called an accumulation point of a sequence $(s_n)_{n=1}^{\infty}$ if every neighborhood of $x$ contains some term of t... | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P8292",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P8292",
"citation": null,
"doi": null,
"arxiv_id": null,
"license": "Selected packets label their content CC0-1.0; original references retain their own terms",
"notes": "Selected problem... | theorem isAccumulationPoint_three_halves_pow :
IsAccumulationPoint answer(sorry) (fun n => Int.fract <| (3 / 2 : ℝ)^n) | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem isAccumulationPoint_three_halves_pow :\n IsAccumulationPoint answer(sorry) (fun n => Int.fract <| (3 / 2 : ℝ)^n)",
"target": null,
"source_url": "https://theoremdb.org/statements/P8292",
"format": "declaration-or-fragment",
"sha256": "bc736b2250472... | [] | TheoremDB P8292 · Candidate awaiting source review | null | Source-reported assessment, retrieved 8 September 2026. Frontier Science has not independently reviewed this status.
Open. No signed Lean verification is attached to this problem or its selected resolution.
This record is in the source backlog; its novelty and unresolved status have not passed source review. | null | published | incremental | hard | [
"source-candidate",
"theoremdb"
] | null | null | null | null | [
{
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"citation": "Source formal statement (not locally verified)",
"url": "https://theoremdb.org/statements/P8292",
"not... | {
"source_record": "P8292",
"imported_at": "2026-09-08",
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"evidence_label": "Source candidate · not reviewed",
"note": "Full public directory statement. Source lane: backlog; source resolution: open. Source verification claims are attributed only; no local Lean verification has be... | {"created_at":"2026-08-17T18:38:28+00:00","display_standard":{"editorial_review":"submitter_supplied","formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":8292,"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": "2026-09-08T03:32:49.579Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"d950a731-b2bb-41b4-9bd0-c3d3d7967007","format_issues":null,"number":3238} |
1.0.0 | problem:theoremdb-p8294 | problem | theoremdb-p8294 | Erdős's squarefree-power-of-two sum problem | {
"main": "math",
"sub": "number theory",
"slug": "number-theory",
"name": "Number theory",
"original_discipline": "mathematics"
} | [] | [
{
"code": "11",
"basis": "community-map",
"evidence": "Inferred from community field: number-theory"
}
] | A positive integer is called squarefree if it is not divisible by the square of any prime. Is every odd integer $n > 1$ equal to the sum of a squarefree number and a power of $2$? | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P8294",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P8294",
"citation": null,
"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 (n : ℕ) (hn : Odd 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 (n : ℕ) (hn : Odd n) (hn' : 1 < n) :\n ∃ k l : ℕ, Squarefree k ∧ n = k + 2 ^ l",
"target": null,
"source_url": "https://theoremdb.org/statements/P8294",
"format": "declaration-or-fragment",
"sha256": "f6875ca7212ad403993cf8b6fa2a51f939c975... | [] | TheoremDB P8294 · Candidate awaiting source review | null | Source-reported assessment, retrieved 8 September 2026. Frontier Science has not independently reviewed this status.
Open. No signed Lean verification is attached to this problem or its selected resolution.
This record is in the source backlog; its novelty and unresolved status have not passed source review. | null | published | incremental | hard | [
"source-candidate",
"theoremdb"
] | null | null | null | null | [
{
"kind": "website",
"citation": "TheoremDB P8294",
"url": "https://theoremdb.org/statements/P8294",
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},
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"citation": "Source formal statement (not locally verified)",
"url": "https://theoremdb.org/statements/P8294",
"not... | {
"source_record": "P8294",
"imported_at": "2026-09-08",
"status_as_of": "2026-09-08",
"evidence_label": "Source candidate · not reviewed",
"note": "Full public directory statement. Source lane: backlog; source resolution: open. Source verification claims are attributed only; no local Lean verification has be... | {"created_at":"2026-08-17T18:38:33+00:00","display_standard":{"editorial_review":"submitter_supplied","formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":8294,"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": "2026-09-08T03:32:56.581Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"dc82584f-046b-45cb-9579-5834e5b565d8","format_issues":null,"number":4771} |
1.0.0 | problem:theoremdb-p8296 | problem | theoremdb-p8296 | Non-existence of real monochromatic quantum graphs with eight 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 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 assignment of a real weight to each edge, where each edge is also assigned one of $D$ colours. For each vertex colouring $\iota : \{1, \dots, N... | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P8296",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P8296",
"citation": null,
"doi": null,
"arxiv_id": null,
"license": "Selected packets label their content CC0-1.0; original references retain their own terms",
"notes": "Selected problem... | theorem eqSystem8_no_solution_d3_real :
answer(sorry) ↔
¬ ∃ W : WeightsN 8 3 ℝ, EqSystemN 8 3 W | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem eqSystem8_no_solution_d3_real :\n answer(sorry) ↔\n ¬ ∃ W : WeightsN 8 3 ℝ, EqSystemN 8 3 W",
"target": null,
"source_url": "https://theoremdb.org/statements/P8296",
"format": "declaration-or-fragment",
"sha256": "0f376caa7c9a5b49e2d3a1c1b747d... | [] | TheoremDB P8296 · Candidate awaiting source review | null | Source-reported assessment, retrieved 8 September 2026. Frontier Science has not independently reviewed this status.
Open. No signed Lean verification is attached to this problem or its selected resolution.
This record is in the source backlog; its novelty and unresolved status have not passed source review. | null | published | incremental | hard | [
"source-candidate",
"theoremdb"
] | null | null | null | null | [
{
"kind": "website",
"citation": "TheoremDB P8296",
"url": "https://theoremdb.org/statements/P8296",
"note": null,
"sort_order": 0
},
{
"kind": "formalization",
"citation": "Source formal statement (not locally verified)",
"url": "https://theoremdb.org/statements/P8296",
"not... | {
"source_record": "P8296",
"imported_at": "2026-09-08",
"status_as_of": "2026-09-08",
"evidence_label": "Source candidate · not reviewed",
"note": "Full public directory statement. Source lane: backlog; source resolution: open. Source verification claims are attributed only; no local Lean verification has be... | {"created_at":"2026-08-17T18:38:42+00:00","display_standard":{"editorial_review":"submitter_supplied","formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":8296,"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": "2026-09-08T03:33:04.415Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"6fb29d24-5602-4b22-a89d-02076ae2629a","format_issues":null,"number":6305} |
1.0.0 | problem:theoremdb-p8298 | problem | theoremdb-p8298 | Erdős's Problem on Sidon Set Growth | {
"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 $F(N)$ denote the maximum cardinality of a Sidon subset of $\{1, 2, \dots, N\}$, that is, a subset in which all pairwise sums of distinct elements are distinct. Is it true that for every integer $k \geq 1$, the inequality
$$
F(N + k) \leq F(N) + 1
$$
holds for all sufficiently large... | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P8298",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P8298",
"citation": null,
"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_155 : answer(sorry) ↔ ∀ k ≥ 1, ∀ᶠ N in atTop, F (N + k) ≤ F N + 1 | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem erdos_155 : answer(sorry) ↔ ∀ k ≥ 1, ∀ᶠ N in atTop, F (N + k) ≤ F N + 1",
"target": null,
"source_url": "https://theoremdb.org/statements/P8298",
"format": "declaration-or-fragment",
"sha256": "088a9d106b715fec0faca43ccf9f4da24a581cc095d4124a7c2c5d333... | [] | TheoremDB P8298 · Candidate awaiting source review | null | Source-reported assessment, retrieved 8 September 2026. Frontier Science has not independently reviewed this status.
Open. No signed Lean verification is attached to this problem or its selected resolution.
This record is in the source backlog; its novelty and unresolved status have not passed source review. | null | published | incremental | hard | [
"source-candidate",
"theoremdb"
] | null | null | null | null | [
{
"kind": "website",
"citation": "TheoremDB P8298",
"url": "https://theoremdb.org/statements/P8298",
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},
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"citation": "Source formal statement (not locally verified)",
"url": "https://theoremdb.org/statements/P8298",
"not... | {
"source_record": "P8298",
"imported_at": "2026-09-08",
"status_as_of": "2026-09-08",
"evidence_label": "Source candidate · not reviewed",
"note": "Full public directory statement. Source lane: backlog; source resolution: open. Source verification claims are attributed only; no local Lean verification has be... | {"created_at":"2026-08-17T18:38:44+00:00","display_standard":{"editorial_review":"submitter_supplied","formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":8298,"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": "2026-09-08T03:32:56.540Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"3cc8d749-727c-425e-a4f1-f2f7bbba82f4","format_issues":null,"number":4762} |
1.0.0 | problem:theoremdb-p8300 | problem | theoremdb-p8300 | Asymptotic behavior of the minimal preimage for divisor-sum representations | {
"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 smallest positive integer $m$ such that $n$ can be expressed as the sum of the $k$ smallest divisors of $m$ for some positive integer $k$. Does there exist a subset $A$ of the natural numbers having natural density $1$ such that $f(n) = o(n)$ as $n \to \infty$ throug... | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P8300",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P8300",
"citation": null,
"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.ii : answer(sorry) ↔ ∃ (A : Set ℕ), A.HasDensity 1 ∧
(fun (n : A) ↦ (f ↑n : ℝ)) =o[atTop] (fun n ↦ (n : ℝ)) | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem erdos_1054.parts.ii : answer(sorry) ↔ ∃ (A : Set ℕ), A.HasDensity 1 ∧\n (fun (n : A) ↦ (f ↑n : ℝ)) =o[atTop] (fun n ↦ (n : ℝ))",
"target": null,
"source_url": "https://theoremdb.org/statements/P8300",
"format": "declaration-or-fragment",
"sha256": ... | [] | TheoremDB P8300 · Candidate awaiting source review | null | Source-reported assessment, retrieved 8 September 2026. Frontier Science has not independently reviewed this status.
Open. No signed Lean verification is attached to this problem or its selected resolution.
This record is in the source backlog; its novelty and unresolved status have not passed source review. | null | published | incremental | hard | [
"source-candidate",
"theoremdb"
] | null | null | null | null | [
{
"kind": "website",
"citation": "TheoremDB P8300",
"url": "https://theoremdb.org/statements/P8300",
"note": null,
"sort_order": 0
},
{
"kind": "formalization",
"citation": "Source formal statement (not locally verified)",
"url": "https://theoremdb.org/statements/P8300",
"not... | {
"source_record": "P8300",
"imported_at": "2026-09-08",
"status_as_of": "2026-09-08",
"evidence_label": "Source candidate · not reviewed",
"note": "Full public directory statement. Source lane: backlog; source resolution: open. Source verification claims are attributed only; no local Lean verification has be... | {"created_at":"2026-08-17T18:38:51+00:00","display_standard":{"editorial_review":"submitter_supplied","formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":8300,"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": "2026-09-08T03:32:49.912Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"2076510e-6377-49ff-b80a-95deed46d532","format_issues":null,"number":3312} |
1.0.0 | problem:theoremdb-p8302 | problem | theoremdb-p8302 | Existence of a projective plane of order 12 | {
"main": "math",
"sub": "combinatorics",
"slug": "combinatorics",
"name": "Combinatorics",
"original_discipline": "mathematics"
} | [] | [
{
"code": "05",
"basis": "community-map",
"evidence": "Inferred from community field: combinatorics"
}
] | A finite projective plane of order $n$ is a geometric structure consisting of a set $P$ of points and a set $L$ of lines (with an incidence relation between points and lines), satisfying the following axioms: any two distinct points lie on exactly one line; any two distinct lines meet in exactly one point; there exist ... | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P8302",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P8302",
"citation": null,
"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_723.variants.eq_12 : answer(sorry) ↔
∃ (P L : Type) (_ : Membership P L) (_ : Fintype P) (_ : Fintype L) (pp : ProjectivePlane P L),
pp.order = 12 | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem erdos_723.variants.eq_12 : answer(sorry) ↔\n ∃ (P L : Type) (_ : Membership P L) (_ : Fintype P) (_ : Fintype L) (pp : ProjectivePlane P L),\n pp.order = 12",
"target": null,
"source_url": "https://theoremdb.org/statements/P8302",
"format": "decla... | [] | TheoremDB P8302 · Candidate awaiting source review | null | Source-reported assessment, retrieved 8 September 2026. Frontier Science has not independently reviewed this status.
Open. No signed Lean verification is attached to this problem or its selected resolution.
This record is in the source backlog; its novelty and unresolved status have not passed source review. | null | published | incremental | hard | [
"source-candidate",
"theoremdb"
] | null | null | null | null | [
{
"kind": "website",
"citation": "TheoremDB P8302",
"url": "https://theoremdb.org/statements/P8302",
"note": null,
"sort_order": 0
},
{
"kind": "formalization",
"citation": "Source formal statement (not locally verified)",
"url": "https://theoremdb.org/statements/P8302",
"not... | {
"source_record": "P8302",
"imported_at": "2026-09-08",
"status_as_of": "2026-09-08",
"evidence_label": "Source candidate · not reviewed",
"note": "Full public directory statement. Source lane: backlog; source resolution: open. Source verification claims are attributed only; no local Lean verification has be... | {"created_at":"2026-08-17T18:38:53+00:00","display_standard":{"editorial_review":"submitter_supplied","formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":8302,"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": "2026-09-08T03:32:58.482Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"05182084-dd03-4569-930d-071d4bea4d59","format_issues":null,"number":5192} |
1.0.0 | problem:theoremdb-p8304 | problem | theoremdb-p8304 | Infinitely many integers with squared shifted factorial dividing a central factorial | {
"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 $m!$ denote the factorial of $m$, defined as the product of all positive integers up to $m$. Let $k = 2$. Does there exist infinitely many natural numbers $n$ such that $((n+k)!)^2$ divides $(2n)!$? Equivalently, is the set
$$
\{n \in \mathbb{N} : ((n+2)!)^2 \mid (2n)!\}
$$
infinite? | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P8304",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P8304",
"citation": null,
"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_727.variants.k_2 :
letI k := 2
answer(sorry) ↔ Set.Infinite {n : ℕ | (Nat.factorial (n + k)) ^ 2 ∣ Nat.factorial (2 * n)} | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem erdos_727.variants.k_2 :\n letI k := 2\n answer(sorry) ↔ Set.Infinite {n : ℕ | (Nat.factorial (n + k)) ^ 2 ∣ Nat.factorial (2 * n)}",
"target": null,
"source_url": "https://theoremdb.org/statements/P8304",
"format": "declaration-or-fragment",
"s... | [] | TheoremDB P8304 · Candidate awaiting source review | null | Source-reported assessment, retrieved 8 September 2026. Frontier Science has not independently reviewed this status.
Open. No signed Lean verification is attached to this problem or its selected resolution.
This record is in 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/P8304",
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},
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"citation": "Source formal statement (not locally verified)",
"url": "https://theoremdb.org/statements/P8304",
"not... | {
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"imported_at": "2026-09-08",
"status_as_of": "2026-09-08",
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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":"2026-08-17T18:38:59+00:00","display_standard":{"editorial_review":"submitter_supplied","formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":8304,"public_status":{"evidence":{"hasFormalization":false,"hasInformalProof":false,"hasRecordedWork":false,"solutionRecord":null},"lean":{... | [
"formalization-contains-placeholder",
"formalization-not-compiled-during-export",
"current-status-unassessed",
"source-full-text-not-present"
] | [] | {
"created_at": "2026-09-08T03:32:15.935Z",
"updated_at": "2026-09-08T03:32:15.935Z",
"published_at": "2026-09-08T03:33:00.926Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"ddc8cf85-7553-4e8e-a4df-50a0103d3dd4","format_issues":null,"number":5620} |
1.0.0 | problem:theoremdb-p8306 | problem | theoremdb-p8306 | Existence of a Small Multiplier for Prime Generation | {
"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$, does there exist a natural number $k$ with $k < n$ such that $nk + 1$ is prime? | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P8306",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P8306",
"citation": null,
"doi": null,
"arxiv_id": null,
"license": "Selected packets label their content CC0-1.0; original references retain their own terms",
"notes": "Selected problem... | theorem exists_k {n : ℕ} (hn : 1 < n) : ∃ k < n, (n * k + 1).Prime | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem exists_k {n : ℕ} (hn : 1 < n) : ∃ k < n, (n * k + 1).Prime",
"target": null,
"source_url": "https://theoremdb.org/statements/P8306",
"format": "declaration-or-fragment",
"sha256": "238827725a3a9d1ce799ffa8972579d77ec7f88172a4a5418e4524c6473dbe91",
... | [] | TheoremDB P8306 · Candidate awaiting source review | null | Source-reported assessment, retrieved 8 September 2026. Frontier Science has not independently reviewed this status.
Open. No signed Lean verification is attached to this problem or its selected resolution.
This record is in the source backlog; its novelty and unresolved status have not passed source review. | null | published | incremental | hard | [
"source-candidate",
"theoremdb"
] | null | null | null | null | [
{
"kind": "website",
"citation": "TheoremDB P8306",
"url": "https://theoremdb.org/statements/P8306",
"note": null,
"sort_order": 0
},
{
"kind": "formalization",
"citation": "Source formal statement (not locally verified)",
"url": "https://theoremdb.org/statements/P8306",
"not... | {
"source_record": "P8306",
"imported_at": "2026-09-08",
"status_as_of": "2026-09-08",
"evidence_label": "Source candidate · not reviewed",
"note": "Full public directory statement. Source lane: backlog; source resolution: open. Source verification claims are attributed only; no local Lean verification has be... | {"created_at":"2026-08-17T18:39:03+00:00","display_standard":{"editorial_review":"submitter_supplied","formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":8306,"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": "2026-09-08T03:32:58.541Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"7ceb7134-ac2f-483f-9f2b-2f801e75b6ea","format_issues":null,"number":5204} |
1.0.0 | problem:theoremdb-p8308 | problem | theoremdb-p8308 | Segal's Conjecture on the Prime Counting 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 $\pi(t)$ denote the number of primes not exceeding $t$, for any real number $t$. Determine whether the following holds: for all sufficiently large real numbers $x$, and for all sufficiently large real numbers $y$ (depending on $x$), we have $\pi(x + y) \leq \pi(x) + \pi(y)$. | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P8308",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P8308",
"citation": null,
"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_855 : answer(sorry) ↔
∀ᶠ x in atTop, ∀ᶠ y in atTop, π (x + y) ≤ π x + π y | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem erdos_855 : answer(sorry) ↔\n ∀ᶠ x in atTop, ∀ᶠ y in atTop, π (x + y) ≤ π x + π y",
"target": null,
"source_url": "https://theoremdb.org/statements/P8308",
"format": "declaration-or-fragment",
"sha256": "40dee67849a2aa74243d13d66a0204cab109d49d79fb... | [] | TheoremDB P8308 · Candidate awaiting source review | null | Source-reported assessment, retrieved 8 September 2026. Frontier Science has not independently reviewed this status.
Open. No signed Lean verification is attached to this problem or its selected resolution.
This record is in the source backlog; its novelty and unresolved status have not passed source review. | null | published | incremental | hard | [
"source-candidate",
"theoremdb"
] | null | null | null | null | [
{
"kind": "website",
"citation": "TheoremDB P8308",
"url": "https://theoremdb.org/statements/P8308",
"note": null,
"sort_order": 0
},
{
"kind": "formalization",
"citation": "Source formal statement (not locally verified)",
"url": "https://theoremdb.org/statements/P8308",
"not... | {
"source_record": "P8308",
"imported_at": "2026-09-08",
"status_as_of": "2026-09-08",
"evidence_label": "Source candidate · not reviewed",
"note": "Full public directory statement. Source lane: backlog; source resolution: open. Source verification claims are attributed only; no local Lean verification has be... | {"created_at":"2026-08-17T18:39:17+00:00","display_standard":{"editorial_review":"submitter_supplied","formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":8308,"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": "2026-09-08T03:33:05.752Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"a42e35b6-c24e-42d6-aa48-49cfcf5a56db","format_issues":null,"number":6595} |
1.0.0 | problem:theoremdb-p8310 | problem | theoremdb-p8310 | Asymptotic upper bound for the maximal length of consecutive integers each with a large 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 each positive integer $n$, define $k(n)$ to be the largest positive integer $k$ for which there exists an integer $m$ with $1 \leq m \leq n$ such that each of the consecutive integers $m+1, m+2, \ldots, m+k$ has a prime divisor strictly greater than $k$. Determine whether
$$
\log k(n) \leq (\log n)^{1/2 + o(1)}
... | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P8310",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P8310",
"citation": null,
"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_962 :
answer(sorry) ↔
∃ ε : ℕ → ℝ,
(∀ δ > 0, ∀ᶠ n in atTop, |ε n| < δ) ∧
∀ᶠ n : ℕ in atTop,
log (k n : ℝ)
≤ rpow (log n) ((1 : ℝ) / 2 + ε n) | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem erdos_962 :\n answer(sorry) ↔\n ∃ ε : ℕ → ℝ,\n (∀ δ > 0, ∀ᶠ n in atTop, |ε n| < δ) ∧\n ∀ᶠ n : ℕ in atTop,\n log (k n : ℝ)\n ≤ rpow (log n) ((1 : ℝ) / 2 + ε n)",
"target": null,
"source_url": "https://theoremdb.org/sta... | [] | TheoremDB P8310 · Candidate awaiting source review | null | Source-reported assessment, retrieved 8 September 2026. Frontier Science has not independently reviewed this status.
Open. No signed Lean verification is attached to this problem or its selected resolution.
This record is in the source backlog; its novelty and unresolved status have not passed source review. | null | published | incremental | hard | [
"source-candidate",
"theoremdb"
] | null | null | null | null | [
{
"kind": "website",
"citation": "TheoremDB P8310",
"url": "https://theoremdb.org/statements/P8310",
"note": null,
"sort_order": 0
},
{
"kind": "formalization",
"citation": "Source formal statement (not locally verified)",
"url": "https://theoremdb.org/statements/P8310",
"not... | {
"source_record": "P8310",
"imported_at": "2026-09-08",
"status_as_of": "2026-09-08",
"evidence_label": "Source candidate · not reviewed",
"note": "Full public directory statement. Source lane: backlog; source resolution: open. Source verification claims are attributed only; no local Lean verification has be... | {"created_at":"2026-08-17T18:39:18+00:00","display_standard":{"editorial_review":"submitter_supplied","formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":8310,"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": "2026-09-08T03:32:50.465Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"44b9d4fd-4e5d-45b3-8eaa-dd8b84b487c5","format_issues":null,"number":3424} |
1.0.0 | problem:theoremdb-p8312 | problem | theoremdb-p8312 | Lower bound for the 3-uniform hypergraph Ramsey number | {
"main": "math",
"sub": "combinatorics",
"slug": "combinatorics",
"name": "Combinatorics",
"original_discipline": "mathematics"
} | [] | [
{
"code": "05",
"basis": "community-map",
"evidence": "Inferred from community field: combinatorics"
}
] | Let $R_3(n)$ denote the minimum number $m$ such that if the edges of the complete $3$-uniform hypergraph on $m$ vertices are colored with two colors, then there exists a monochromatic copy of the complete $3$-uniform hypergraph on $n$ vertices. Does there exist a constant $c > 0$ such that for all sufficiently large po... | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P8312",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P8312",
"citation": null,
"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_564 : answer(sorry) ↔
∃ c > 0, ∀ᶠ n in atTop, (2 : ℝ)^(2 : ℝ)^(c * n) ≤ hypergraphRamsey 3 n | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem erdos_564 : answer(sorry) ↔\n ∃ c > 0, ∀ᶠ n in atTop, (2 : ℝ)^(2 : ℝ)^(c * n) ≤ hypergraphRamsey 3 n",
"target": null,
"source_url": "https://theoremdb.org/statements/P8312",
"format": "declaration-or-fragment",
"sha256": "ce8afeeeea49ffb6b6be8bd6e... | [] | TheoremDB P8312 · Candidate awaiting source review | null | Source-reported assessment, retrieved 8 September 2026. Frontier Science has not independently reviewed this status.
Open. No signed Lean verification is attached to this problem or its selected resolution.
This record is in the source backlog; its novelty and unresolved status have not passed source review. | null | published | incremental | hard | [
"source-candidate",
"theoremdb"
] | null | null | null | null | [
{
"kind": "website",
"citation": "TheoremDB P8312",
"url": "https://theoremdb.org/statements/P8312",
"note": null,
"sort_order": 0
},
{
"kind": "formalization",
"citation": "Source formal statement (not locally verified)",
"url": "https://theoremdb.org/statements/P8312",
"not... | {
"source_record": "P8312",
"imported_at": "2026-09-08",
"status_as_of": "2026-09-08",
"evidence_label": "Source candidate · not reviewed",
"note": "Full public directory statement. Source lane: backlog; source resolution: open. Source verification claims are attributed only; no local Lean verification has be... | {"created_at":"2026-08-17T18:39:33+00:00","display_standard":{"editorial_review":"submitter_supplied","formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":8312,"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": "2026-09-08T03:33:03.433Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"177532d9-823e-4d3e-b10a-60b9be94e56a","format_issues":null,"number":6095} |
1.0.0 | problem:theoremdb-p8314 | problem | theoremdb-p8314 | Little-o asymptotic for representations as prime plus a power of two | {
"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 $f(n)$ denote the number of pairs $(p,k)$ consisting of a prime number $p$ and a nonnegative integer $k$ such that $n = p + 2^k$. Determine whether $f(n) = o(\log n)$ as $n \to \infty$. Equivalently, determine whether for every $\varepsilon > 0$ there exists $N$ such that $f(n) \leq \... | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P8314",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P8314",
"citation": null,
"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_236: (fun n => (f n : ℝ)) =o[atTop] (fun n => Real.log (n : ℝ)) | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem erdos_236: (fun n => (f n : ℝ)) =o[atTop] (fun n => Real.log (n : ℝ))",
"target": null,
"source_url": "https://theoremdb.org/statements/P8314",
"format": "declaration-or-fragment",
"sha256": "bbadbbaf1b29d1bf47d0e7099000390b98cb1a8b277eee7a72f13a64299... | [] | TheoremDB P8314 · Candidate awaiting source review | null | Source-reported assessment, retrieved 8 September 2026. Frontier Science has not independently reviewed this status.
Open. No signed Lean verification is attached to this problem or its selected resolution.
This record is in the source backlog; its novelty and unresolved status have not passed source review. | null | published | incremental | hard | [
"source-candidate",
"theoremdb"
] | null | null | null | null | [
{
"kind": "website",
"citation": "TheoremDB P8314",
"url": "https://theoremdb.org/statements/P8314",
"note": null,
"sort_order": 0
},
{
"kind": "formalization",
"citation": "Source formal statement (not locally verified)",
"url": "https://theoremdb.org/statements/P8314",
"not... | {
"source_record": "P8314",
"imported_at": "2026-09-08",
"status_as_of": "2026-09-08",
"evidence_label": "Source candidate · not reviewed",
"note": "Full public directory statement. Source lane: backlog; source resolution: open. Source verification claims are attributed only; no local Lean verification has be... | {"created_at":"2026-08-17T18:39:34+00:00","display_standard":{"editorial_review":"submitter_supplied","formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":8314,"public_status":{"evidence":{"hasFormalization":false,"hasInformalProof":false,"hasRecordedWork":false,"solutionRecord":null},"lean":{... | [
"formalization-not-compiled-during-export",
"current-status-unassessed",
"source-full-text-not-present"
] | [] | {
"created_at": "2026-09-08T03:32:15.935Z",
"updated_at": "2026-09-08T03:32:15.935Z",
"published_at": "2026-09-08T03:33:03.278Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"528334ec-3c43-43b0-9445-a7bd85830975","format_issues":null,"number":6061} |
1.0.0 | problem:theoremdb-p8316 | problem | theoremdb-p8316 | Convergence of an alternating series involving 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 $p_n$ denote the $n$-th prime number, with $p_1 = 2, p_2 = 3, p_3 = 5,\ldots$. Consider the infinite series
$$
\sum_{n=1}^{\infty} (-1)^n \frac{n}{p_n}.
$$
Is this series convergent? | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P8316",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P8316",
"citation": null,
"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_15 : answer(sorry) ↔
Summable (fun k : ℕ => (-1 : ℚ) ^ (k + 1) * (k + 1) / (k.nth Nat.Prime)) | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem erdos_15 : answer(sorry) ↔\n Summable (fun k : ℕ => (-1 : ℚ) ^ (k + 1) * (k + 1) / (k.nth Nat.Prime))",
"target": null,
"source_url": "https://theoremdb.org/statements/P8316",
"format": "declaration-or-fragment",
"sha256": "276dbfe05799bc3c3afff5e3... | [] | TheoremDB P8316 · Candidate awaiting source review | null | Source-reported assessment, retrieved 8 September 2026. Frontier Science has not independently reviewed this status.
Open. No signed Lean verification is attached to this problem or its selected resolution.
This record is in the source backlog; its novelty and unresolved status have not passed source review. | null | published | incremental | hard | [
"source-candidate",
"theoremdb"
] | null | null | null | null | [
{
"kind": "website",
"citation": "TheoremDB P8316",
"url": "https://theoremdb.org/statements/P8316",
"note": null,
"sort_order": 0
},
{
"kind": "formalization",
"citation": "Source formal statement (not locally verified)",
"url": "https://theoremdb.org/statements/P8316",
"not... | {
"source_record": "P8316",
"imported_at": "2026-09-08",
"status_as_of": "2026-09-08",
"evidence_label": "Source candidate · not reviewed",
"note": "Full public directory statement. Source lane: backlog; source resolution: open. Source verification claims are attributed only; no local Lean verification has be... | {"created_at":"2026-08-17T18:39:34+00:00","display_standard":{"editorial_review":"submitter_supplied","formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":8316,"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": "2026-09-08T03:32:52.663Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"bc2293c9-4818-4800-897f-4b6408c1cf36","format_issues":null,"number":3909} |
1.0.0 | problem:theoremdb-p8318 | problem | theoremdb-p8318 | Lower bound for numbers representable as sums of $k$ nonnegative $k$th powers | {
"main": "math",
"sub": "number theory",
"slug": "number-theory",
"name": "Number theory",
"original_discipline": "mathematics"
} | [] | [
{
"code": "11",
"basis": "community-map",
"evidence": "Inferred from community field: number-theory"
}
] | For positive integers $k$, $m$, and $x$ with $1 \leq m \leq k$, 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, that is, $n = \sum_{i=1}^{m} v_i^k$ for some nonnegative integers $v_1, \ldots, v_m$. Determine whether for every integer $k \geq 1$... | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P8318",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P8318",
"citation": null,
"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.i :
answer(sorry) ↔ ∀ k ≥ 1, ∀ ε > (0 : ℝ),
(fun (x : ℕ) ↦ (x : ℝ) ^ (1 - ε)) =O[atTop] (fun (x : ℕ) ↦ (f k k x : ℝ)) | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem erdos_323.parts.i :\n answer(sorry) ↔ ∀ k ≥ 1, ∀ ε > (0 : ℝ),\n (fun (x : ℕ) ↦ (x : ℝ) ^ (1 - ε)) =O[atTop] (fun (x : ℕ) ↦ (f k k x : ℝ))",
"target": null,
"source_url": "https://theoremdb.org/statements/P8318",
"format": "declaration-or-fragment"... | [] | TheoremDB P8318 · Candidate awaiting source review | null | Source-reported assessment, retrieved 8 September 2026. Frontier Science has not independently reviewed this status.
Open. No signed Lean verification is attached to this problem or its selected resolution.
This record is in the source backlog; its novelty and unresolved status have not passed source review. | null | published | incremental | hard | [
"source-candidate",
"theoremdb"
] | null | null | null | null | [
{
"kind": "website",
"citation": "TheoremDB P8318",
"url": "https://theoremdb.org/statements/P8318",
"note": null,
"sort_order": 0
},
{
"kind": "formalization",
"citation": "Source formal statement (not locally verified)",
"url": "https://theoremdb.org/statements/P8318",
"not... | {
"source_record": "P8318",
"imported_at": "2026-09-08",
"status_as_of": "2026-09-08",
"evidence_label": "Source candidate · not reviewed",
"note": "Full public directory statement. Source lane: backlog; source resolution: open. Source verification claims are attributed only; no local Lean verification has be... | {"created_at":"2026-08-17T18:39:35+00:00","display_standard":{"editorial_review":"submitter_supplied","formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":8318,"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": "2026-09-08T03:33:03.410Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"3d7e780c-77bc-4450-8a23-99652840488a","format_issues":null,"number":6090} |
1.0.0 | problem:theoremdb-p8320 | problem | theoremdb-p8320 | De Giorgi's conjecture in dimension four | {
"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"
}
] | Consider the semilinear elliptic equation $\Delta u + u - u^3 = 0$ on $\mathbb{R}^n$, where $\Delta$ denotes the Laplacian. A function $u : \mathbb{R}^n \to \mathbb{R}$ is called a bounded classical solution if $u$ is twice continuously differentiable, there exists a constant $C$ such that $|u(x)| \leq C$ for all $x \i... | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P8320",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P8320",
"citation": null,
"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_four : (DeGiorgi_conclusion 4) | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem DeGiorgi_four : (DeGiorgi_conclusion 4)",
"target": null,
"source_url": "https://theoremdb.org/statements/P8320",
"format": "declaration-or-fragment",
"sha256": "256717ce8965fa75c86b1bf7935befa1bfcec315852bc68fe8d89ce282580a5c",
"license": "Select... | [] | TheoremDB P8320 · Candidate awaiting source review | null | Source-reported assessment, retrieved 8 September 2026. Frontier Science has not independently reviewed this status.
Open. No signed Lean verification is attached to this problem or its selected resolution.
This record is in the source backlog; its novelty and unresolved status have not passed source review. | null | published | incremental | hard | [
"source-candidate",
"theoremdb"
] | null | null | null | null | [
{
"kind": "website",
"citation": "TheoremDB P8320",
"url": "https://theoremdb.org/statements/P8320",
"note": null,
"sort_order": 0
},
{
"kind": "formalization",
"citation": "Source formal statement (not locally verified)",
"url": "https://theoremdb.org/statements/P8320",
"not... | {
"source_record": "P8320",
"imported_at": "2026-09-08",
"status_as_of": "2026-09-08",
"evidence_label": "Source candidate · not reviewed",
"note": "Full public directory statement. Source lane: backlog; source resolution: open. Source verification claims are attributed only; no local Lean verification has be... | {"created_at":"2026-08-17T18:39:35+00:00","display_standard":{"editorial_review":"submitter_supplied","formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":8320,"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": "2026-09-08T03:32:52.910Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"c199360b-ee45-4f43-a090-ca371189a7ad","format_issues":null,"number":3962} |
1.0.0 | problem:theoremdb-p8322 | problem | theoremdb-p8322 | Growth rate of the additive convolution of powerful numbers | {
"main": "math",
"sub": "number theory",
"slug": "number-theory",
"name": "Number theory",
"original_discipline": "mathematics"
} | [] | [
{
"code": "11",
"basis": "community-map",
"evidence": "Inferred from community field: number-theory"
}
] | A positive integer is called powerful if for every prime $p$ dividing it, $p^2$ also divides it. Let $A$ denote the set of powerful numbers, and let $1_A$ denote its indicator function. Define the additive convolution $(1_A \ast 1_A)(n) = \sum_{a + b = n} 1_A(a) \cdot 1_A(b)$, counting the number of representations of ... | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P8322",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P8322",
"citation": null,
"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_943 : answer(sorry) ↔
∃ (o : ℕ → ℝ), o =o[atTop] (1 : ℕ → ℝ) ∧ ∀ᶠ n in atTop, (sumRep Powerful n) ≤ (n : ℝ)^(o n) | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem erdos_943 : answer(sorry) ↔\n ∃ (o : ℕ → ℝ), o =o[atTop] (1 : ℕ → ℝ) ∧ ∀ᶠ n in atTop, (sumRep Powerful n) ≤ (n : ℝ)^(o n)",
"target": null,
"source_url": "https://theoremdb.org/statements/P8322",
"format": "declaration-or-fragment",
"sha256": "a93d... | [] | TheoremDB P8322 · Candidate awaiting source review | null | Source-reported assessment, retrieved 8 September 2026. Frontier Science has not independently reviewed this status.
Open. No signed Lean verification is attached to this problem or its selected resolution.
This record is in the source backlog; its novelty and unresolved status have not passed source review. | null | published | incremental | hard | [
"source-candidate",
"theoremdb"
] | null | null | null | null | [
{
"kind": "website",
"citation": "TheoremDB P8322",
"url": "https://theoremdb.org/statements/P8322",
"note": null,
"sort_order": 0
},
{
"kind": "formalization",
"citation": "Source formal statement (not locally verified)",
"url": "https://theoremdb.org/statements/P8322",
"not... | {
"source_record": "P8322",
"imported_at": "2026-09-08",
"status_as_of": "2026-09-08",
"evidence_label": "Source candidate · not reviewed",
"note": "Full public directory statement. Source lane: backlog; source resolution: open. Source verification claims are attributed only; no local Lean verification has be... | {"created_at":"2026-08-17T18:39:38+00:00","display_standard":{"editorial_review":"submitter_supplied","formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":8322,"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": "2026-09-08T03:32:59.506Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"9df69437-b237-46e4-8d61-6768e9c423d8","format_issues":null,"number":5410} |
1.0.0 | problem:theoremdb-p8324 | problem | theoremdb-p8324 | Boundedness of the sum of reciprocals of non-dividing primes of central 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"
}
] | For each positive integer $n$, define
$$
f(n) \;=\; \sum_{\substack{p \leq n \\ p \nmid \binom{2n}{n}}} \frac{1}{p},
$$
where the sum ranges over all primes $p$ not exceeding $n$ that do not divide the central binomial coefficient $\binom{2n}{n}$. Does there exist an absolute constant $C > 0$ such that $f(n) \leq C... | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P8324",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P8324",
"citation": null,
"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_377 : answer(sorry) ↔
∃ C > (0 : ℝ), ∀ (n : ℕ), sumInvPrimesNotDvdCentralBinom n ≤ C | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem erdos_377 : answer(sorry) ↔\n ∃ C > (0 : ℝ), ∀ (n : ℕ), sumInvPrimesNotDvdCentralBinom n ≤ C",
"target": null,
"source_url": "https://theoremdb.org/statements/P8324",
"format": "declaration-or-fragment",
"sha256": "7e166f3a088ba458d6e6066031cf4f6ec... | [] | TheoremDB P8324 · Candidate awaiting source review | null | Source-reported assessment, retrieved 8 September 2026. Frontier Science has not independently reviewed this status.
Open. No signed Lean verification is attached to this problem or its selected resolution.
This record is in the source backlog; its novelty and unresolved status have not passed source review. | null | published | incremental | hard | [
"source-candidate",
"theoremdb"
] | null | null | null | null | [
{
"kind": "website",
"citation": "TheoremDB P8324",
"url": "https://theoremdb.org/statements/P8324",
"note": null,
"sort_order": 0
},
{
"kind": "formalization",
"citation": "Source formal statement (not locally verified)",
"url": "https://theoremdb.org/statements/P8324",
"not... | {
"source_record": "P8324",
"imported_at": "2026-09-08",
"status_as_of": "2026-09-08",
"evidence_label": "Source candidate · not reviewed",
"note": "Full public directory statement. Source lane: backlog; source resolution: open. Source verification claims are attributed only; no local Lean verification has be... | {"created_at":"2026-08-17T18:39:39+00:00","display_standard":{"editorial_review":"submitter_supplied","formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":8324,"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": "2026-09-08T03:32:50.739Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"2d67394e-b3dc-4533-bab3-9f1f4e40acf1","format_issues":null,"number":3482} |
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