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1.0.0 | problem:theoremdb-p9136 | problem | theoremdb-p9136 | Polynomial logarithmic bound for the Sylvester–Schur 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 $k$, let $f(k)$ denote the smallest positive integer $n$ such that every set of $n$ consecutive integers greater than $k$ contains at least one integer that is not $(k+1)$-smooth, meaning it is divisible by some prime $p > k$. A number is $(k+1)$-smooth if all of its prime factors are at most ... | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P9136",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P9136",
"citation": null,
"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_961 : answer(sorry) ↔ ∃ C : ℕ, ∀ᶠ k : ℕ in atTop, f k < log k ^ C | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem erdos_961 : answer(sorry) ↔ ∃ C : ℕ, ∀ᶠ k : ℕ in atTop, f k < log k ^ C",
"target": null,
"source_url": "https://theoremdb.org/statements/P9136",
"format": "declaration-or-fragment",
"sha256": "bcc8c08ce039947273136346c8bceddd7959b517e3315341c281cac19... | [] | TheoremDB P9136 · Candidate awaiting source review | null | Source-reported assessment, retrieved 8 September 2026. Frontier Science has not independently reviewed this status.
Open. No signed Lean verification is attached to this problem or its selected resolution.
This record is in the source backlog; its novelty and unresolved status have not passed source review. | null | published | incremental | hard | [
"source-candidate",
"theoremdb"
] | null | null | null | null | [
{
"kind": "website",
"citation": "TheoremDB P9136",
"url": "https://theoremdb.org/statements/P9136",
"note": null,
"sort_order": 0
},
{
"kind": "formalization",
"citation": "Source formal statement (not locally verified)",
"url": "https://theoremdb.org/statements/P9136",
"not... | {
"source_record": "P9136",
"imported_at": "2026-09-08",
"status_as_of": "2026-09-08",
"evidence_label": "Source candidate · not reviewed",
"note": "Full public directory statement. Source lane: backlog; source resolution: open. Source verification claims are attributed only; no local Lean verification has be... | {"created_at":"2026-08-17T19:47:46+00:00","display_standard":{"formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":9136,"public_status":{"evidence":{"hasFormalization":false,"hasInformalProof":false,"hasRecordedWork":false,"solutionRecord":null},"lean":{"kernelAccepted":false,"label":"","reaso... | [
"formalization-contains-placeholder",
"formalization-not-compiled-during-export",
"current-status-unassessed",
"source-full-text-not-present"
] | [] | {
"created_at": "2026-09-08T03:32:15.935Z",
"updated_at": "2026-09-08T03:32:15.935Z",
"published_at": "2026-09-08T03:33:04.894Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"5d209663-061d-4c3c-b58b-b5222174217c","format_issues":null,"number":6409} |
1.0.0 | problem:theoremdb-p9138 | problem | theoremdb-p9138 | Natural density of indices with non-increasing consecutive prime gaps | {
"main": "math",
"sub": "number theory",
"slug": "number-theory",
"name": "Number theory",
"original_discipline": "mathematics"
} | [] | [
{
"code": "11",
"basis": "community-map",
"evidence": "Inferred from community field: number-theory"
}
] | For each positive integer $n$, let $p_n$ denote the $n$-th prime number, and define the $n$-th prime gap as $g_n = p_{n+1} - p_n$. Determine whether the set
$$
\{ n \in \mathbb{N} : g_{n+1} \leq g_n \}
$$
has natural density equal to $\frac{1}{2}$. | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P9138",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P9138",
"citation": null,
"doi": null,
"arxiv_id": null,
"license": "Selected packets label their content CC0-1.0; original references retain their own terms",
"notes": "Selected problem... | theorem erdos_218.variants.ge : {n | primeGap (n + 1) ≤ primeGap n}.HasDensity <| 1 / 2 | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem erdos_218.variants.ge : {n | primeGap (n + 1) ≤ primeGap n}.HasDensity <| 1 / 2",
"target": null,
"source_url": "https://theoremdb.org/statements/P9138",
"format": "declaration-or-fragment",
"sha256": "cc5444b8b7db739b982b63b814441049d76c440402379ee06... | [] | TheoremDB P9138 · Candidate awaiting source review | null | Source-reported assessment, retrieved 8 September 2026. Frontier Science has not independently reviewed this status.
Open. No signed Lean verification is attached to this problem or its selected resolution.
This record is in the source backlog; its novelty and unresolved status have not passed source review. | null | published | incremental | hard | [
"source-candidate",
"theoremdb"
] | null | null | null | null | [
{
"kind": "website",
"citation": "TheoremDB P9138",
"url": "https://theoremdb.org/statements/P9138",
"note": null,
"sort_order": 0
},
{
"kind": "formalization",
"citation": "Source formal statement (not locally verified)",
"url": "https://theoremdb.org/statements/P9138",
"not... | {
"source_record": "P9138",
"imported_at": "2026-09-08",
"status_as_of": "2026-09-08",
"evidence_label": "Source candidate · not reviewed",
"note": "Full public directory statement. Source lane: backlog; source resolution: open. Source verification claims are attributed only; no local Lean verification has be... | {"created_at":"2026-08-17T19:47:46+00:00","display_standard":{"editorial_review":"submitter_supplied","formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":9138,"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": "2026-09-08T03:33:04.194Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"80970d54-ba22-4745-b9aa-ba582aba5d43","format_issues":null,"number":6259} |
1.0.0 | problem:theoremdb-p9140 | problem | theoremdb-p9140 | Pillai's conjecture on finiteness of solutions to generalized Catalan-type equations | {
"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 $a$, $b$, and $c$ be positive integers. Consider the Diophantine equation
$$
a x^n - b y^m = c
$$
in positive integers $x$, $y$, $m$, $n$ with $x > 1$, $y > 1$, $m > 1$, $n > 1$, and $(m, n) \neq (2, 2)$. Prove that the set of all solutions $(x, y, m, n)$ is finite. | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P9140",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P9140",
"citation": null,
"doi": null,
"arxiv_id": null,
"license": "Selected packets label their content CC0-1.0; original references retain their own terms",
"notes": "Selected problem... | theorem pillais_conjecture (a b c : ℕ) (ha : 0 < a) (hb : 0 < b) (hc : 0 < c) :
{ (x, y, m, n) : (ℕ × ℕ × ℕ × ℕ) |
1 < x ∧ 1 < y ∧ 1 < m ∧ 1 < n ∧ (m, n) ≠ (2, 2) ∧
a * x^n - b * y^m = c }.Finite | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem pillais_conjecture (a b c : ℕ) (ha : 0 < a) (hb : 0 < b) (hc : 0 < c) :\n { (x, y, m, n) : (ℕ × ℕ × ℕ × ℕ) |\n 1 < x ∧ 1 < y ∧ 1 < m ∧ 1 < n ∧ (m, n) ≠ (2, 2) ∧\n a * x^n - b * y^m = c }.Finite",
"target": null,
"source_url": "https://theoremdb.o... | [] | TheoremDB P9140 · Candidate awaiting source review | null | Source-reported assessment, retrieved 8 September 2026. Frontier Science has not independently reviewed this status.
Open. No signed Lean verification is attached to this problem or its selected resolution.
This record is in the source backlog; its novelty and unresolved status have not passed source review. | null | published | incremental | hard | [
"source-candidate",
"theoremdb"
] | null | null | null | null | [
{
"kind": "website",
"citation": "TheoremDB P9140",
"url": "https://theoremdb.org/statements/P9140",
"note": null,
"sort_order": 0
},
{
"kind": "formalization",
"citation": "Source formal statement (not locally verified)",
"url": "https://theoremdb.org/statements/P9140",
"not... | {
"source_record": "P9140",
"imported_at": "2026-09-08",
"status_as_of": "2026-09-08",
"evidence_label": "Source candidate · not reviewed",
"note": "Full public directory statement. Source lane: backlog; source resolution: open. Source verification claims are attributed only; no local Lean verification has be... | {"created_at":"2026-08-17T19:47:47+00:00","display_standard":{"editorial_review":"submitter_supplied","formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":9140,"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": "2026-09-08T03:33:04.837Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"09813194-ac6c-492f-8c32-3c4b3af7909f","format_issues":null,"number":6397} |
1.0.0 | problem:theoremdb-p9142 | problem | theoremdb-p9142 | Margulis's conjecture on compact diagonal orbits in homogeneous spaces | {
"main": "math",
"sub": "number theory",
"slug": "number-theory",
"name": "Number theory",
"original_discipline": "mathematics"
} | [] | [
{
"code": "11",
"basis": "community-map",
"evidence": "Inferred from community field: number-theory"
}
] | Let $n$ be a natural number with $n \geq 3$. Let $\mathrm{SL}(n, \mathbb{R})$ denote the special linear group of $n \times n$ real matrices with determinant 1, and let $\mathrm{SL}(n, \mathbb{Z})$ be its subgroup of integer matrices. Consider the homogeneous space $\mathrm{SL}(n, \mathbb{R}) / \mathrm{SL}(n, \mathbb{Z}... | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P9142",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P9142",
"citation": null,
"doi": null,
"arxiv_id": null,
"license": "Selected packets label their content CC0-1.0; original references retain their own terms",
"notes": "Selected problem... | theorem conjecture_1_1 {n : ℕ} (hn : 3 ≤ n)
(g : SL(n, ℝ) ⧸ Subgroup.map (map (Int.castRingHom ℝ)) ⊤)
(hg : IsCompact <| closure (MulAction.orbit (diagonalSubgroup (Fin n) ℝ) g)) :
IsClosed <| MulAction.orbit (diagonalSubgroup (Fin n) ℝ) g | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem conjecture_1_1 {n : ℕ} (hn : 3 ≤ n)\n (g : SL(n, ℝ) ⧸ Subgroup.map (map (Int.castRingHom ℝ)) ⊤)\n (hg : IsCompact <| closure (MulAction.orbit (diagonalSubgroup (Fin n) ℝ) g)) :\n IsClosed <| MulAction.orbit (diagonalSubgroup (Fin n) ℝ) g",
"target": null... | [] | TheoremDB P9142 · Candidate awaiting source review | null | Source-reported assessment, retrieved 8 September 2026. Frontier Science has not independently reviewed this status.
Open. No signed Lean verification is attached to this problem or its selected resolution.
This record is in the source backlog; its novelty and unresolved status have not passed source review. | null | published | incremental | hard | [
"source-candidate",
"theoremdb"
] | null | null | null | null | [
{
"kind": "website",
"citation": "TheoremDB P9142",
"url": "https://theoremdb.org/statements/P9142",
"note": null,
"sort_order": 0
},
{
"kind": "formalization",
"citation": "Source formal statement (not locally verified)",
"url": "https://theoremdb.org/statements/P9142",
"not... | {
"source_record": "P9142",
"imported_at": "2026-09-08",
"status_as_of": "2026-09-08",
"evidence_label": "Source candidate · not reviewed",
"note": "Full public directory statement. Source lane: backlog; source resolution: open. Source verification claims are attributed only; no local Lean verification has be... | {"created_at":"2026-08-17T19:47:48+00:00","display_standard":{"formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":9142,"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": "2026-09-08T03:33:03.618Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"a1de509e-1b1d-43e2-9936-766595a24829","format_issues":null,"number":6133} |
1.0.0 | problem:theoremdb-p9144 | problem | theoremdb-p9144 | Zhi-Wei Sun's Conjecture on Prime-Valued 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 every integer $n > 1$, do there exist positive integers $x$ and $y$ with $n = x + y$ such that both $x + ny$ and $x^2 + ny^2$ are prime? | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P9144",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P9144",
"citation": null,
"doi": null,
"arxiv_id": null,
"license": "Selected packets label their content CC0-1.0; original references retain their own terms",
"notes": "Selected problem... | theorem conjecture (n : ℕ) (hn : 1 < n) : A n | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem conjecture (n : ℕ) (hn : 1 < n) : A n",
"target": null,
"source_url": "https://theoremdb.org/statements/P9144",
"format": "declaration-or-fragment",
"sha256": "9c3552ea3b574a89c5f6b8cd5a92ea437c055998af76a271622edaaf8ee3d6e7",
"license": "Selected... | [] | TheoremDB P9144 · Candidate awaiting source review | null | Source-reported assessment, retrieved 8 September 2026. Frontier Science has not independently reviewed this status.
Open. No signed Lean verification is attached to this problem or its selected resolution.
This record is in the source backlog; its novelty and unresolved status have not passed source review. | null | published | incremental | hard | [
"source-candidate",
"theoremdb"
] | null | null | null | null | [
{
"kind": "website",
"citation": "TheoremDB P9144",
"url": "https://theoremdb.org/statements/P9144",
"note": null,
"sort_order": 0
},
{
"kind": "formalization",
"citation": "Source formal statement (not locally verified)",
"url": "https://theoremdb.org/statements/P9144",
"not... | {
"source_record": "P9144",
"imported_at": "2026-09-08",
"status_as_of": "2026-09-08",
"evidence_label": "Source candidate · not reviewed",
"note": "Full public directory statement. Source lane: backlog; source resolution: open. Source verification claims are attributed only; no local Lean verification has be... | {"created_at":"2026-08-17T19:47:48+00:00","display_standard":{"formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":9144,"public_status":{"evidence":{"hasFormalization":false,"hasInformalProof":false,"hasRecordedWork":false,"solutionRecord":null},"lean":{"kernelAccepted":false,"label":"","reaso... | [
"formalization-not-compiled-during-export",
"current-status-unassessed",
"source-full-text-not-present"
] | [] | {
"created_at": "2026-09-08T03:32:15.935Z",
"updated_at": "2026-09-08T03:32:15.935Z",
"published_at": "2026-09-08T03:33:07.649Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"9f1a5d37-a0b4-4950-8e62-33966899818e","format_issues":null,"number":7005} |
1.0.0 | problem:theoremdb-p9146 | problem | theoremdb-p9146 | Asymptotic order of the minimum size of a universal arithmetic progression cover | {
"main": "math",
"sub": "combinatorics",
"slug": "combinatorics",
"name": "Combinatorics",
"original_discipline": "mathematics"
} | [] | [
{
"code": "05",
"basis": "community-map",
"evidence": "Inferred from community field: combinatorics"
}
] | For a fixed integer $k \ge 1$ and a positive integer $N$, let $m(N,k)$ denote the minimum cardinality of a set $A \subseteq \mathbb{N}$ such that for every common difference $d \in \{1, 2, \ldots, N\}$, the set $A$ contains an arithmetic progression of length $k$ with common difference $d$. Determine a function $f: \ma... | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P9146",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P9146",
"citation": null,
"doi": null,
"arxiv_id": null,
"license": "Selected packets label their content CC0-1.0; original references retain their own terms",
"notes": "Selected problem... | theorem green_37_theta (k : ℕ) :
(fun N ↦ (m N k : ℝ)) =Θ[atTop] (answer(sorry) : ℕ → ℝ) | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem green_37_theta (k : ℕ) :\n (fun N ↦ (m N k : ℝ)) =Θ[atTop] (answer(sorry) : ℕ → ℝ)",
"target": null,
"source_url": "https://theoremdb.org/statements/P9146",
"format": "declaration-or-fragment",
"sha256": "c23cf35e1b808c320d328c94459e179bcc525f3ba04... | [] | TheoremDB P9146 · Candidate awaiting source review | null | Source-reported assessment, retrieved 8 September 2026. Frontier Science has not independently reviewed this status.
Open. No signed Lean verification is attached to this problem or its selected resolution.
This record is in the source backlog; its novelty and unresolved status have not passed source review. | null | published | incremental | hard | [
"source-candidate",
"theoremdb"
] | null | null | null | null | [
{
"kind": "website",
"citation": "TheoremDB P9146",
"url": "https://theoremdb.org/statements/P9146",
"note": null,
"sort_order": 0
},
{
"kind": "formalization",
"citation": "Source formal statement (not locally verified)",
"url": "https://theoremdb.org/statements/P9146",
"not... | {
"source_record": "P9146",
"imported_at": "2026-09-08",
"status_as_of": "2026-09-08",
"evidence_label": "Source candidate · not reviewed",
"note": "Full public directory statement. Source lane: backlog; source resolution: open. Source verification claims are attributed only; no local Lean verification has be... | {"created_at":"2026-08-17T19:47:49+00:00","display_standard":{"editorial_review":"submitter_supplied","formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":9146,"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": "2026-09-08T03:32:50.368Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"05b8ceff-b88b-4369-96ff-94d1f9a3eff4","format_issues":null,"number":3405} |
1.0.0 | problem:theoremdb-p9148 | problem | theoremdb-p9148 | Brocard's Conjecture on Prime Gaps Between Consecutive Prime Squares | {
"main": "math",
"sub": "number theory",
"slug": "number-theory",
"name": "Number theory",
"original_discipline": "mathematics"
} | [] | [
{
"code": "11",
"basis": "community-map",
"evidence": "Inferred from community field: number-theory"
}
] | For every natural number $n \geq 1$, let $p_n$ denote the $n$-th prime number, and let $p_{n+1}$ denote the $(n+1)$-th prime number. Consider the open interval $(p_n^2, p_{n+1}^2)$ of natural numbers strictly between $p_n^2$ and $p_{n+1}^2$. Let $\pi(x)$ denote the prime-counting function, which gives the number of pri... | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P9148",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P9148",
"citation": null,
"doi": null,
"arxiv_id": null,
"license": "Selected packets label their content CC0-1.0; original references retain their own terms",
"notes": "Selected problem... | theorem brocard_conjecture (n : ℕ) (hn : 1 ≤ n) :
letI prev := n.nth Nat.Prime;
letI next := (n+1).nth Nat.Prime;
4 ≤ ((Ioo (prev^2) (next^2)).filter Nat.Prime).card | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem brocard_conjecture (n : ℕ) (hn : 1 ≤ n) :\n letI prev := n.nth Nat.Prime;\n letI next := (n+1).nth Nat.Prime;\n 4 ≤ ((Ioo (prev^2) (next^2)).filter Nat.Prime).card",
"target": null,
"source_url": "https://theoremdb.org/statements/P9148",
"format"... | [] | TheoremDB P9148 · Candidate awaiting source review | null | Source-reported assessment, retrieved 8 September 2026. Frontier Science has not independently reviewed this status.
Open. No signed Lean verification is attached to this problem or its selected resolution.
This record is in the source backlog; its novelty and unresolved status have not passed source review. | null | published | incremental | hard | [
"source-candidate",
"theoremdb"
] | null | null | null | null | [
{
"kind": "website",
"citation": "TheoremDB P9148",
"url": "https://theoremdb.org/statements/P9148",
"note": null,
"sort_order": 0
},
{
"kind": "formalization",
"citation": "Source formal statement (not locally verified)",
"url": "https://theoremdb.org/statements/P9148",
"not... | {
"source_record": "P9148",
"imported_at": "2026-09-08",
"status_as_of": "2026-09-08",
"evidence_label": "Source candidate · not reviewed",
"note": "Full public directory statement. Source lane: backlog; source resolution: open. Source verification claims are attributed only; no local Lean verification has be... | {"created_at":"2026-08-17T19:47:49+00:00","display_standard":{"editorial_review":"submitter_supplied","formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":9148,"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": "2026-09-08T03:32:50.782Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"cec3d41c-5bf9-496d-a1b2-d4768b62f853","format_issues":null,"number":3491} |
1.0.0 | problem:theoremdb-p9150 | problem | theoremdb-p9150 | Asymptotic upper bound for minimum arithmetic-progression covering sets | {
"main": "math",
"sub": "combinatorics",
"slug": "combinatorics",
"name": "Combinatorics",
"original_discipline": "mathematics"
} | [] | [
{
"code": "05",
"basis": "community-map",
"evidence": "Inferred from community field: combinatorics"
}
] | For each pair of natural numbers $N$ and $k$, let $m(N,k)$ denote the minimum cardinality of a set $A \subseteq \mathbb{N}$ such that for every integer $d$ with $1 \leq d \leq N$, the set $A$ contains an arithmetic progression of length $k$ with common difference $d$. A set $A$ is said to contain an arithmetic progress... | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P9150",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P9150",
"citation": null,
"doi": null,
"arxiv_id": null,
"license": "Selected packets label their content CC0-1.0; original references retain their own terms",
"notes": "Selected problem... | theorem green_37_bigO (k : ℕ) :
(fun N ↦ (m N k : ℝ)) =O[atTop] (answer(sorry) : ℕ → ℝ) | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem green_37_bigO (k : ℕ) :\n (fun N ↦ (m N k : ℝ)) =O[atTop] (answer(sorry) : ℕ → ℝ)",
"target": null,
"source_url": "https://theoremdb.org/statements/P9150",
"format": "declaration-or-fragment",
"sha256": "31128cde2f511edd9d65dc5d83599f95d0ec3b90567c... | [] | TheoremDB P9150 · Candidate awaiting source review | null | Source-reported assessment, retrieved 8 September 2026. Frontier Science has not independently reviewed this status.
Open. No signed Lean verification is attached to this problem or its selected resolution.
This record is in the source backlog; its novelty and unresolved status have not passed source review. | null | published | incremental | hard | [
"source-candidate",
"theoremdb"
] | null | null | null | null | [
{
"kind": "website",
"citation": "TheoremDB P9150",
"url": "https://theoremdb.org/statements/P9150",
"note": null,
"sort_order": 0
},
{
"kind": "formalization",
"citation": "Source formal statement (not locally verified)",
"url": "https://theoremdb.org/statements/P9150",
"not... | {
"source_record": "P9150",
"imported_at": "2026-09-08",
"status_as_of": "2026-09-08",
"evidence_label": "Source candidate · not reviewed",
"note": "Full public directory statement. Source lane: backlog; source resolution: open. Source verification claims are attributed only; no local Lean verification has be... | {"created_at":"2026-08-17T19:47:49+00:00","display_standard":{"editorial_review":"submitter_supplied","formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":9150,"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": "2026-09-08T03:32:50.428Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"d56fe855-1745-4973-930b-b2f1188f6377","format_issues":null,"number":3418} |
1.0.0 | problem:theoremdb-p9152 | problem | theoremdb-p9152 | Greatest common divisor of factorial and left 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 every integer $n > 2$, define the left factorial of $n$ by
$$
!n = \sum_{m=0}^{n-1} m! = 0! + 1! + 2! + \cdots + (n-1)!.
$$
Determine whether $\gcd(n!, !n) = 2$. | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P9152",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P9152",
"citation": null,
"doi": null,
"arxiv_id": null,
"license": "Selected packets label their content CC0-1.0; original references retain their own terms",
"notes": "Selected problem... | theorem kurepa_conjecture.variants.gcd (n : ℕ) : 2 < n → (n !).gcd (! n) = 2 | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem kurepa_conjecture.variants.gcd (n : ℕ) : 2 < n → (n !).gcd (! n) = 2",
"target": null,
"source_url": "https://theoremdb.org/statements/P9152",
"format": "declaration-or-fragment",
"sha256": "ec0a80bfc8faf01072cfecd51965a1fbfc2d852a66d2d8fc4ce19c5ac896... | [] | TheoremDB P9152 · Candidate awaiting source review | null | Source-reported assessment, retrieved 8 September 2026. Frontier Science has not independently reviewed this status.
Open. No signed Lean verification is attached to this problem or its selected resolution.
This record is in the source backlog; its novelty and unresolved status have not passed source review. | null | published | incremental | hard | [
"source-candidate",
"theoremdb"
] | null | null | null | null | [
{
"kind": "website",
"citation": "TheoremDB P9152",
"url": "https://theoremdb.org/statements/P9152",
"note": null,
"sort_order": 0
},
{
"kind": "formalization",
"citation": "Source formal statement (not locally verified)",
"url": "https://theoremdb.org/statements/P9152",
"not... | {
"source_record": "P9152",
"imported_at": "2026-09-08",
"status_as_of": "2026-09-08",
"evidence_label": "Source candidate · not reviewed",
"note": "Full public directory statement. Source lane: backlog; source resolution: open. Source verification claims are attributed only; no local Lean verification has be... | {"created_at":"2026-08-17T19:47:50+00:00","display_standard":{"editorial_review":"submitter_supplied","formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":9152,"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": "2026-09-08T03:32:59.406Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"41990abd-56fc-4a57-894d-1c47368cdad0","format_issues":null,"number":5388} |
1.0.0 | problem:theoremdb-p9154 | problem | theoremdb-p9154 | Tunnell's even converse for squarefree congruent 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 $n$ be a squarefree even natural number. Define $C_n$ to be the set of integer triples $(x,y,z)$ such that $n = 8x^2 + 2y^2 + 64z^2$, and define $D_n$ to be the set of integer triples $(x,y,z)$ such that $n = 8x^2 + 2y^2 + 16z^2$. A natural number $n$ is called a congruent number if there exists a right triangle wi... | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P9154",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P9154",
"citation": null,
"doi": null,
"arxiv_id": null,
"license": "Selected packets label their content CC0-1.0; original references retain their own terms",
"notes": "Selected problem... | theorem Tunnell_even_converse (n : ℕ) (hsqf : Squarefree n) (heven : Even n) :
2 * (C n).ncard = (D n).ncard → congruentNumber n | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem Tunnell_even_converse (n : ℕ) (hsqf : Squarefree n) (heven : Even n) :\n 2 * (C n).ncard = (D n).ncard → congruentNumber n",
"target": null,
"source_url": "https://theoremdb.org/statements/P9154",
"format": "declaration-or-fragment",
"sha256": "2b6... | [] | TheoremDB P9154 · Candidate awaiting source review | null | Source-reported assessment, retrieved 8 September 2026. Frontier Science has not independently reviewed this status.
Open. No signed Lean verification is attached to this problem or its selected resolution.
This record is in the source backlog; its novelty and unresolved status have not passed source review. | null | published | incremental | hard | [
"source-candidate",
"theoremdb"
] | null | null | null | null | [
{
"kind": "website",
"citation": "TheoremDB P9154",
"url": "https://theoremdb.org/statements/P9154",
"note": null,
"sort_order": 0
},
{
"kind": "formalization",
"citation": "Source formal statement (not locally verified)",
"url": "https://theoremdb.org/statements/P9154",
"not... | {
"source_record": "P9154",
"imported_at": "2026-09-08",
"status_as_of": "2026-09-08",
"evidence_label": "Source candidate · not reviewed",
"note": "Full public directory statement. Source lane: backlog; source resolution: open. Source verification claims are attributed only; no local Lean verification has be... | {"created_at":"2026-08-17T19:47:50+00:00","display_standard":{"editorial_review":"submitter_supplied","formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":9154,"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": "2026-09-08T03:33:07.048Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"2e33034f-54d3-4bfa-9310-202ee05746b7","format_issues":null,"number":6874} |
1.0.0 | problem:theoremdb-p9156 | problem | theoremdb-p9156 | The tenth Dedekind number | {
"main": "math",
"sub": "combinatorics",
"slug": "combinatorics",
"name": "Combinatorics",
"original_discipline": "mathematics"
} | [] | [
{
"code": "05",
"basis": "community-map",
"evidence": "Inferred from community field: combinatorics"
}
] | For each non-negative integer $n$, let $M(n)$ denote the number of monotone Boolean functions on $n$ variables, i.e., functions $f : \{0,1\}^n \to \{0,1\}$ such that $f(x) \leq f(y)$ whenever $x_i \leq y_i$ for all $i = 1, \ldots, n$. Equivalently, $M(n)$ is the number of antichains (also called Sperner families) in th... | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P9156",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P9156",
"citation": null,
"doi": null,
"arxiv_id": null,
"license": "Selected packets label their content CC0-1.0; original references retain their own terms",
"notes": "Selected problem... | theorem M_eq : M = answer(sorry) | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem M_eq : M = answer(sorry)",
"target": null,
"source_url": "https://theoremdb.org/statements/P9156",
"format": "declaration-or-fragment",
"sha256": "ff77ae7a16fb5904c342d419fbe43044df28138960e889f813e7e317a7ff065c",
"license": "Selected packets labe... | [] | TheoremDB P9156 · Candidate awaiting source review | null | Source-reported assessment, retrieved 8 September 2026. Frontier Science has not independently reviewed this status.
Open. No signed Lean verification is attached to this problem or its selected resolution.
This record is in the source backlog; its novelty and unresolved status have not passed source review. | null | published | incremental | hard | [
"source-candidate",
"theoremdb"
] | null | null | null | null | [
{
"kind": "website",
"citation": "TheoremDB P9156",
"url": "https://theoremdb.org/statements/P9156",
"note": null,
"sort_order": 0
},
{
"kind": "formalization",
"citation": "Source formal statement (not locally verified)",
"url": "https://theoremdb.org/statements/P9156",
"not... | {
"source_record": "P9156",
"imported_at": "2026-09-08",
"status_as_of": "2026-09-08",
"evidence_label": "Source candidate · not reviewed",
"note": "Full public directory statement. Source lane: backlog; source resolution: open. Source verification claims are attributed only; no local Lean verification has be... | {"created_at":"2026-08-17T19:47:51+00:00","display_standard":{"editorial_review":"submitter_supplied","formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":9156,"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": "2026-09-08T03:33:06.808Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"bb6f0957-7b95-4582-b2f5-ca7cd2266e9f","format_issues":null,"number":6822} |
1.0.0 | problem:theoremdb-p9158 | problem | theoremdb-p9158 | Erdős Problem 414 on the Orbit Intersection of Iterated Divisor-Sum Functions | {
"main": "math",
"sub": "number theory",
"slug": "number-theory",
"name": "Number theory",
"original_discipline": "mathematics"
} | [] | [
{
"code": "11",
"basis": "community-map",
"evidence": "Inferred from community field: number-theory"
}
] | Define the function $h : \mathbb{N} \to \mathbb{N}$ by $h(n) = n + \tau(n)$, where $\tau(n)$ denotes the number of positive divisors of $n$. For $k \geq 1$, define the $k$-fold iterate $h_k$ by $h_1(n) = h(n)$ and $h_k(n) = h(h_{k-1}(n))$ for $k \geq 2$. For any positive integers $m$ and $n$, do there exist non-negativ... | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P9158",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P9158",
"citation": null,
"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_414 : answer(sorry) ↔ ∀ᵉ (m > 0) (n > 0), ∃ i j, h^[i] m = h^[j] n | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem erdos_414 : answer(sorry) ↔ ∀ᵉ (m > 0) (n > 0), ∃ i j, h^[i] m = h^[j] n",
"target": null,
"source_url": "https://theoremdb.org/statements/P9158",
"format": "declaration-or-fragment",
"sha256": "503515ec73bddc063e761cb970191f4c18696694ff036015665e4d2... | [] | TheoremDB P9158 · Candidate awaiting source review | null | Source-reported assessment, retrieved 8 September 2026. Frontier Science has not independently reviewed this status.
Open. No signed Lean verification is attached to this problem or its selected resolution.
This record is in the source backlog; its novelty and unresolved status have not passed source review. | null | published | incremental | hard | [
"source-candidate",
"theoremdb"
] | null | null | null | null | [
{
"kind": "website",
"citation": "TheoremDB P9158",
"url": "https://theoremdb.org/statements/P9158",
"note": null,
"sort_order": 0
},
{
"kind": "formalization",
"citation": "Source formal statement (not locally verified)",
"url": "https://theoremdb.org/statements/P9158",
"not... | {
"source_record": "P9158",
"imported_at": "2026-09-08",
"status_as_of": "2026-09-08",
"evidence_label": "Source candidate · not reviewed",
"note": "Full public directory statement. Source lane: backlog; source resolution: open. Source verification claims are attributed only; no local Lean verification has be... | {"created_at":"2026-08-17T19:47:52+00:00","display_standard":{"formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":9158,"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": "2026-09-08T03:32:55.452Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"9c54775c-fa17-4c5d-8ca3-9a5d60bd069c","format_issues":null,"number":4522} |
1.0.0 | problem:theoremdb-p9160 | problem | theoremdb-p9160 | Non-existence of monochromatic quantum graphs with ten vertices and nine colours | {
"main": "math",
"sub": "combinatorics",
"slug": "combinatorics",
"name": "Combinatorics",
"original_discipline": "mathematics"
} | [] | [
{
"code": "05",
"basis": "community-map",
"evidence": "Inferred from community field: combinatorics"
}
] | Let $N$ and $D$ be positive integers. A quantum graph with $N$ vertices and $D$ colours consists of a complete graph on $N$ vertices together with a weight function $W$ that assigns to every unordered pair of distinct vertices $\{i,j\}$ and every colour $c \in \{1,\dots,D\}$ a complex number $W(\{i,j\}, c)$. For each v... | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P9160",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P9160",
"citation": null,
"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_d9 :
answer(sorry) ↔
¬ ∃ W : WeightsN 10 9 ℂ, EqSystemN 10 9 W | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem eqSystem10_no_solution_d9 :\n answer(sorry) ↔\n ¬ ∃ W : WeightsN 10 9 ℂ, EqSystemN 10 9 W",
"target": null,
"source_url": "https://theoremdb.org/statements/P9160",
"format": "declaration-or-fragment",
"sha256": "b2f0d7a15353c19cd1ae47bac4cb436... | [] | TheoremDB P9160 · Candidate awaiting source review | null | Source-reported assessment, retrieved 8 September 2026. Frontier Science has not independently reviewed this status.
Open. No signed Lean verification is attached to this problem or its selected resolution.
This record is in the source backlog; its novelty and unresolved status have not passed source review. | null | published | incremental | hard | [
"source-candidate",
"theoremdb"
] | null | null | null | null | [
{
"kind": "website",
"citation": "TheoremDB P9160",
"url": "https://theoremdb.org/statements/P9160",
"note": null,
"sort_order": 0
},
{
"kind": "formalization",
"citation": "Source formal statement (not locally verified)",
"url": "https://theoremdb.org/statements/P9160",
"not... | {
"source_record": "P9160",
"imported_at": "2026-09-08",
"status_as_of": "2026-09-08",
"evidence_label": "Source candidate · not reviewed",
"note": "Full public directory statement. Source lane: backlog; source resolution: open. Source verification claims are attributed only; no local Lean verification has be... | {"created_at":"2026-08-17T19:47:53+00:00","display_standard":{"editorial_review":"submitter_supplied","formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":9160,"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": "2026-09-08T03:33:04.376Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"374e14cf-5f5f-41b5-9419-fc1d3926f08d","format_issues":null,"number":6297} |
1.0.0 | problem:theoremdb-p9162 | problem | theoremdb-p9162 | Erdős distance problem for convex polygons | {
"main": "math",
"sub": "geometry and topology",
"slug": "geometry-and-topology",
"name": "Geometry and topology",
"original_discipline": "mathematics"
} | [] | [] | Let $n \geq 3$ be a natural number and let $p_0, p_1, \ldots, p_{n-1}$ be $n$ distinct points in $\mathbb{R}^2$ that form a convex polygon. For each vertex $p_i$, let $D_i$ denote the set of distances from $p_i$ to the other vertices, that is, $D_i = \{\|p_i - p_j\| : j \neq i\}$. Does there exist a vertex $p_i$ such t... | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P9162",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P9162",
"citation": null,
"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_982 (n : ℕ) (hn : 3 ≤ n) (p : Fin n → ℝ²) (hp : Function.Injective p)
(hp' : EuclideanGeometry.IsConvexPolygon p) :
∃ (i : Fin n), { d : ℝ | ∃ j : Fin n, j ≠ i ∧ d = dist (p i) (p j) }.ncard ≥ n / 2 | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem erdos_982 (n : ℕ) (hn : 3 ≤ n) (p : Fin n → ℝ²) (hp : Function.Injective p)\n (hp' : EuclideanGeometry.IsConvexPolygon p) :\n ∃ (i : Fin n), { d : ℝ | ∃ j : Fin n, j ≠ i ∧ d = dist (p i) (p j) }.ncard ≥ n / 2",
"target": null,
"source_url": "https://the... | [] | TheoremDB P9162 · Candidate awaiting source review | null | Source-reported assessment, retrieved 8 September 2026. Frontier Science has not independently reviewed this status.
Open. No signed Lean verification is attached to this problem or its selected resolution.
This record is in the source backlog; its novelty and unresolved status have not passed source review. | null | published | incremental | hard | [
"source-candidate",
"theoremdb"
] | null | null | null | null | [
{
"kind": "website",
"citation": "TheoremDB P9162",
"url": "https://theoremdb.org/statements/P9162",
"note": null,
"sort_order": 0
},
{
"kind": "formalization",
"citation": "Source formal statement (not locally verified)",
"url": "https://theoremdb.org/statements/P9162",
"not... | {
"source_record": "P9162",
"imported_at": "2026-09-08",
"status_as_of": "2026-09-08",
"evidence_label": "Source candidate · not reviewed",
"note": "Full public directory statement. Source lane: backlog; source resolution: open. Source verification claims are attributed only; no local Lean verification has be... | {"created_at":"2026-08-17T19:47:53+00:00","display_standard":{"formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":9162,"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": "2026-09-08T03:32:54.802Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"59b29839-57a9-4b04-8e1e-be4795573646","format_issues":null,"number":4381} |
1.0.0 | problem:theoremdb-p9164 | problem | theoremdb-p9164 | Infinitude of Coprime Harmonic Numerator-Denominator Pairs | {
"main": "math",
"sub": "number theory",
"slug": "number-theory",
"name": "Number theory",
"original_discipline": "mathematics"
} | [] | [
{
"code": "11",
"basis": "community-map",
"evidence": "Inferred from community field: number-theory"
}
] | For each positive integer $n$, let $L_n$ denote the least common multiple of the set $\{1, 2, \ldots, n\}$, and let $a_n$ be the unique positive integer defined by the equation
$$
\sum_{k=1}^{n} \frac{1}{k} = \frac{a_n}{L_n}.
$$
Determine whether the set $\{n \in \mathbb{N} : \gcd(a_n, L_n) = 1\}$ is infinite. | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P9164",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P9164",
"citation": null,
"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.parts.i :
answer(sorry) ↔
{ n : ℕ | Nat.gcd (a n) (L n) = 1 }.Infinite | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem erdos_291.parts.i :\n answer(sorry) ↔\n { n : ℕ | Nat.gcd (a n) (L n) = 1 }.Infinite",
"target": null,
"source_url": "https://theoremdb.org/statements/P9164",
"format": "declaration-or-fragment",
"sha256": "2402ef7d9826f6c29f22aa4a2929a3f892db... | [] | TheoremDB P9164 · Candidate awaiting source review | null | Source-reported assessment, retrieved 8 September 2026. Frontier Science has not independently reviewed this status.
Open. No signed Lean verification is attached to this problem or its selected resolution.
This record is in the source backlog; its novelty and unresolved status have not passed source review. | null | published | incremental | hard | [
"source-candidate",
"theoremdb"
] | null | null | null | null | [
{
"kind": "website",
"citation": "TheoremDB P9164",
"url": "https://theoremdb.org/statements/P9164",
"note": null,
"sort_order": 0
},
{
"kind": "formalization",
"citation": "Source formal statement (not locally verified)",
"url": "https://theoremdb.org/statements/P9164",
"not... | {
"source_record": "P9164",
"imported_at": "2026-09-08",
"status_as_of": "2026-09-08",
"evidence_label": "Source candidate · not reviewed",
"note": "Full public directory statement. Source lane: backlog; source resolution: open. Source verification claims are attributed only; no local Lean verification has be... | {"created_at":"2026-08-17T19:47:53+00:00","display_standard":{"editorial_review":"submitter_supplied","formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":9164,"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": "2026-09-08T03:33:01.141Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"589f67ba-38ec-4f64-af8f-e9d9bb1c491b","format_issues":null,"number":5666} |
1.0.0 | problem:theoremdb-p9166 | problem | theoremdb-p9166 | Existence of an infinite abundancy club | {
"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 $m$ and $n$, define the abundancy index of $n$ as $\sigma(n)/n$, where $\sigma(n) = \sum_{d \mid n} d$ is the sum-of-divisors function. Two positive integers $m$ and $n$ are called \emph{friendly} if they have the same abundancy index, that is, $\sigma(m)/m = \sigma(n)/n$. A \emph{club} is an equi... | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P9166",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P9166",
"citation": null,
"doi": null,
"arxiv_id": null,
"license": "Selected packets label their content CC0-1.0; original references retain their own terms",
"notes": "Selected problem... | theorem infinite_club_exists :
answer(sorry) ↔ ∃ n, 0 < n ∧ {m : ℕ | Friendly m n}.Infinite | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem infinite_club_exists :\n answer(sorry) ↔ ∃ n, 0 < n ∧ {m : ℕ | Friendly m n}.Infinite",
"target": null,
"source_url": "https://theoremdb.org/statements/P9166",
"format": "declaration-or-fragment",
"sha256": "34937ee2d1715599561e23b1d530aa3d0571e3e6... | [] | TheoremDB P9166 · Candidate awaiting source review | null | Source-reported assessment, retrieved 8 September 2026. Frontier Science has not independently reviewed this status.
Open. No signed Lean verification is attached to this problem or its selected resolution.
This record is in the source backlog; its novelty and unresolved status have not passed source review. | null | published | incremental | hard | [
"source-candidate",
"theoremdb"
] | null | null | null | null | [
{
"kind": "website",
"citation": "TheoremDB P9166",
"url": "https://theoremdb.org/statements/P9166",
"note": null,
"sort_order": 0
},
{
"kind": "formalization",
"citation": "Source formal statement (not locally verified)",
"url": "https://theoremdb.org/statements/P9166",
"not... | {
"source_record": "P9166",
"imported_at": "2026-09-08",
"status_as_of": "2026-09-08",
"evidence_label": "Source candidate · not reviewed",
"note": "Full public directory statement. Source lane: backlog; source resolution: open. Source verification claims are attributed only; no local Lean verification has be... | {"created_at":"2026-08-17T19:47:54+00:00","display_standard":{"editorial_review":"submitter_supplied","formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":9166,"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": "2026-09-08T03:32:58.687Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"0fc66e76-ea60-42a0-88c3-930df7309292","format_issues":null,"number":5234} |
1.0.0 | problem:theoremdb-p9168 | problem | theoremdb-p9168 | Maximum Size of a Solution-Free Subset for a Linear Equation | {
"main": "math",
"sub": "combinatorics",
"slug": "combinatorics",
"name": "Combinatorics",
"original_discipline": "mathematics"
} | [] | [
{
"code": "05",
"basis": "community-map",
"evidence": "Inferred from community field: combinatorics"
}
] | For a positive integer $N$, let $[N]$ denote the set $\{1, 2, \ldots, N\}$. A finite set $A$ of natural numbers is called solution-free if there do not exist distinct elements $x, y, z, w \in A$ such that $x + 3y = 2z + 2w$. Let $f(N)$ denote the maximum cardinality of a solution-free subset of $[N]$. Determine the exa... | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P9168",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P9168",
"citation": null,
"doi": null,
"arxiv_id": null,
"license": "Selected packets label their content CC0-1.0; original references retain their own terms",
"notes": "Selected problem... | theorem green_16 (N : ℕ) :
∃ A : Finset ℕ, A ⊆ Icc 1 N ∧ SolutionFree A ∧
A.card = answer(sorry) ∧
MaximalFor (fun B => B ⊆ Icc 1 N ∧ SolutionFree B) Finset.card A | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem green_16 (N : ℕ) :\n ∃ A : Finset ℕ, A ⊆ Icc 1 N ∧ SolutionFree A ∧\n A.card = answer(sorry) ∧\n MaximalFor (fun B => B ⊆ Icc 1 N ∧ SolutionFree B) Finset.card A",
"target": null,
"source_url": "https://theoremdb.org/statements/P9168",
"forma... | [] | TheoremDB P9168 · Candidate awaiting source review | null | Source-reported assessment, retrieved 8 September 2026. Frontier Science has not independently reviewed this status.
Open. No signed Lean verification is attached to this problem or its selected resolution.
This record is in the source backlog; its novelty and unresolved status have not passed source review. | null | published | incremental | hard | [
"source-candidate",
"theoremdb"
] | null | null | null | null | [
{
"kind": "website",
"citation": "TheoremDB P9168",
"url": "https://theoremdb.org/statements/P9168",
"note": null,
"sort_order": 0
},
{
"kind": "formalization",
"citation": "Source formal statement (not locally verified)",
"url": "https://theoremdb.org/statements/P9168",
"not... | {
"source_record": "P9168",
"imported_at": "2026-09-08",
"status_as_of": "2026-09-08",
"evidence_label": "Source candidate · not reviewed",
"note": "Full public directory statement. Source lane: backlog; source resolution: open. Source verification claims are attributed only; no local Lean verification has be... | {"created_at":"2026-08-17T19:47:55+00:00","display_standard":{"formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":9168,"public_status":{"evidence":{"hasFormalization":false,"hasInformalProof":false,"hasRecordedWork":false,"solutionRecord":null},"lean":{"kernelAccepted":false,"label":"","reaso... | [
"formalization-contains-placeholder",
"formalization-not-compiled-during-export",
"current-status-unassessed",
"source-full-text-not-present"
] | [] | {
"created_at": "2026-09-08T03:32:15.935Z",
"updated_at": "2026-09-08T03:32:15.935Z",
"published_at": "2026-09-08T03:33:03.844Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"c56740f9-689e-4e07-a3f1-b565013a5e90","format_issues":null,"number":6183} |
1.0.0 | problem:theoremdb-p9170 | problem | theoremdb-p9170 | Asymptotic growth of the least multiplier yielding a prime | {
"main": "math",
"sub": "number theory",
"slug": "number-theory",
"name": "Number theory",
"original_discipline": "mathematics"
} | [] | [
{
"code": "11",
"basis": "community-map",
"evidence": "Inferred from community field: number-theory"
}
] | For each positive integer $n$, let $a(n)$ denote the smallest natural number $k$ such that $kn + 1$ is prime. Determine whether $a(n) = O(\log n \cdot \log \log n)$, i.e., whether there exist constants $C > 0$ and $N$ such that $a(n) \leq C \cdot \log n \cdot \log \log n$ for all $n \geq N$. | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P9170",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P9170",
"citation": null,
"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_isBigO : (fun n ↦ (a n : ℝ)) =O[atTop] (fun n ↦ Real.log n * Real.log (Real.log n)) | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem a_isBigO : (fun n ↦ (a n : ℝ)) =O[atTop] (fun n ↦ Real.log n * Real.log (Real.log n))",
"target": null,
"source_url": "https://theoremdb.org/statements/P9170",
"format": "declaration-or-fragment",
"sha256": "ec030374b8d467195139f0aa00626808af1fc7b35fc... | [] | TheoremDB P9170 · Candidate awaiting source review | null | Source-reported assessment, retrieved 8 September 2026. Frontier Science has not independently reviewed this status.
Open. No signed Lean verification is attached to this problem or its selected resolution.
This record is in the source backlog; its novelty and unresolved status have not passed source review. | null | published | incremental | hard | [
"source-candidate",
"theoremdb"
] | null | null | null | null | [
{
"kind": "website",
"citation": "TheoremDB P9170",
"url": "https://theoremdb.org/statements/P9170",
"note": null,
"sort_order": 0
},
{
"kind": "formalization",
"citation": "Source formal statement (not locally verified)",
"url": "https://theoremdb.org/statements/P9170",
"not... | {
"source_record": "P9170",
"imported_at": "2026-09-08",
"status_as_of": "2026-09-08",
"evidence_label": "Source candidate · not reviewed",
"note": "Full public directory statement. Source lane: backlog; source resolution: open. Source verification claims are attributed only; no local Lean verification has be... | {"created_at":"2026-08-17T19:47:55+00:00","display_standard":{"editorial_review":"submitter_supplied","formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":9170,"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": "2026-09-08T03:32:50.193Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"99442518-ddad-4db0-8e3e-f71c5d625864","format_issues":null,"number":3370} |
1.0.0 | problem:theoremdb-p9172 | problem | theoremdb-p9172 | Prime existence between blocks with identical prime-factor sets | {
"main": "math",
"sub": "number theory",
"slug": "number-theory",
"name": "Number theory",
"original_discipline": "mathematics"
} | [] | [
{
"code": "11",
"basis": "community-map",
"evidence": "Inferred from community field: number-theory"
}
] | Let $k_1, k_2, n_1, n_2$ be natural numbers satisfying $k_2 \leq k_1$, $k_2 \geq 3$, and $n_1 + k_1 \leq n_2$. Suppose the sets of prime factors of the products $\prod_{i=1}^{k_1} (n_1 + i)$ and $\prod_{j=1}^{k_2} (n_2 + j)$ are equal. Prove that there exists a prime number $p$ such that $n_1 \leq p \leq n_2$. | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P9172",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P9172",
"citation": null,
"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.exists_prime (k₁ k₂ n₁ n₂ : ℕ) (h₁ : k₂ ≤ k₁) (h₂ : 3 ≤ k₂)
(h₃ : n₁ + k₁ ≤ n₂) (h₄ : (∏ i ∈ Finset.Icc 1 k₁, (n₁ + i)).primeFactors =
(∏ j ∈ Finset.Icc 1 k₂, (n₂ + j)).primeFactors) :
∃ (p : ℕ), p.Prime ∧ n₁ ≤ p ∧ p ≤ n₂ | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem erdos_931.variants.exists_prime (k₁ k₂ n₁ n₂ : ℕ) (h₁ : k₂ ≤ k₁) (h₂ : 3 ≤ k₂)\n (h₃ : n₁ + k₁ ≤ n₂) (h₄ : (∏ i ∈ Finset.Icc 1 k₁, (n₁ + i)).primeFactors =\n (∏ j ∈ Finset.Icc 1 k₂, (n₂ + j)).primeFactors) :\n ∃ (p : ℕ), p.Prime ∧ n₁ ≤ p ∧ p ≤ n₂",
"ta... | [] | TheoremDB P9172 · Candidate awaiting source review | null | Source-reported assessment, retrieved 8 September 2026. Frontier Science has not independently reviewed this status.
Open. No signed Lean verification is attached to this problem or its selected resolution.
This record is in the source backlog; its novelty and unresolved status have not passed source review. | null | published | incremental | hard | [
"source-candidate",
"theoremdb"
] | null | null | null | null | [
{
"kind": "website",
"citation": "TheoremDB P9172",
"url": "https://theoremdb.org/statements/P9172",
"note": null,
"sort_order": 0
},
{
"kind": "formalization",
"citation": "Source formal statement (not locally verified)",
"url": "https://theoremdb.org/statements/P9172",
"not... | {
"source_record": "P9172",
"imported_at": "2026-09-08",
"status_as_of": "2026-09-08",
"evidence_label": "Source candidate · not reviewed",
"note": "Full public directory statement. Source lane: backlog; source resolution: open. Source verification claims are attributed only; no local Lean verification has be... | {"created_at":"2026-08-17T19:47:56+00:00","display_standard":{"formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":9172,"public_status":{"evidence":{"hasFormalization":false,"hasInformalProof":false,"hasRecordedWork":false,"solutionRecord":null},"lean":{"kernelAccepted":false,"label":"","reaso... | [
"formalization-not-compiled-during-export",
"current-status-unassessed",
"source-full-text-not-present"
] | [] | {
"created_at": "2026-09-08T03:32:15.935Z",
"updated_at": "2026-09-08T03:32:15.935Z",
"published_at": "2026-09-08T03:33:05.106Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"f8a87576-afd3-4bab-9f8b-a7213ac90144","format_issues":null,"number":6455} |
1.0.0 | problem:theoremdb-p9174 | problem | theoremdb-p9174 | Stronger upper bound for prime-producing linear forms | {
"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 positive integer $n$, does there exist a natural number $k$ such that $k < 1 + n^{3/4}$ and $nk + 1$ is prime? | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P9174",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P9174",
"citation": null,
"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_stronger {n : ℕ} (hn : 0 < n) : ∃ k : ℕ,
k < 1 + (Real.nthRoot 4 n) ^ 3 ∧ (n * k + 1).Prime | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem exists_k_stronger {n : ℕ} (hn : 0 < n) : ∃ k : ℕ,\n k < 1 + (Real.nthRoot 4 n) ^ 3 ∧ (n * k + 1).Prime",
"target": null,
"source_url": "https://theoremdb.org/statements/P9174",
"format": "declaration-or-fragment",
"sha256": "06b5f66ae08ad1a087bcbcc... | [] | TheoremDB P9174 · Candidate awaiting source review | null | Source-reported assessment, retrieved 8 September 2026. Frontier Science has not independently reviewed this status.
Open. No signed Lean verification is attached to this problem or its selected resolution.
This record is in the source backlog; its novelty and unresolved status have not passed source review. | null | published | incremental | hard | [
"source-candidate",
"theoremdb"
] | null | null | null | null | [
{
"kind": "website",
"citation": "TheoremDB P9174",
"url": "https://theoremdb.org/statements/P9174",
"note": null,
"sort_order": 0
},
{
"kind": "formalization",
"citation": "Source formal statement (not locally verified)",
"url": "https://theoremdb.org/statements/P9174",
"not... | {
"source_record": "P9174",
"imported_at": "2026-09-08",
"status_as_of": "2026-09-08",
"evidence_label": "Source candidate · not reviewed",
"note": "Full public directory statement. Source lane: backlog; source resolution: open. Source verification claims are attributed only; no local Lean verification has be... | {"created_at":"2026-08-17T19:47:56+00:00","display_standard":{"editorial_review":"submitter_supplied","formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":9174,"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": "2026-09-08T03:33:06.132Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"1c31d7c4-a57d-4fc0-80ee-a2b8538ad575","format_issues":null,"number":6677} |
1.0.0 | problem:theoremdb-p9176 | problem | theoremdb-p9176 | Raising the lower bound for Green's Problem 25 on restricted sumsets in partitions | {
"main": "math",
"sub": "combinatorics",
"slug": "combinatorics",
"name": "Combinatorics",
"original_discipline": "mathematics"
} | [] | [
{
"code": "05",
"basis": "community-map",
"evidence": "Inferred from community field: combinatorics"
}
] | For positive integers $k$ and $N$ with $1 \leq k \leq N$, say that Property 25 holds for $(k,N)$ if for every partition of $\{1,2,\dots,N\}$ into $k$ parts $A_1 \cup \dots \cup A_k$, the union of the restricted sumsets satisfies
$$
\left|\bigcup_{i=1}^{k} (A_i \hat{+} A_i)\right| \geq \frac{1}{10} N,
$$
where the r... | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P9176",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P9176",
"citation": null,
"doi": null,
"arxiv_id": null,
"license": "Selected packets label their content CC0-1.0; original references retain their own terms",
"notes": "Selected problem... | theorem green_25.lower :
let ans := (answer(sorry) : ℕ → ℝ)
(bestLower =o[atTop] ans) ∧
(∀ᶠ N in atTop, 1 ≤ ans N ∧ ans N ≤ N) ∧
∀ k : ℕ → ℕ,
(∀ᶠ N in atTop, 1 ≤ k N ∧ k N ≤ N) ∧
((fun N => (k N : ℝ)) ≪ ans) →
∀ᶠ N in atTop, Property25 (k N) N | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem green_25.lower :\n let ans := (answer(sorry) : ℕ → ℝ)\n (bestLower =o[atTop] ans) ∧\n (∀ᶠ N in atTop, 1 ≤ ans N ∧ ans N ≤ N) ∧\n ∀ k : ℕ → ℕ,\n (∀ᶠ N in atTop, 1 ≤ k N ∧ k N ≤ N) ∧\n ((fun N => (k N : ℝ)) ≪ ans) →\n ∀ᶠ N in atTop, Property2... | [] | TheoremDB P9176 · Candidate awaiting source review | null | Source-reported assessment, retrieved 8 September 2026. Frontier Science has not independently reviewed this status.
Open. No signed Lean verification is attached to this problem or its selected resolution.
This record is in the source backlog; its novelty and unresolved status have not passed source review. | null | published | incremental | hard | [
"source-candidate",
"theoremdb"
] | null | null | null | null | [
{
"kind": "website",
"citation": "TheoremDB P9176",
"url": "https://theoremdb.org/statements/P9176",
"note": null,
"sort_order": 0
},
{
"kind": "formalization",
"citation": "Source formal statement (not locally verified)",
"url": "https://theoremdb.org/statements/P9176",
"not... | {
"source_record": "P9176",
"imported_at": "2026-09-08",
"status_as_of": "2026-09-08",
"evidence_label": "Source candidate · not reviewed",
"note": "Full public directory statement. Source lane: backlog; source resolution: open. Source verification claims are attributed only; no local Lean verification has be... | {"created_at":"2026-08-17T19:47:57+00:00","display_standard":{"editorial_review":"submitter_supplied","formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":9176,"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": "2026-09-08T03:33:05.381Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"11a8ba04-c461-4d6b-b208-c20465a299c9","format_issues":null,"number":6515} |
1.0.0 | problem:theoremdb-p9178 | problem | theoremdb-p9178 | Infinitely many near-misses to the divisor function maximum bound | {
"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 $\tau(n)$ denote the number of positive divisors of $n$. For each positive integer $k$, define $S_k$ to be the set of positive integers $n$ such that
$$
\max_{n-k < m < n} \bigl(m + \tau(m)\bigr) \leq n + 2.
$$
Does $S_k$ contain infinitely many elements for every positive integer $k... | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P9178",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P9178",
"citation": null,
"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_647.variants.infinite :
answer(sorry) ↔ ∀ k, { n | ⨆ m : Set.Ioo (n - k) n, ↑m + σ 0 m ≤ n + 2 }.Infinite | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem erdos_647.variants.infinite :\n answer(sorry) ↔ ∀ k, { n | ⨆ m : Set.Ioo (n - k) n, ↑m + σ 0 m ≤ n + 2 }.Infinite",
"target": null,
"source_url": "https://theoremdb.org/statements/P9178",
"format": "declaration-or-fragment",
"sha256": "4efbce9df824... | [] | TheoremDB P9178 · Candidate awaiting source review | null | Source-reported assessment, retrieved 8 September 2026. Frontier Science has not independently reviewed this status.
Open. No signed Lean verification is attached to this problem or its selected resolution.
This record is in the source backlog; its novelty and unresolved status have not passed source review. | null | published | incremental | hard | [
"source-candidate",
"theoremdb"
] | null | null | null | null | [
{
"kind": "website",
"citation": "TheoremDB P9178",
"url": "https://theoremdb.org/statements/P9178",
"note": null,
"sort_order": 0
},
{
"kind": "formalization",
"citation": "Source formal statement (not locally verified)",
"url": "https://theoremdb.org/statements/P9178",
"not... | {
"source_record": "P9178",
"imported_at": "2026-09-08",
"status_as_of": "2026-09-08",
"evidence_label": "Source candidate · not reviewed",
"note": "Full public directory statement. Source lane: backlog; source resolution: open. Source verification claims are attributed only; no local Lean verification has be... | {"created_at":"2026-08-17T19:48:31+00:00","display_standard":{"editorial_review":"submitter_supplied","formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":9178,"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": "2026-09-08T03:33:00.953Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"89498cb6-7f5a-4dbd-b741-dffbdd699b7e","format_issues":null,"number":5626} |
1.0.0 | problem:theoremdb-p9180 | problem | theoremdb-p9180 | Infinitude of Prime Euclid 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 Euclid number is defined as $\operatorname{Euclid}(n) = 1 + \prod_{i=0}^{n-1} p_i$, where $p_i$ denotes the $i$-th prime number (with $p_0 = 2$). Determine whether the set $\{n \in \mathbb{N} \mid \operatorname{Euclid}(n) \text{ is prime}\}$ is infinite. | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P9180",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P9180",
"citation": null,
"doi": null,
"arxiv_id": null,
"license": "Selected packets label their content CC0-1.0; original references retain their own terms",
"notes": "Selected problem... | theorem infinite_prime_euclid_numbers : answer(sorry) ↔ {n | (Euclid n).Prime}.Infinite | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem infinite_prime_euclid_numbers : answer(sorry) ↔ {n | (Euclid n).Prime}.Infinite",
"target": null,
"source_url": "https://theoremdb.org/statements/P9180",
"format": "declaration-or-fragment",
"sha256": "07bc4c3cdfe1cbab13376feebc1e96f439517e5bc5fabb929... | [] | TheoremDB P9180 · Candidate awaiting source review | null | Source-reported assessment, retrieved 8 September 2026. Frontier Science has not independently reviewed this status.
Open. No signed Lean verification is attached to this problem or its selected resolution.
This record is in the source backlog; its novelty and unresolved status have not passed source review. | null | published | incremental | hard | [
"source-candidate",
"theoremdb"
] | null | null | null | null | [
{
"kind": "website",
"citation": "TheoremDB P9180",
"url": "https://theoremdb.org/statements/P9180",
"note": null,
"sort_order": 0
},
{
"kind": "formalization",
"citation": "Source formal statement (not locally verified)",
"url": "https://theoremdb.org/statements/P9180",
"not... | {
"source_record": "P9180",
"imported_at": "2026-09-08",
"status_as_of": "2026-09-08",
"evidence_label": "Source candidate · not reviewed",
"note": "Full public directory statement. Source lane: backlog; source resolution: open. Source verification claims are attributed only; no local Lean verification has be... | {"created_at":"2026-08-17T19:49:00+00:00","display_standard":{"editorial_review":"submitter_supplied","formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":9180,"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": "2026-09-08T03:33:01.194Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"be757ef5-0334-4bbd-8d98-539334f46426","format_issues":null,"number":5678} |
1.0.0 | problem:theoremdb-p9182 | problem | theoremdb-p9182 | Erdős's conjectured upper bound for sets avoiding constant pairwise GCD | {
"main": "math",
"sub": "combinatorics",
"slug": "combinatorics",
"name": "Combinatorics",
"original_discipline": "mathematics"
} | [] | [
{
"code": "05",
"basis": "community-map",
"evidence": "Inferred from community field: combinatorics"
}
] | For each integer $r \geq 3$, let $f_r(N)$ denote the maximum cardinality of a subset $A \subseteq \{1, \ldots, N\}$ such that no $r$-element subset of $A$ has constant pairwise greatest common divisor; that is, there does not exist $d > 0$ such that $\gcd(a,b) = d$ for all distinct $a, b \in S$ whenever $S \subseteq A$... | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P9182",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P9182",
"citation": null,
"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_535 : ∀ r ≥ 3, ∃ c > (0 : ℝ),
∀ᶠ (N : ℕ) in atTop,
(f r N : ℝ) ≤ (N : ℝ) ^ (c / log (log (N : ℝ))) | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem erdos_535 : ∀ r ≥ 3, ∃ c > (0 : ℝ),\n ∀ᶠ (N : ℕ) in atTop,\n (f r N : ℝ) ≤ (N : ℝ) ^ (c / log (log (N : ℝ)))",
"target": null,
"source_url": "https://theoremdb.org/statements/P9182",
"format": "declaration-or-fragment",
"sha256": "00cc1ccbbc55... | [] | TheoremDB P9182 · Candidate awaiting source review | null | Source-reported assessment, retrieved 8 September 2026. Frontier Science has not independently reviewed this status.
Open. No signed Lean verification is attached to this problem or its selected resolution.
This record is in the source backlog; its novelty and unresolved status have not passed source review. | null | published | incremental | hard | [
"source-candidate",
"theoremdb"
] | null | null | null | null | [
{
"kind": "website",
"citation": "TheoremDB P9182",
"url": "https://theoremdb.org/statements/P9182",
"note": null,
"sort_order": 0
},
{
"kind": "formalization",
"citation": "Source formal statement (not locally verified)",
"url": "https://theoremdb.org/statements/P9182",
"not... | {
"source_record": "P9182",
"imported_at": "2026-09-08",
"status_as_of": "2026-09-08",
"evidence_label": "Source candidate · not reviewed",
"note": "Full public directory statement. Source lane: backlog; source resolution: open. Source verification claims are attributed only; no local Lean verification has be... | {"created_at":"2026-08-17T19:49:02+00:00","display_standard":{"editorial_review":"submitter_supplied","formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":9182,"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": "2026-09-08T03:32:56.361Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"a573c769-4fa8-46e9-84bc-5e923e876e9f","format_issues":null,"number":4724} |
1.0.0 | problem:theoremdb-p9184 | problem | theoremdb-p9184 | Irrationality of the Factorial Reciprocal Minus One Series | {
"main": "math",
"sub": "number theory",
"slug": "number-theory",
"name": "Number theory",
"original_discipline": "mathematics"
} | [] | [
{
"code": "11",
"basis": "community-map",
"evidence": "Inferred from community field: number-theory"
}
] | Consider the infinite series
$$
S = \sum_{n=2}^{\infty} \frac{1}{n! - 1}.
$$
Is $S$ irrational? | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P9184",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P9184",
"citation": null,
"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_68 :
answer(sorry) ↔ Irrational (∑' n : ℕ, 1 / ((n + 2).factorial - 1 : ℝ)) | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem erdos_68 :\n answer(sorry) ↔ Irrational (∑' n : ℕ, 1 / ((n + 2).factorial - 1 : ℝ))",
"target": null,
"source_url": "https://theoremdb.org/statements/P9184",
"format": "declaration-or-fragment",
"sha256": "3ed926caabd8031236d1c1f3c0054af6a674d20267... | [] | TheoremDB P9184 · Candidate awaiting source review | null | Source-reported assessment, retrieved 8 September 2026. Frontier Science has not independently reviewed this status.
Open. No signed Lean verification is attached to this problem or its selected resolution.
This record is in the source backlog; its novelty and unresolved status have not passed source review. | null | published | incremental | hard | [
"source-candidate",
"theoremdb"
] | null | null | null | null | [
{
"kind": "website",
"citation": "TheoremDB P9184",
"url": "https://theoremdb.org/statements/P9184",
"note": null,
"sort_order": 0
},
{
"kind": "formalization",
"citation": "Source formal statement (not locally verified)",
"url": "https://theoremdb.org/statements/P9184",
"not... | {
"source_record": "P9184",
"imported_at": "2026-09-08",
"status_as_of": "2026-09-08",
"evidence_label": "Source candidate · not reviewed",
"note": "Full public directory statement. Source lane: backlog; source resolution: open. Source verification claims are attributed only; no local Lean verification has be... | {"created_at":"2026-08-17T19:49:03+00:00","display_standard":{"editorial_review":"submitter_supplied","formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":9184,"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": "2026-09-08T03:33:01.775Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"74eceb62-b1ee-44de-9c43-2ac24ceadd76","format_issues":null,"number":5732} |
1.0.0 | problem:theoremdb-p9186 | problem | theoremdb-p9186 | Finiteness of high-quality abc triples | {
"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 radical of a positive integer $n$, denoted $\operatorname{rad}(n)$, be the product of the distinct prime factors of $n$. For positive integers $a$, $b$, $c$, define the quality $q(a,b,c)$ by
$$
q(a,b,c) = \frac{\log c}{\log(\operatorname{rad}(abc))}.
$$
Let $\varepsilon$ be a positive real number. Must the se... | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P9186",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P9186",
"citation": null,
"doi": null,
"arxiv_id": null,
"license": "Selected packets label their content CC0-1.0; original references retain their own terms",
"notes": "Selected problem... | theorem abc.variants.quality (ε : ℝ) (hε : 0 < ε) :
{(a, b, c) : ℕ × ℕ × ℕ | 0 < a ∧ 0 < b ∧ 0 < c ∧ ({a, b, c} : Set ℕ).Pairwise Nat.Coprime ∧
a + b = c ∧ quality a b c > (1 + ε)}.Finite | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem abc.variants.quality (ε : ℝ) (hε : 0 < ε) :\n {(a, b, c) : ℕ × ℕ × ℕ | 0 < a ∧ 0 < b ∧ 0 < c ∧ ({a, b, c} : Set ℕ).Pairwise Nat.Coprime ∧\n a + b = c ∧ quality a b c > (1 + ε)}.Finite",
"target": null,
"source_url": "https://theoremdb.org/statements/P91... | [] | TheoremDB P9186 · Candidate awaiting source review | null | Source-reported assessment, retrieved 8 September 2026. Frontier Science has not independently reviewed this status.
Open. No signed Lean verification is attached to this problem or its selected resolution.
This record is in the source backlog; its novelty and unresolved status have not passed source review. | null | published | incremental | hard | [
"source-candidate",
"theoremdb"
] | null | null | null | null | [
{
"kind": "website",
"citation": "TheoremDB P9186",
"url": "https://theoremdb.org/statements/P9186",
"note": null,
"sort_order": 0
},
{
"kind": "formalization",
"citation": "Source formal statement (not locally verified)",
"url": "https://theoremdb.org/statements/P9186",
"not... | {
"source_record": "P9186",
"imported_at": "2026-09-08",
"status_as_of": "2026-09-08",
"evidence_label": "Source candidate · not reviewed",
"note": "Full public directory statement. Source lane: backlog; source resolution: open. Source verification claims are attributed only; no local Lean verification has be... | {"created_at":"2026-08-17T19:49:04+00:00","display_standard":{"editorial_review":"submitter_supplied","formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":9186,"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": "2026-09-08T03:32:59.048Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"5c90cb90-9e1d-4fd3-85a4-06fc05e942a6","format_issues":null,"number":5311} |
1.0.0 | problem:theoremdb-p9188 | problem | theoremdb-p9188 | Asymptotic upper bound for the σ-iteration prime-reaching time | {
"main": "math",
"sub": "number theory",
"slug": "number-theory",
"name": "Number theory",
"original_discipline": "mathematics"
} | [] | [
{
"code": "11",
"basis": "community-map",
"evidence": "Inferred from community field: number-theory"
}
] | For each integer $n > 1$, let $c(n)$ denote the minimum number of iterations of the map $n \mapsto \sigma(n) - 1$ needed to reach a prime number, where $\sigma(n)$ denotes the sum-of-divisors function $\sigma(n) = \sum_{d \mid n} d$. That is, $c(n)$ is the least non-negative integer $i$ such that the $i$-fold iterate o... | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P9188",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P9188",
"citation": null,
"doi": null,
"arxiv_id": null,
"license": "Selected packets label their content CC0-1.0; original references retain their own terms",
"notes": "Selected problem... | theorem erdos_409.variants.sigma_isBigO (c : ℕ → ℕ)
(h : ∀ n > 1, IsLeast { i | (σ 1 · - 1)^[i] n |>.Prime } (c n)) :
(fun n => (c n : ℝ)) =O[atTop] (answer(sorry) : ℕ → ℝ) | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem erdos_409.variants.sigma_isBigO (c : ℕ → ℕ)\n (h : ∀ n > 1, IsLeast { i | (σ 1 · - 1)^[i] n |>.Prime } (c n)) :\n (fun n => (c n : ℝ)) =O[atTop] (answer(sorry) : ℕ → ℝ)",
"target": null,
"source_url": "https://theoremdb.org/statements/P9188",
"forma... | [] | TheoremDB P9188 · Candidate awaiting source review | null | Source-reported assessment, retrieved 8 September 2026. Frontier Science has not independently reviewed this status.
Open. No signed Lean verification is attached to this problem or its selected resolution.
This record is in the source backlog; its novelty and unresolved status have not passed source review. | null | published | incremental | hard | [
"source-candidate",
"theoremdb"
] | null | null | null | null | [
{
"kind": "website",
"citation": "TheoremDB P9188",
"url": "https://theoremdb.org/statements/P9188",
"note": null,
"sort_order": 0
},
{
"kind": "formalization",
"citation": "Source formal statement (not locally verified)",
"url": "https://theoremdb.org/statements/P9188",
"not... | {
"source_record": "P9188",
"imported_at": "2026-09-08",
"status_as_of": "2026-09-08",
"evidence_label": "Source candidate · not reviewed",
"note": "Full public directory statement. Source lane: backlog; source resolution: open. Source verification claims are attributed only; no local Lean verification has be... | {"created_at":"2026-08-17T19:49:04+00:00","display_standard":{"editorial_review":"submitter_supplied","formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":9188,"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": "2026-09-08T03:32:50.505Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"caf89179-1099-4805-a871-28fb57e62615","format_issues":null,"number":3430} |
1.0.0 | problem:theoremdb-p9190 | problem | theoremdb-p9190 | Upper bound on the maximal length of a 2-increasing sequence of triples | {
"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 $[n]$ denote the set $\{1, 2, \ldots, n\}$. For triples $a = (a_1, a_2, a_3)$ and $b = (b_1, b_2, b_3)$ of integers, we say that $a$ is $2$-less than $b$, denoted $a <_2 b$, if $a_i < b_i$ for at least two coordinates $i \in \{1, 2, 3\}$. A sequence of triples is called $2$-increasing... | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P9190",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P9190",
"citation": null,
"doi": null,
"arxiv_id": null,
"license": "Selected packets label their content CC0-1.0; original references retain their own terms",
"notes": "Selected problem... | theorem maximalLength_le_strong (n : ℕ) : F n ≤ Real.sqrt n ^ 3 | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem maximalLength_le_strong (n : ℕ) : F n ≤ Real.sqrt n ^ 3",
"target": null,
"source_url": "https://theoremdb.org/statements/P9190",
"format": "declaration-or-fragment",
"sha256": "9dd6277c0897c1be778da6f2a65cee7fc31390b9a584d9f6f82e4d2b1a2c052e",
"l... | [] | TheoremDB P9190 · Candidate awaiting source review | null | Source-reported assessment, retrieved 8 September 2026. Frontier Science has not independently reviewed this status.
Open. No signed Lean verification is attached to this problem or its selected resolution.
This record is in the source backlog; its novelty and unresolved status have not passed source review. | null | published | incremental | hard | [
"source-candidate",
"theoremdb"
] | null | null | null | null | [
{
"kind": "website",
"citation": "TheoremDB P9190",
"url": "https://theoremdb.org/statements/P9190",
"note": null,
"sort_order": 0
},
{
"kind": "formalization",
"citation": "Source formal statement (not locally verified)",
"url": "https://theoremdb.org/statements/P9190",
"not... | {
"source_record": "P9190",
"imported_at": "2026-09-08",
"status_as_of": "2026-09-08",
"evidence_label": "Source candidate · not reviewed",
"note": "Full public directory statement. Source lane: backlog; source resolution: open. Source verification claims are attributed only; no local Lean verification has be... | {"created_at":"2026-08-17T19:49:06+00:00","display_standard":{"editorial_review":"submitter_supplied","formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":9190,"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": "2026-09-08T03:33:07.385Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"32d10258-40cb-430f-b68d-2b82d2f9492f","format_issues":null,"number":6948} |
1.0.0 | problem:theoremdb-p9192 | problem | theoremdb-p9192 | The Feit–Thompson conjecture on prime divisibility of cyclotomic-like 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"
}
] | Let $p$ and $q$ be prime numbers with $p < q$. Define the integers
$$
A = \frac{q^p - 1}{q - 1} = 1 + q + q^2 + \cdots + q^{p-1}
$$
and
$$
B = \frac{p^q - 1}{p - 1} = 1 + p + p^2 + \cdots + p^{q-1}.
$$
Prove that $A$ does not divide $B$. | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P9192",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P9192",
"citation": null,
"doi": null,
"arxiv_id": null,
"license": "Selected packets label their content CC0-1.0; original references retain their own terms",
"notes": "Selected problem... | theorem feit_thompson_primes (p q : ℕ) (hp : p.Prime) (hq : q.Prime) (h : p < q) :
¬ (q ^ p - 1) / (q - 1) ∣ (p ^ q - 1) / (p - 1) | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem feit_thompson_primes (p q : ℕ) (hp : p.Prime) (hq : q.Prime) (h : p < q) :\n ¬ (q ^ p - 1) / (q - 1) ∣ (p ^ q - 1) / (p - 1)",
"target": null,
"source_url": "https://theoremdb.org/statements/P9192",
"format": "declaration-or-fragment",
"sha256": "5... | [] | TheoremDB P9192 · Candidate awaiting source review | null | Source-reported assessment, retrieved 8 September 2026. Frontier Science has not independently reviewed this status.
Open. No signed Lean verification is attached to this problem or its selected resolution.
This record is in the source backlog; its novelty and unresolved status have not passed source review. | null | published | incremental | hard | [
"source-candidate",
"theoremdb"
] | null | null | null | null | [
{
"kind": "website",
"citation": "TheoremDB P9192",
"url": "https://theoremdb.org/statements/P9192",
"note": null,
"sort_order": 0
},
{
"kind": "formalization",
"citation": "Source formal statement (not locally verified)",
"url": "https://theoremdb.org/statements/P9192",
"not... | {
"source_record": "P9192",
"imported_at": "2026-09-08",
"status_as_of": "2026-09-08",
"evidence_label": "Source candidate · not reviewed",
"note": "Full public directory statement. Source lane: backlog; source resolution: open. Source verification claims are attributed only; no local Lean verification has be... | {"created_at":"2026-08-17T19:49:09+00:00","display_standard":{"formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":9192,"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": "2026-09-08T03:33:06.489Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"e97b8b99-1957-4a12-8d12-2e2786fdaa0e","format_issues":null,"number":6753} |
1.0.0 | problem:theoremdb-p9194 | problem | theoremdb-p9194 | Infinitely many pairs of consecutive integers with equal prime divisor sets 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"
}
] | Let the central binomial coefficient be $\binom{2n}{n}$ for a positive integer $n$. Define $S$ to be the set of ordered pairs $(n, m)$ of natural numbers with $n < m$ such that $\binom{2n}{n}$ and $\binom{2m}{m}$ have exactly the same set of prime divisors. Is $S$ an infinite set? | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P9194",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P9194",
"citation": null,
"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_730 : answer(sorry) ↔ S.Infinite | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem erdos_730 : answer(sorry) ↔ S.Infinite",
"target": null,
"source_url": "https://theoremdb.org/statements/P9194",
"format": "declaration-or-fragment",
"sha256": "773c84b7b897bcd2a5d6d0461cd10bbcb9ddb655dee26ddb38a948b2afed22aa",
"license": "Selecte... | [] | TheoremDB P9194 · Candidate awaiting source review | null | Source-reported assessment, retrieved 8 September 2026. Frontier Science has not independently reviewed this status.
Open. No signed Lean verification is attached to this problem or its selected resolution.
This record is in the source backlog; its novelty and unresolved status have not passed source review. | null | published | incremental | hard | [
"source-candidate",
"theoremdb"
] | null | null | null | null | [
{
"kind": "website",
"citation": "TheoremDB P9194",
"url": "https://theoremdb.org/statements/P9194",
"note": null,
"sort_order": 0
},
{
"kind": "formalization",
"citation": "Source formal statement (not locally verified)",
"url": "https://theoremdb.org/statements/P9194",
"not... | {
"source_record": "P9194",
"imported_at": "2026-09-08",
"status_as_of": "2026-09-08",
"evidence_label": "Source candidate · not reviewed",
"note": "Full public directory statement. Source lane: backlog; source resolution: open. Source verification claims are attributed only; no local Lean verification has be... | {"created_at":"2026-08-17T19:49:10+00:00","display_standard":{"editorial_review":"submitter_supplied","formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":9194,"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": "2026-09-08T03:33:00.961Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"9c4635da-fa31-4425-b6a1-5e65dc89e458","format_issues":null,"number":5628} |
1.0.0 | problem:theoremdb-p9196 | problem | theoremdb-p9196 | Polynomial bound on the diameter of alternating groups | {
"main": "math",
"sub": "combinatorics",
"slug": "combinatorics",
"name": "Combinatorics",
"original_discipline": "mathematics"
} | [] | [
{
"code": "05",
"basis": "community-map",
"evidence": "Inferred from community field: combinatorics"
}
] | For each finite group $G$ and each generating set $S \subseteq G$, the undirected Cayley graph $\Gamma(G,S)$ is the simple graph whose vertices are the elements of $G$, with two distinct vertices $g,h \in G$ adjacent whenever $g^{-1}h \in S$ or $h^{-1}g \in S$. The diameter of a finite group $G$, denoted $\operatorname... | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P9196",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P9196",
"citation": null,
"doi": null,
"arxiv_id": null,
"license": "Selected packets label their content CC0-1.0; original references retain their own terms",
"notes": "Selected problem... | theorem babai_seress_conjecture_alternating :
∃ C : ℕ, ∀ n : ℕ,
(groupDiam (alternatingGroup (Fin n)) : ℝ) ≤ (n : ℝ) ^ C | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem babai_seress_conjecture_alternating :\n ∃ C : ℕ, ∀ n : ℕ,\n (groupDiam (alternatingGroup (Fin n)) : ℝ) ≤ (n : ℝ) ^ C",
"target": null,
"source_url": "https://theoremdb.org/statements/P9196",
"format": "declaration-or-fragment",
"sha256": "f1c726... | [] | TheoremDB P9196 · Candidate awaiting source review | null | Source-reported assessment, retrieved 8 September 2026. Frontier Science has not independently reviewed this status.
Open. No signed Lean verification is attached to this problem or its selected resolution.
This record is in the source backlog; its novelty and unresolved status have not passed source review. | null | published | incremental | hard | [
"source-candidate",
"theoremdb"
] | null | null | null | null | [
{
"kind": "website",
"citation": "TheoremDB P9196",
"url": "https://theoremdb.org/statements/P9196",
"note": null,
"sort_order": 0
},
{
"kind": "formalization",
"citation": "Source formal statement (not locally verified)",
"url": "https://theoremdb.org/statements/P9196",
"not... | {
"source_record": "P9196",
"imported_at": "2026-09-08",
"status_as_of": "2026-09-08",
"evidence_label": "Source candidate · not reviewed",
"note": "Full public directory statement. Source lane: backlog; source resolution: open. Source verification claims are attributed only; no local Lean verification has be... | {"created_at":"2026-08-17T19:49:10+00:00","display_standard":{"formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":9196,"public_status":{"evidence":{"hasFormalization":false,"hasInformalProof":false,"hasRecordedWork":false,"solutionRecord":null},"lean":{"kernelAccepted":false,"label":"","reaso... | [
"formalization-not-compiled-during-export",
"current-status-unassessed",
"source-full-text-not-present"
] | [] | {
"created_at": "2026-09-08T03:32:15.935Z",
"updated_at": "2026-09-08T03:32:15.935Z",
"published_at": "2026-09-08T03:33:04.871Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"0d05f28d-01af-43d9-a7a0-b0d05bef72bd","format_issues":null,"number":6404} |
1.0.0 | problem:theoremdb-p9198 | problem | theoremdb-p9198 | Asymptotic density of integers where the least totient multiple exceeds the least prime in a residue class | {
"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 $p_n$ to be the smallest prime number congruent to $1$ modulo $n$, and define $m_n$ to be the smallest positive integer $k$ such that $n$ divides Euler's totient function $\phi(k)$. Determine whether $m_n < p_n$ holds for almost all positive integers $n$, in the sense that the set ... | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P9198",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P9198",
"citation": null,
"doi": null,
"arxiv_id": null,
"license": "Selected packets label their content CC0-1.0; original references retain their own terms",
"notes": "Selected problem... | theorem erdos_456.parts.i :
answer(sorry) ↔
Tendsto (fun N ↦ (count { n | m n < p n } N : ℝ) / (N : ℝ)) atTop (𝓝 1) | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem erdos_456.parts.i :\n answer(sorry) ↔\n Tendsto (fun N ↦ (count { n | m n < p n } N : ℝ) / (N : ℝ)) atTop (𝓝 1)",
"target": null,
"source_url": "https://theoremdb.org/statements/P9198",
"format": "declaration-or-fragment",
"sha256": "3a012632... | [] | TheoremDB P9198 · Candidate awaiting source review | null | Source-reported assessment, retrieved 8 September 2026. Frontier Science has not independently reviewed this status.
Open. No signed Lean verification is attached to this problem or its selected resolution.
This record is in the source backlog; its novelty and unresolved status have not passed source review. | null | published | incremental | hard | [
"source-candidate",
"theoremdb"
] | null | null | null | null | [
{
"kind": "website",
"citation": "TheoremDB P9198",
"url": "https://theoremdb.org/statements/P9198",
"note": null,
"sort_order": 0
},
{
"kind": "formalization",
"citation": "Source formal statement (not locally verified)",
"url": "https://theoremdb.org/statements/P9198",
"not... | {
"source_record": "P9198",
"imported_at": "2026-09-08",
"status_as_of": "2026-09-08",
"evidence_label": "Source candidate · not reviewed",
"note": "Full public directory statement. Source lane: backlog; source resolution: open. Source verification claims are attributed only; no local Lean verification has be... | {"created_at":"2026-08-17T19:49:13+00:00","display_standard":{"editorial_review":"submitter_supplied","formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":9198,"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": "2026-09-08T03:32:50.006Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"1da0333f-0f63-4499-b074-5a9f6b179002","format_issues":null,"number":3331} |
1.0.0 | problem:theoremdb-p9200 | problem | theoremdb-p9200 | Unique representation by sums from a set and cubic values | {
"main": "math",
"sub": "algebra",
"slug": "algebra",
"name": "Algebra",
"original_discipline": "mathematics"
} | [] | [] | Let $f(x)=x^3$. For any set $A\subseteq\mathbb{Z}$, must there exist an integer $z$ that cannot be written uniquely as $z=a+f(n)$ with $a\in A$ and $n$ a positive integer? In other words, does there exist $z\in\mathbb{Z}$ such that there is not exactly one pair $(a,n)$ with $a\in A$, $n\in\mathbb{Z}_{>0}$, and $z=a+n^3... | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P9200",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P9200",
"citation": null,
"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_477.variants.X_pow_three :
letI f := X ^ 3
∀ A : Set ℤ, ∃ z, ¬ ∃! a ∈ A ×ˢ (Set.range f.eval), z = a.1 + a.2 | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem erdos_477.variants.X_pow_three :\n letI f := X ^ 3\n ∀ A : Set ℤ, ∃ z, ¬ ∃! a ∈ A ×ˢ (Set.range f.eval), z = a.1 + a.2",
"target": null,
"source_url": "https://theoremdb.org/statements/P9200",
"format": "declaration-or-fragment",
"sha256": "abf8... | [] | TheoremDB P9200 · Candidate awaiting source review | null | Source-reported assessment, retrieved 8 September 2026. Frontier Science has not independently reviewed this status.
Open. No signed Lean verification is attached to this problem or its selected resolution.
This record is in the source backlog; its novelty and unresolved status have not passed source review. | null | published | incremental | hard | [
"source-candidate",
"theoremdb"
] | null | null | null | null | [
{
"kind": "website",
"citation": "TheoremDB P9200",
"url": "https://theoremdb.org/statements/P9200",
"note": null,
"sort_order": 0
},
{
"kind": "formalization",
"citation": "Source formal statement (not locally verified)",
"url": "https://theoremdb.org/statements/P9200",
"not... | {
"source_record": "P9200",
"imported_at": "2026-09-08",
"status_as_of": "2026-09-08",
"evidence_label": "Source candidate · not reviewed",
"note": "Full public directory statement. Source lane: backlog; source resolution: open. Source verification claims are attributed only; no local Lean verification has be... | {"created_at":"2026-08-17T19:49:13+00:00","display_standard":{"editorial_review":"submitter_supplied","formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":9200,"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": "2026-09-08T03:33:07.233Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"f3355bea-ddda-4485-ae6b-b6e2e092cd81","format_issues":null,"number":6915} |
1.0.0 | problem:theoremdb-p9202 | problem | theoremdb-p9202 | Exact rank of Elkies's rank-28 elliptic curve | {
"main": "math",
"sub": "number theory",
"slug": "number-theory",
"name": "Number theory",
"original_discipline": "mathematics"
} | [] | [
{
"code": "11",
"basis": "community-map",
"evidence": "Inferred from community field: number-theory"
}
] | Let $E$ be the elliptic curve over $\mathbb{Q}$ defined by the Weierstrass equation
$$
y^2 + xy - y = x^3 - x^2 - 20067762415575526585033208209338542750930230312178956502\,x + 34481611795030556467032985690390720374855944359319180361266008296291939448732243429.
$$
The discriminant of $E$ is nonzero, so $E$ is indeed a... | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P9202",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P9202",
"citation": null,
"doi": null,
"arxiv_id": null,
"license": "Selected packets label their content CC0-1.0; original references retain their own terms",
"notes": "Selected problem... | theorem rank_elkies28 : finrank ℤ elkies28⟮ℚ⟯ = 28 | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem rank_elkies28 : finrank ℤ elkies28⟮ℚ⟯ = 28",
"target": null,
"source_url": "https://theoremdb.org/statements/P9202",
"format": "declaration-or-fragment",
"sha256": "2d14a75a83746e73b3ffbca668b1cc8ef084840eb24e4f7a8df1936f7359b314",
"license": "Sel... | [] | TheoremDB P9202 · Candidate awaiting source review | null | Source-reported assessment, retrieved 8 September 2026. Frontier Science has not independently reviewed this status.
Open. No signed Lean verification is attached to this problem or its selected resolution.
This record is in the source backlog; its novelty and unresolved status have not passed source review. | null | published | incremental | hard | [
"source-candidate",
"theoremdb"
] | null | null | null | null | [
{
"kind": "website",
"citation": "TheoremDB P9202",
"url": "https://theoremdb.org/statements/P9202",
"note": null,
"sort_order": 0
},
{
"kind": "formalization",
"citation": "Source formal statement (not locally verified)",
"url": "https://theoremdb.org/statements/P9202",
"not... | {
"source_record": "P9202",
"imported_at": "2026-09-08",
"status_as_of": "2026-09-08",
"evidence_label": "Source candidate · not reviewed",
"note": "Full public directory statement. Source lane: backlog; source resolution: open. Source verification claims are attributed only; no local Lean verification has be... | {"created_at":"2026-08-17T19:49:14+00:00","display_standard":{"formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":9202,"public_status":{"evidence":{"hasFormalization":false,"hasInformalProof":false,"hasRecordedWork":false,"solutionRecord":null},"lean":{"kernelAccepted":false,"label":"","reaso... | [
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"current-status-unassessed",
"source-full-text-not-present"
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"created_at": "2026-09-08T03:32:15.935Z",
"updated_at": "2026-09-08T03:32:15.935Z",
"published_at": "2026-09-08T03:32:57.315Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"88acb588-5f3e-4e09-a522-8aecc211f9e4","format_issues":null,"number":4935} |
1.0.0 | problem:theoremdb-p9204 | problem | theoremdb-p9204 | Infinitude of balanced primes of arbitrary order | {
"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 $p_k$ denote the $k$-th prime number. For each positive integer $k$, a prime $p_n$ is said to be a balanced prime of order $k$ if $k \leq n$ and
$$
2 k \cdot p_n = \sum_{i=1}^{k} \bigl(p_{n-i} + p_{n+i}\bigr).
$$
Does the set of balanced primes of order $k$ contain infinitely many elements for every positive inte... | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P9204",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P9204",
"citation": null,
"doi": null,
"arxiv_id": null,
"license": "Selected packets label their content CC0-1.0; original references retain their own terms",
"notes": "Selected problem... | theorem balanced_primes_order :
answer(sorry) ↔ ∀ k > 0, {n : ℕ | k ≤ n ∧ 2 * k * Nat.nth Prime n = ∑ i ∈ .Ioc 0 k,
((n - i).nth Prime + (n + i).nth Prime)}.Infinite | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem balanced_primes_order :\n answer(sorry) ↔ ∀ k > 0, {n : ℕ | k ≤ n ∧ 2 * k * Nat.nth Prime n = ∑ i ∈ .Ioc 0 k,\n ((n - i).nth Prime + (n + i).nth Prime)}.Infinite",
"target": null,
"source_url": "https://theoremdb.org/statements/P9204",
"format": "... | [] | TheoremDB P9204 · Candidate awaiting source review | null | Source-reported assessment, retrieved 8 September 2026. Frontier Science has not independently reviewed this status.
Open. No signed Lean verification is attached to this problem or its selected resolution.
This record is in the source backlog; its novelty and unresolved status have not passed source review. | null | published | incremental | hard | [
"source-candidate",
"theoremdb"
] | null | null | null | null | [
{
"kind": "website",
"citation": "TheoremDB P9204",
"url": "https://theoremdb.org/statements/P9204",
"note": null,
"sort_order": 0
},
{
"kind": "formalization",
"citation": "Source formal statement (not locally verified)",
"url": "https://theoremdb.org/statements/P9204",
"not... | {
"source_record": "P9204",
"imported_at": "2026-09-08",
"status_as_of": "2026-09-08",
"evidence_label": "Source candidate · not reviewed",
"note": "Full public directory statement. Source lane: backlog; source resolution: open. Source verification claims are attributed only; no local Lean verification has be... | {"created_at":"2026-08-17T19:49:14+00:00","display_standard":{"editorial_review":"submitter_supplied","formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":9204,"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": "2026-09-08T03:33:01.086Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"bfbffe1f-10b6-4bd0-8077-e25b51669b70","format_issues":null,"number":5654} |
1.0.0 | problem:theoremdb-p9206 | problem | theoremdb-p9206 | Equality of the Köthe radical and the matrix ideal over the Köthe radical | {
"main": "math",
"sub": "algebra",
"slug": "algebra",
"name": "Algebra",
"original_discipline": "mathematics"
} | [] | [] | Let $R$ be a ring. A subset $I$ of $R$ is called nilpotent if every element of $I$ is nilpotent, i.e., for every $i \in I$ there exists a positive integer $k$ such that $i^k = 0$. The Köthe radical of $R$, denoted $\mathrm{Nil}^*(R)$, is defined as the sum of all two-sided nil ideals of $R$. For a finite type $n$ and a... | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P9206",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P9206",
"citation": null,
"doi": null,
"arxiv_id": null,
"license": "Selected packets label their content CC0-1.0; original references retain their own terms",
"notes": "Selected problem... | theorem KotherConjecture.variants.matrixOver_KotherRadical
{I : TwoSidedIdeal R} (hI : IsNil I) (n : Type*) [Fintype n] :
matrix n (Nil* R) = Nil* (Matrix n n R) | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem KotherConjecture.variants.matrixOver_KotherRadical\n {I : TwoSidedIdeal R} (hI : IsNil I) (n : Type*) [Fintype n] :\n matrix n (Nil* R) = Nil* (Matrix n n R)",
"target": null,
"source_url": "https://theoremdb.org/statements/P9206",
"format": "declar... | [] | TheoremDB P9206 · Candidate awaiting source review | null | Source-reported assessment, retrieved 8 September 2026. Frontier Science has not independently reviewed this status.
Open. No signed Lean verification is attached to this problem or its selected resolution.
This record is in the source backlog; its novelty and unresolved status have not passed source review. | null | published | incremental | hard | [
"source-candidate",
"theoremdb"
] | null | null | null | null | [
{
"kind": "website",
"citation": "TheoremDB P9206",
"url": "https://theoremdb.org/statements/P9206",
"note": null,
"sort_order": 0
},
{
"kind": "formalization",
"citation": "Source formal statement (not locally verified)",
"url": "https://theoremdb.org/statements/P9206",
"not... | {
"source_record": "P9206",
"imported_at": "2026-09-08",
"status_as_of": "2026-09-08",
"evidence_label": "Source candidate · not reviewed",
"note": "Full public directory statement. Source lane: backlog; source resolution: open. Source verification claims are attributed only; no local Lean verification has be... | {"created_at":"2026-08-17T19:49:17+00:00","display_standard":{"formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":9206,"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": "2026-09-08T03:32:54.767Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"eadada8a-d9c5-402a-8016-396aefef4003","format_issues":null,"number":4374} |
1.0.0 | problem:theoremdb-p9208 | problem | theoremdb-p9208 | Asymptotic upper bound for totient-prime iteration length | {
"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 smallest non-negative 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 minimum $i \geq 0$ for which $f^{(i)}(n)$ is prime, ... | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P9208",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P9208",
"citation": null,
"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.isBigO (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.isBigO (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/P9208",
"format": "decl... | [] | TheoremDB P9208 · Candidate awaiting source review | null | Source-reported assessment, retrieved 8 September 2026. Frontier Science has not independently reviewed this status.
Open. No signed Lean verification is attached to this problem or its selected resolution.
This record is in the source backlog; its novelty and unresolved status have not passed source review. | null | published | incremental | hard | [
"source-candidate",
"theoremdb"
] | null | null | null | null | [
{
"kind": "website",
"citation": "TheoremDB P9208",
"url": "https://theoremdb.org/statements/P9208",
"note": null,
"sort_order": 0
},
{
"kind": "formalization",
"citation": "Source formal statement (not locally verified)",
"url": "https://theoremdb.org/statements/P9208",
"not... | {
"source_record": "P9208",
"imported_at": "2026-09-08",
"status_as_of": "2026-09-08",
"evidence_label": "Source candidate · not reviewed",
"note": "Full public directory statement. Source lane: backlog; source resolution: open. Source verification claims are attributed only; no local Lean verification has be... | {"created_at":"2026-08-17T19:49:22+00:00","display_standard":{"editorial_review":"submitter_supplied","formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":9208,"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": "2026-09-08T03:32:50.510Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"4fb0634d-4ca4-4306-988b-c58573e3dd33","format_issues":null,"number":3431} |
1.0.0 | problem:theoremdb-p9210 | problem | theoremdb-p9210 | Erdős–Selfridge liminf problem for restricted prime factor counts in consecutive integers | {
"main": "math",
"sub": "number theory",
"slug": "number-theory",
"name": "Number theory",
"original_discipline": "mathematics"
} | [] | [
{
"code": "11",
"basis": "community-map",
"evidence": "Inferred from community field: number-theory"
}
] | For positive integers $k$ and $n$, let $\omega_k(n)$ denote the number of distinct prime factors of $n$ that are strictly greater than $k$. Determine whether, for every integer $k \geq 1$,
$$
\liminf_{n \to \infty} \sum_{i=0}^{k-1} \omega_k(n+i) \leq k.
$$
Equivalently, does there exist, for each $k \geq 1$, a sequ... | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P9210",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P9210",
"citation": null,
"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_890.parts.a :
answer(sorry) ↔
∀ k ≥ 1, liminf (fun n ↦ (∑ i ∈ range k, (ω_gt k (n + i) : EReal))) atTop ≤ k | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem erdos_890.parts.a :\n answer(sorry) ↔\n ∀ k ≥ 1, liminf (fun n ↦ (∑ i ∈ range k, (ω_gt k (n + i) : EReal))) atTop ≤ k",
"target": null,
"source_url": "https://theoremdb.org/statements/P9210",
"format": "declaration-or-fragment",
"sha256": "ff555... | [] | TheoremDB P9210 · Candidate awaiting source review | null | Source-reported assessment, retrieved 8 September 2026. Frontier Science has not independently reviewed this status.
Open. No signed Lean verification is attached to this problem or its selected resolution.
This record is in the source backlog; its novelty and unresolved status have not passed source review. | null | published | incremental | hard | [
"source-candidate",
"theoremdb"
] | null | null | null | null | [
{
"kind": "website",
"citation": "TheoremDB P9210",
"url": "https://theoremdb.org/statements/P9210",
"note": null,
"sort_order": 0
},
{
"kind": "formalization",
"citation": "Source formal statement (not locally verified)",
"url": "https://theoremdb.org/statements/P9210",
"not... | {
"source_record": "P9210",
"imported_at": "2026-09-08",
"status_as_of": "2026-09-08",
"evidence_label": "Source candidate · not reviewed",
"note": "Full public directory statement. Source lane: backlog; source resolution: open. Source verification claims are attributed only; no local Lean verification has be... | {"created_at":"2026-08-17T19:49:28+00:00","display_standard":{"editorial_review":"submitter_supplied","formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":9210,"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": "2026-09-08T03:32:56.705Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"6ffbcbbc-9518-42b9-8b06-4f6da50a3c46","format_issues":null,"number":4797} |
1.0.0 | problem:theoremdb-p9212 | problem | theoremdb-p9212 | Maximum admissible constant for Sidon subsets in linear sets with restricted difference sets | {
"main": "math",
"sub": "combinatorics",
"slug": "combinatorics",
"name": "Combinatorics",
"original_discipline": "mathematics"
} | [] | [
{
"code": "05",
"basis": "community-map",
"evidence": "Inferred from community field: combinatorics"
}
] | A real number $c$ is called admissible if for every finite set $A \subset \mathbb{R}$ satisfying the property that every 4-element subset $B \subseteq A$ has exactly 11 distinct differences (that is, $|B - B| = 11$), there exists a Sidon subset $S \subseteq A$ with $|S| \geq c \cdot |A|$. Here, a Sidon set is a set in ... | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P9212",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P9212",
"citation": null,
"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_757 {A : Set ℝ} :
answer(sorry) = sSup {c | IsAdmissible c} | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem erdos_757 {A : Set ℝ} :\n answer(sorry) = sSup {c | IsAdmissible c}",
"target": null,
"source_url": "https://theoremdb.org/statements/P9212",
"format": "declaration-or-fragment",
"sha256": "05687bb13e004d4f88764b54fcdb0fcb21621766a53210c77ddfa0e03e... | [] | TheoremDB P9212 · Candidate awaiting source review | null | Source-reported assessment, retrieved 8 September 2026. Frontier Science has not independently reviewed this status.
Open. No signed Lean verification is attached to this problem or its selected resolution.
This record is in the source backlog; its novelty and unresolved status have not passed source review. | null | published | incremental | hard | [
"source-candidate",
"theoremdb"
] | null | null | null | null | [
{
"kind": "website",
"citation": "TheoremDB P9212",
"url": "https://theoremdb.org/statements/P9212",
"note": null,
"sort_order": 0
},
{
"kind": "formalization",
"citation": "Source formal statement (not locally verified)",
"url": "https://theoremdb.org/statements/P9212",
"not... | {
"source_record": "P9212",
"imported_at": "2026-09-08",
"status_as_of": "2026-09-08",
"evidence_label": "Source candidate · not reviewed",
"note": "Full public directory statement. Source lane: backlog; source resolution: open. Source verification claims are attributed only; no local Lean verification has be... | {"created_at":"2026-08-17T19:49:32+00:00","display_standard":{"formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":9212,"public_status":{"evidence":{"hasFormalization":false,"hasInformalProof":false,"hasRecordedWork":false,"solutionRecord":null},"lean":{"kernelAccepted":false,"label":"","reaso... | [
"formalization-contains-placeholder",
"formalization-not-compiled-during-export",
"current-status-unassessed",
"source-full-text-not-present"
] | [] | {
"created_at": "2026-09-08T03:32:15.935Z",
"updated_at": "2026-09-08T03:32:15.935Z",
"published_at": "2026-09-08T03:33:03.670Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"c0901c9d-55fa-41b5-9597-6a3edc7241be","format_issues":null,"number":6144} |
1.0.0 | problem:theoremdb-p9214 | problem | theoremdb-p9214 | Nilpotence of matrix ideals over nil ideals | {
"main": "math",
"sub": "algebra",
"slug": "algebra",
"name": "Algebra",
"original_discipline": "mathematics"
} | [] | [] | Let $R$ be a ring. A subset $I \subseteq R$ is called nil if every element of $I$ is nilpotent, meaning for each $i \in I$ there exists some positive integer $k$ such that $i^k = 0$. Let $I$ be a two-sided ideal of $R$ that is nil, and let $n$ be a finite type indexing a set of size $n$. The matrix ideal $M_n(I)$ consi... | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P9214",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P9214",
"citation": null,
"doi": null,
"arxiv_id": null,
"license": "Selected packets label their content CC0-1.0; original references retain their own terms",
"notes": "Selected problem... | theorem KotherConjecture.variants.general_matrix {I : TwoSidedIdeal R} (hI : IsNil I)
(n : Type*) [Fintype n] : IsNil (matrix n I) | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem KotherConjecture.variants.general_matrix {I : TwoSidedIdeal R} (hI : IsNil I)\n (n : Type*) [Fintype n] : IsNil (matrix n I)",
"target": null,
"source_url": "https://theoremdb.org/statements/P9214",
"format": "declaration-or-fragment",
"sha256": "e... | [] | TheoremDB P9214 · Candidate awaiting source review | null | Source-reported assessment, retrieved 8 September 2026. Frontier Science has not independently reviewed this status.
Open. No signed Lean verification is attached to this problem or its selected resolution.
This record is in the source backlog; its novelty and unresolved status have not passed source review. | null | published | incremental | hard | [
"source-candidate",
"theoremdb"
] | null | null | null | null | [
{
"kind": "website",
"citation": "TheoremDB P9214",
"url": "https://theoremdb.org/statements/P9214",
"note": null,
"sort_order": 0
},
{
"kind": "formalization",
"citation": "Source formal statement (not locally verified)",
"url": "https://theoremdb.org/statements/P9214",
"not... | {
"source_record": "P9214",
"imported_at": "2026-09-08",
"status_as_of": "2026-09-08",
"evidence_label": "Source candidate · not reviewed",
"note": "Full public directory statement. Source lane: backlog; source resolution: open. Source verification claims are attributed only; no local Lean verification has be... | {"created_at":"2026-08-17T19:49:36+00:00","display_standard":{"editorial_review":"submitter_supplied","formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":9214,"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": "2026-09-08T03:33:04.239Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"939f96dc-0cc4-4608-9381-65aa8f24a6b5","format_issues":null,"number":6269} |
1.0.0 | problem:theoremdb-p9216 | problem | theoremdb-p9216 | Irrationality of the Riemann zeta value at 9 | {
"main": "math",
"sub": "number theory",
"slug": "number-theory",
"name": "Number theory",
"original_discipline": "mathematics"
} | [] | [
{
"code": "11",
"basis": "community-map",
"evidence": "Inferred from community field: number-theory"
}
] | The Riemann zeta function $\zeta(s)$ is defined for $\operatorname{Re}(s) > 1$ by $\zeta(s) = \sum_{n=1}^{\infty} n^{-s}$. Determine whether $\zeta(9)$ is irrational. That is, does there exist an irrational number $x$ such that $\zeta(9) = x$? | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P9216",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P9216",
"citation": null,
"doi": null,
"arxiv_id": null,
"license": "Selected packets label their content CC0-1.0; original references retain their own terms",
"notes": "Selected problem... | theorem irrational_nine : ∃ x, Irrational x ∧ riemannZeta 9 = x | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem irrational_nine : ∃ x, Irrational x ∧ riemannZeta 9 = x",
"target": null,
"source_url": "https://theoremdb.org/statements/P9216",
"format": "declaration-or-fragment",
"sha256": "26589278fa3e7fc69db7560aa41e577c79fa201db04b36207ad0e771487414d1",
"l... | [] | TheoremDB P9216 · Candidate awaiting source review | null | Source-reported assessment, retrieved 8 September 2026. Frontier Science has not independently reviewed this status.
Open. No signed Lean verification is attached to this problem or its selected resolution.
This record is in the source backlog; its novelty and unresolved status have not passed source review. | null | published | incremental | hard | [
"source-candidate",
"theoremdb"
] | null | null | null | null | [
{
"kind": "website",
"citation": "TheoremDB P9216",
"url": "https://theoremdb.org/statements/P9216",
"note": null,
"sort_order": 0
},
{
"kind": "formalization",
"citation": "Source formal statement (not locally verified)",
"url": "https://theoremdb.org/statements/P9216",
"not... | {
"source_record": "P9216",
"imported_at": "2026-09-08",
"status_as_of": "2026-09-08",
"evidence_label": "Source candidate · not reviewed",
"note": "Full public directory statement. Source lane: backlog; source resolution: open. Source verification claims are attributed only; no local Lean verification has be... | {"created_at":"2026-08-17T19:49:37+00:00","display_standard":{"editorial_review":"submitter_supplied","formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":9216,"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": "2026-09-08T03:33:01.826Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"650829fa-89fc-42c6-b7dd-bcc5210f5b5e","format_issues":null,"number":5743} |
1.0.0 | problem:theoremdb-p9218 | problem | theoremdb-p9218 | Transcendence of the fourth iterated power tower of π | {
"main": "math",
"sub": "number theory",
"slug": "number-theory",
"name": "Number theory",
"original_discipline": "mathematics"
} | [] | [
{
"code": "11",
"basis": "community-map",
"evidence": "Inferred from community field: number-theory"
}
] | A real number is called transcendental if it is not a root of any nonzero polynomial with rational coefficients. Let $\pi$ denote the ratio of the circumference of a circle to its diameter. Determine whether $\pi^{\pi^{\pi^{\pi}}}$ is transcendental over $\mathbb{Q}$. | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P9218",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P9218",
"citation": null,
"doi": null,
"arxiv_id": null,
"license": "Selected packets label their content CC0-1.0; original references retain their own terms",
"notes": "Selected problem... | theorem pi_pow_pi_pow_pi_pow_pi_transcendental : Transcendental ℚ (π ^ (π ^ (π ^ π))) | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem pi_pow_pi_pow_pi_pow_pi_transcendental : Transcendental ℚ (π ^ (π ^ (π ^ π)))",
"target": null,
"source_url": "https://theoremdb.org/statements/P9218",
"format": "declaration-or-fragment",
"sha256": "297ecf73d62f490607a559574b36bd3d5bd6831938c62ad2192... | [] | TheoremDB P9218 · Candidate awaiting source review | null | Source-reported assessment, retrieved 8 September 2026. Frontier Science has not independently reviewed this status.
Open. No signed Lean verification is attached to this problem or its selected resolution.
This record is in the source backlog; its novelty and unresolved status have not passed source review. | null | published | incremental | hard | [
"source-candidate",
"theoremdb"
] | null | null | null | null | [
{
"kind": "website",
"citation": "TheoremDB P9218",
"url": "https://theoremdb.org/statements/P9218",
"note": null,
"sort_order": 0
},
{
"kind": "formalization",
"citation": "Source formal statement (not locally verified)",
"url": "https://theoremdb.org/statements/P9218",
"not... | {
"source_record": "P9218",
"imported_at": "2026-09-08",
"status_as_of": "2026-09-08",
"evidence_label": "Source candidate · not reviewed",
"note": "Full public directory statement. Source lane: backlog; source resolution: open. Source verification claims are attributed only; no local Lean verification has be... | {"created_at":"2026-08-17T19:49:39+00:00","display_standard":{"editorial_review":"submitter_supplied","formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":9218,"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": "2026-09-08T03:33:07.003Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"a1e23d86-f7dd-48ba-b27a-798fb1aabdff","format_issues":null,"number":6864} |
1.0.0 | problem:theoremdb-p9220 | problem | theoremdb-p9220 | Non-existence of a ternary-integer monochromatic quantum graph for ten vertices and three colours | {
"main": "math",
"sub": "combinatorics",
"slug": "combinatorics",
"name": "Combinatorics",
"original_discipline": "mathematics"
} | [] | [
{
"code": "05",
"basis": "community-map",
"evidence": "Inferred from community field: combinatorics"
}
] | Let $N$ be a positive even integer and let $D$ be a positive integer. An edge of the complete graph $K_N$ is an unordered pair of distinct vertices, and a $D$-colouring of an edge is an assignment of a colour from the set $\{1,\dots,D\}$ to that edge. A weight function on the $D$-coloured edges of $K_N$ with values in ... | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P9220",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P9220",
"citation": null,
"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_trinary_int :
answer(sorry) ↔
¬ ∃ W : WeightsN 10 3 ℤ,
(∀ e, W e = (-1 : ℤ) ∨ W e = 0 ∨ W e = 1) ∧
EqSystemN 10 3 W | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem eqSystem10_no_solution_d3_trinary_int :\n answer(sorry) ↔\n ¬ ∃ W : WeightsN 10 3 ℤ,\n (∀ e, W e = (-1 : ℤ) ∨ W e = 0 ∨ W e = 1) ∧\n EqSystemN 10 3 W",
"target": null,
"source_url": "https://theoremdb.org/statements/P9220",
"fo... | [] | TheoremDB P9220 · Candidate awaiting source review | null | Source-reported assessment, retrieved 8 September 2026. Frontier Science has not independently reviewed this status.
Open. No signed Lean verification is attached to this problem or its selected resolution.
This record is in the source backlog; its novelty and unresolved status have not passed source review. | null | published | incremental | hard | [
"source-candidate",
"theoremdb"
] | null | null | null | null | [
{
"kind": "website",
"citation": "TheoremDB P9220",
"url": "https://theoremdb.org/statements/P9220",
"note": null,
"sort_order": 0
},
{
"kind": "formalization",
"citation": "Source formal statement (not locally verified)",
"url": "https://theoremdb.org/statements/P9220",
"not... | {
"source_record": "P9220",
"imported_at": "2026-09-08",
"status_as_of": "2026-09-08",
"evidence_label": "Source candidate · not reviewed",
"note": "Full public directory statement. Source lane: backlog; source resolution: open. Source verification claims are attributed only; no local Lean verification has be... | {"created_at":"2026-08-17T19:49:41+00:00","display_standard":{"editorial_review":"submitter_supplied","formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":9220,"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": "2026-09-08T03:33:04.275Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"d661142f-945d-4fa6-8caf-a7f9fbf96053","format_issues":null,"number":6277} |
1.0.0 | problem:theoremdb-p9222 | problem | theoremdb-p9222 | Furstenberg's times $p$, times $q$ conjecture | {
"main": "math",
"sub": "dynamical systems",
"slug": "dynamical-systems",
"name": "Dynamical systems",
"original_discipline": "mathematics"
} | [] | [
{
"code": "37",
"basis": "community-map",
"evidence": "Inferred from community field: dynamical-systems"
}
] | Let $p$ and $q$ be natural numbers with $p \geq 2$ and $q \geq 2$. Two integers $p, q \geq 2$ are called multiplicatively independent if $\log p / \log q$ is irrational. Let $\mathbb{T}$ denote the unit additive circle. For each natural number $n$, define the map $T_n : \mathbb{T} \to \mathbb{T}$ by $T_n(x) = nx \pmod{... | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P9222",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P9222",
"citation": null,
"doi": null,
"arxiv_id": null,
"license": "Selected packets label their content CC0-1.0; original references retain their own terms",
"notes": "Selected problem... | theorem conjecture_1_3 {p q : ℕ} (hp : 2 <= p) (hq : 2 <= q) (hpq : MultiplicativelyIndependent p q)
{μ : Measure 𝕋} [IsProbabilityMeasure μ] [MeasureTheory.IsAtomLess μ]
(hmup : MeasurePreserving (Tn p) μ μ)
(hmuq : MeasurePreserving (Tn q) μ μ) :
μ = volume | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem conjecture_1_3 {p q : ℕ} (hp : 2 <= p) (hq : 2 <= q) (hpq : MultiplicativelyIndependent p q)\n {μ : Measure 𝕋} [IsProbabilityMeasure μ] [MeasureTheory.IsAtomLess μ]\n (hmup : MeasurePreserving (Tn p) μ μ)\n (hmuq : MeasurePreserving (Tn q) μ μ) :\n μ = v... | [] | TheoremDB P9222 · Candidate awaiting source review | null | Source-reported assessment, retrieved 8 September 2026. Frontier Science has not independently reviewed this status.
Open. No signed Lean verification is attached to this problem or its selected resolution.
This record is in the source backlog; its novelty and unresolved status have not passed source review. | null | published | incremental | hard | [
"source-candidate",
"theoremdb"
] | null | null | null | null | [
{
"kind": "website",
"citation": "TheoremDB P9222",
"url": "https://theoremdb.org/statements/P9222",
"note": null,
"sort_order": 0
},
{
"kind": "formalization",
"citation": "Source formal statement (not locally verified)",
"url": "https://theoremdb.org/statements/P9222",
"not... | {
"source_record": "P9222",
"imported_at": "2026-09-08",
"status_as_of": "2026-09-08",
"evidence_label": "Source candidate · not reviewed",
"note": "Full public directory statement. Source lane: backlog; source resolution: open. Source verification claims are attributed only; no local Lean verification has be... | {"created_at":"2026-08-17T19:49:43+00:00","display_standard":{"editorial_review":"submitter_supplied","formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":9222,"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": "2026-09-08T03:32:59.234Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"30c69c58-48be-40d1-ae60-4708596032ba","format_issues":null,"number":5352} |
1.0.0 | problem:theoremdb-p9224 | problem | theoremdb-p9224 | Maximum cardinality of an isosceles set in Euclidean space | {
"main": "math",
"sub": "geometry and topology",
"slug": "geometry-and-topology",
"name": "Geometry and topology",
"original_discipline": "mathematics"
} | [] | [] | For each natural number $n$, let an isosceles set be a subset $A \subseteq \mathbb{R}^n$ such that for any three distinct points $x, y, z \in A$, at least two of the three distances $\|x - y\|$, $\|y - z\|$, $\|x - z\|$ are equal. Determine the maximum possible cardinality of an isosceles set in $\mathbb{R}^n$. | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P9224",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P9224",
"citation": null,
"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_503 (n : ℕ) :
IsGreatest {(A.ncard) | (A : Set (ℝ^n)) (hA : A.IsIsosceles)} answer(sorry) | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem erdos_503 (n : ℕ) :\n IsGreatest {(A.ncard) | (A : Set (ℝ^n)) (hA : A.IsIsosceles)} answer(sorry)",
"target": null,
"source_url": "https://theoremdb.org/statements/P9224",
"format": "declaration-or-fragment",
"sha256": "d7387dafef46bbc771c8b8512814... | [] | TheoremDB P9224 · Candidate awaiting source review | null | Source-reported assessment, retrieved 8 September 2026. Frontier Science has not independently reviewed this status.
Open. No signed Lean verification is attached to this problem or its selected resolution.
This record is in the source backlog; its novelty and unresolved status have not passed source review. | null | published | incremental | hard | [
"source-candidate",
"theoremdb"
] | null | null | null | null | [
{
"kind": "website",
"citation": "TheoremDB P9224",
"url": "https://theoremdb.org/statements/P9224",
"note": null,
"sort_order": 0
},
{
"kind": "formalization",
"citation": "Source formal statement (not locally verified)",
"url": "https://theoremdb.org/statements/P9224",
"not... | {
"source_record": "P9224",
"imported_at": "2026-09-08",
"status_as_of": "2026-09-08",
"evidence_label": "Source candidate · not reviewed",
"note": "Full public directory statement. Source lane: backlog; source resolution: open. Source verification claims are attributed only; no local Lean verification has be... | {"created_at":"2026-08-17T19:49:46+00:00","display_standard":{"formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":9224,"public_status":{"evidence":{"hasFormalization":false,"hasInformalProof":false,"hasRecordedWork":false,"solutionRecord":null},"lean":{"kernelAccepted":false,"label":"","reaso... | [
"formalization-contains-placeholder",
"formalization-not-compiled-during-export",
"current-status-unassessed",
"source-full-text-not-present"
] | [] | {
"created_at": "2026-09-08T03:32:15.935Z",
"updated_at": "2026-09-08T03:32:15.935Z",
"published_at": "2026-09-08T03:33:03.674Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"227dff99-c86a-4396-996e-a247678f7620","format_issues":null,"number":6145} |
1.0.0 | problem:theoremdb-p9226 | problem | theoremdb-p9226 | Asymptotic lower bound for the maximal covering exponent in Erdős's prime congruence problem | {
"main": "math",
"sub": "number theory",
"slug": "number-theory",
"name": "Number theory",
"original_discipline": "mathematics"
} | [] | [
{
"code": "11",
"basis": "community-map",
"evidence": "Inferred from community field: number-theory"
}
] | For each positive integer $n$, define $\epsilon_n$ to be the supremum of all real numbers $\epsilon$ for which there exists a choice of residue class $a_p$ for every prime $p$ with $n^\epsilon < p \leq n$ such that every integer $m$ with $1 \leq m \leq n$ satisfies at least one congruence $m \equiv a_p \pmod{p}$. Deter... | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P9226",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P9226",
"citation": null,
"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_688.parts.i.lower_bound :
(answer(sorry) : ℕ → ℝ) =O[atTop] epsilonFunction | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem erdos_688.parts.i.lower_bound :\n (answer(sorry) : ℕ → ℝ) =O[atTop] epsilonFunction",
"target": null,
"source_url": "https://theoremdb.org/statements/P9226",
"format": "declaration-or-fragment",
"sha256": "b81850c5e3c922e0201c1158232e5acaf0c7cea559... | [] | TheoremDB P9226 · Candidate awaiting source review | null | Source-reported assessment, retrieved 8 September 2026. Frontier Science has not independently reviewed this status.
Open. No signed Lean verification is attached to this problem or its selected resolution.
This record is in the source backlog; its novelty and unresolved status have not passed source review. | null | published | incremental | hard | [
"source-candidate",
"theoremdb"
] | null | null | null | null | [
{
"kind": "website",
"citation": "TheoremDB P9226",
"url": "https://theoremdb.org/statements/P9226",
"note": null,
"sort_order": 0
},
{
"kind": "formalization",
"citation": "Source formal statement (not locally verified)",
"url": "https://theoremdb.org/statements/P9226",
"not... | {
"source_record": "P9226",
"imported_at": "2026-09-08",
"status_as_of": "2026-09-08",
"evidence_label": "Source candidate · not reviewed",
"note": "Full public directory statement. Source lane: backlog; source resolution: open. Source verification claims are attributed only; no local Lean verification has be... | {"created_at":"2026-08-17T19:49:52+00:00","display_standard":{"editorial_review":"submitter_supplied","formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":9226,"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": "2026-09-08T03:32:50.327Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"e7759be8-acdf-4fb3-b1ba-526273f860d1","format_issues":null,"number":3397} |
1.0.0 | problem:theoremdb-p9228 | problem | theoremdb-p9228 | Asymptotic distribution function for a fractional-part 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 any real number $\alpha$ with $0 < \alpha < 1$ and any integer $n \geq 1$, define
$$
f(\alpha,n) = \frac{1}{\log n}\sum_{k=1}^{n}\left(\frac{1}{2} - \{\alpha k\}\right),
$$
where $\{x\} = x - \lfloor x \rfloor$ denotes the fractional part of $x$. Does there exist a non-decreasing function $g : \mathbb{R} \to \mat... | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P9228",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P9228",
"citation": null,
"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_1002 :
answer(sorry) ↔
∃ g : ℝ → ℝ, Monotone g ∧
Tendsto g atBot (𝓝 0) ∧
Tendsto g atTop (𝓝 1) ∧
letI f := fun (α : ℝ) (n : ℕ) ↦
(1 / log n) * ∑ k ∈ Icc (1 : ℕ) n, (1 / 2 - Int.fract (α * k))
∀ c : ℝ, Tendsto (fun (n : ℕ) ↦ (volume { α | α ∈ Ioo (0 : ℝ) 1 ∧ f α... | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem erdos_1002 :\n answer(sorry) ↔\n ∃ g : ℝ → ℝ, Monotone g ∧\n Tendsto g atBot (𝓝 0) ∧\n Tendsto g atTop (𝓝 1) ∧\n letI f := fun (α : ℝ) (n : ℕ) ↦\n (1 / log n) * ∑ k ∈ Icc (1 : ℕ) n, (1 / 2 - Int.fract (α * k))\n ∀ c : ℝ, Tendsto ... | [] | TheoremDB P9228 · Candidate awaiting source review | null | Source-reported assessment, retrieved 8 September 2026. Frontier Science has not independently reviewed this status.
Open. No signed Lean verification is attached to this problem or its selected resolution.
This record is in the source backlog; its novelty and unresolved status have not passed source review. | null | published | incremental | hard | [
"source-candidate",
"theoremdb"
] | null | null | null | null | [
{
"kind": "website",
"citation": "TheoremDB P9228",
"url": "https://theoremdb.org/statements/P9228",
"note": null,
"sort_order": 0
},
{
"kind": "formalization",
"citation": "Source formal statement (not locally verified)",
"url": "https://theoremdb.org/statements/P9228",
"not... | {
"source_record": "P9228",
"imported_at": "2026-09-08",
"status_as_of": "2026-09-08",
"evidence_label": "Source candidate · not reviewed",
"note": "Full public directory statement. Source lane: backlog; source resolution: open. Source verification claims are attributed only; no local Lean verification has be... | {"created_at":"2026-08-17T19:49:58+00:00","display_standard":{"editorial_review":"submitter_supplied","formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":9228,"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": "2026-09-08T03:32:50.020Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"14776ee7-2859-4528-ac19-f38b45bafccf","format_issues":null,"number":3334} |
1.0.0 | problem:theoremdb-p9230 | problem | theoremdb-p9230 | Erdős–Rosenfeld divisor bound in short intervals above the square root | {
"main": "math",
"sub": "number theory",
"slug": "number-theory",
"name": "Number theory",
"original_discipline": "mathematics"
} | [] | [
{
"code": "11",
"basis": "community-map",
"evidence": "Inferred from community field: number-theory"
}
] | Let $n$ be a positive integer and let $\sqrt{n}$ denote its positive square root. For a real number $x$, let $\lfloor x \rfloor$ denote the floor of $x$ (the greatest integer less than or equal to $x$) and $\lceil x \rceil$ denote the ceiling of $x$ (the least integer greater than or equal to $x$). For a real number $C... | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P9230",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P9230",
"citation": null,
"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_887.parts.i : ∀ C > (0 : ℝ), ∀ᶠ n in atTop,
#{ d ∈ Ioo ⌊√n⌋₊ ⌈√n + C * n^((1 : ℝ) / 4)⌉₊ | d ∣ n } ≤ answer(sorry) | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem erdos_887.parts.i : ∀ C > (0 : ℝ), ∀ᶠ n in atTop,\n #{ d ∈ Ioo ⌊√n⌋₊ ⌈√n + C * n^((1 : ℝ) / 4)⌉₊ | d ∣ n } ≤ answer(sorry)",
"target": null,
"source_url": "https://theoremdb.org/statements/P9230",
"format": "declaration-or-fragment",
"sha256": "50f... | [] | TheoremDB P9230 · Candidate awaiting source review | null | Source-reported assessment, retrieved 8 September 2026. Frontier Science has not independently reviewed this status.
Open. No signed Lean verification is attached to this problem or its selected resolution.
This record is in the source backlog; its novelty and unresolved status have not passed source review. | null | published | incremental | hard | [
"source-candidate",
"theoremdb"
] | null | null | null | null | [
{
"kind": "website",
"citation": "TheoremDB P9230",
"url": "https://theoremdb.org/statements/P9230",
"note": null,
"sort_order": 0
},
{
"kind": "formalization",
"citation": "Source formal statement (not locally verified)",
"url": "https://theoremdb.org/statements/P9230",
"not... | {
"source_record": "P9230",
"imported_at": "2026-09-08",
"status_as_of": "2026-09-08",
"evidence_label": "Source candidate · not reviewed",
"note": "Full public directory statement. Source lane: backlog; source resolution: open. Source verification claims are attributed only; no local Lean verification has be... | {"created_at":"2026-08-17T19:50:02+00:00","display_standard":{"editorial_review":"submitter_supplied","formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":9230,"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": "2026-09-08T03:32:56.701Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"d919a427-1e92-47e5-9a47-a54ce3595c46","format_issues":null,"number":4796} |
1.0.0 | problem:theoremdb-p9232 | problem | theoremdb-p9232 | Asymptotic constant for the number of distinct exponents in prime factorizations of factorials | {
"main": "math",
"sub": "number theory",
"slug": "number-theory",
"name": "Number theory",
"original_discipline": "mathematics"
} | [] | [
{
"code": "11",
"basis": "community-map",
"evidence": "Inferred from community field: number-theory"
}
] | Let $n! = \prod_{i} p_i^{k_i}$ denote the prime factorization of $n$ factorial into distinct primes $p_i$ with exponents $k_i$. Define $h(n)$ to be the number of distinct values among these exponents $k_i$. It is known that $h(n) \asymp \left(\frac{n}{\log n}\right)^{1/2}$ as $n \to \infty$. Prove that
$$
h(n) \sim \s... | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P9232",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P9232",
"citation": null,
"doi": null,
"arxiv_id": null,
"license": "Selected packets label their content CC0-1.0; original references retain their own terms",
"notes": "Selected problem... | theorem erdos_912.variants.tao :
(fun n => (h n : ℝ)) ~[atTop] (fun n => √(2 * Real.pi) * (n / Real.log n) ^ (1 / 2 : ℝ)) | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem erdos_912.variants.tao :\n (fun n => (h n : ℝ)) ~[atTop] (fun n => √(2 * Real.pi) * (n / Real.log n) ^ (1 / 2 : ℝ))",
"target": null,
"source_url": "https://theoremdb.org/statements/P9232",
"format": "declaration-or-fragment",
"sha256": "4f906d6c1b... | [] | TheoremDB P9232 · Candidate awaiting source review | null | Source-reported assessment, retrieved 8 September 2026. Frontier Science has not independently reviewed this status.
Open. No signed Lean verification is attached to this problem or its selected resolution.
This record is in the source backlog; its novelty and unresolved status have not passed source review. | null | published | incremental | hard | [
"source-candidate",
"theoremdb"
] | null | null | null | null | [
{
"kind": "website",
"citation": "TheoremDB P9232",
"url": "https://theoremdb.org/statements/P9232",
"note": null,
"sort_order": 0
},
{
"kind": "formalization",
"citation": "Source formal statement (not locally verified)",
"url": "https://theoremdb.org/statements/P9232",
"not... | {
"source_record": "P9232",
"imported_at": "2026-09-08",
"status_as_of": "2026-09-08",
"evidence_label": "Source candidate · not reviewed",
"note": "Full public directory statement. Source lane: backlog; source resolution: open. Source verification claims are attributed only; no local Lean verification has be... | {"created_at":"2026-08-17T19:50:06+00:00","display_standard":{"editorial_review":"submitter_supplied","formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":9232,"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": "2026-09-08T03:32:49.963Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"273680bb-ce73-4724-8858-58055dab72a9","format_issues":null,"number":3322} |
1.0.0 | problem:theoremdb-p9234 | problem | theoremdb-p9234 | Equality of Consecutive Integer Least Common Multiples | {
"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$, let $M(n, k)$ denote the least common multiple of the set of $k$ consecutive integers $\{n+1, n+2, \dotsc, n+k\}$. Is it true that for all natural numbers $m$, $n$, and $k$ with $k > 0$ and $m \geq n + k$, we have $M(m, k) \neq M(n, k)$? | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P9234",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P9234",
"citation": null,
"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_677 :
∀ (m n k : ℕ), k > 0 → m ≥ n + k → lcmInterval m k ≠ lcmInterval n k | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem erdos_677 :\n ∀ (m n k : ℕ), k > 0 → m ≥ n + k → lcmInterval m k ≠ lcmInterval n k",
"target": null,
"source_url": "https://theoremdb.org/statements/P9234",
"format": "declaration-or-fragment",
"sha256": "94458e9dcf06594121bf13520e22843a7418e281809... | [] | TheoremDB P9234 · Candidate awaiting source review | null | Source-reported assessment, retrieved 8 September 2026. Frontier Science has not independently reviewed this status.
Open. No signed Lean verification is attached to this problem or its selected resolution.
This record is in the source backlog; its novelty and unresolved status have not passed source review. | null | published | incremental | hard | [
"source-candidate",
"theoremdb"
] | null | null | null | null | [
{
"kind": "website",
"citation": "TheoremDB P9234",
"url": "https://theoremdb.org/statements/P9234",
"note": null,
"sort_order": 0
},
{
"kind": "formalization",
"citation": "Source formal statement (not locally verified)",
"url": "https://theoremdb.org/statements/P9234",
"not... | {
"source_record": "P9234",
"imported_at": "2026-09-08",
"status_as_of": "2026-09-08",
"evidence_label": "Source candidate · not reviewed",
"note": "Full public directory statement. Source lane: backlog; source resolution: open. Source verification claims are attributed only; no local Lean verification has be... | {"created_at":"2026-08-17T19:50:13+00:00","display_standard":{"formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":9234,"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": "2026-09-08T03:32:54.749Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"498973c3-6921-4e65-9d01-c2ad6515bbfa","format_issues":null,"number":4370} |
1.0.0 | problem:theoremdb-p9236 | problem | theoremdb-p9236 | Lower bound for the van der Waerden number W(3,26) | {
"main": "math",
"sub": "combinatorics",
"slug": "combinatorics",
"name": "Combinatorics",
"original_discipline": "mathematics"
} | [] | [
{
"code": "05",
"basis": "community-map",
"evidence": "Inferred from community field: combinatorics"
}
] | For positive integers $k$ and $r$, let $W(k,r)$ denote the least positive integer $N$ such that every coloring of $\{1, 2, \ldots, N\}$ with two colors contains either a monochromatic arithmetic progression of length $k$ in the first color or a monochromatic arithmetic progression of length $r$ in the second color. Det... | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P9236",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P9236",
"citation": null,
"doi": null,
"arxiv_id": null,
"license": "Selected packets label their content CC0-1.0; original references retain their own terms",
"notes": "Selected problem... | theorem W_3_26_lower : answer(sorry) ↔ W 3 26 ≥ 727 | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem W_3_26_lower : answer(sorry) ↔ W 3 26 ≥ 727",
"target": null,
"source_url": "https://theoremdb.org/statements/P9236",
"format": "declaration-or-fragment",
"sha256": "991d6b5f985a19766d3c8d55d8a8e791084129d7db23aef6edd757a1ac514092",
"license": "Se... | [] | TheoremDB P9236 · Candidate awaiting source review | null | Source-reported assessment, retrieved 8 September 2026. Frontier Science has not independently reviewed this status.
Open. No signed Lean verification is attached to this problem or its selected resolution.
This record is in the source backlog; its novelty and unresolved status have not passed source review. | null | published | incremental | hard | [
"source-candidate",
"theoremdb"
] | null | null | null | null | [
{
"kind": "website",
"citation": "TheoremDB P9236",
"url": "https://theoremdb.org/statements/P9236",
"note": null,
"sort_order": 0
},
{
"kind": "formalization",
"citation": "Source formal statement (not locally verified)",
"url": "https://theoremdb.org/statements/P9236",
"not... | {
"source_record": "P9236",
"imported_at": "2026-09-08",
"status_as_of": "2026-09-08",
"evidence_label": "Source candidate · not reviewed",
"note": "Full public directory statement. Source lane: backlog; source resolution: open. Source verification claims are attributed only; no local Lean verification has be... | {"created_at":"2026-08-17T19:50:15+00:00","display_standard":{"formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":9236,"public_status":{"evidence":{"hasFormalization":false,"hasInformalProof":false,"hasRecordedWork":false,"solutionRecord":null},"lean":{"kernelAccepted":false,"label":"","reaso... | [
"formalization-contains-placeholder",
"formalization-not-compiled-during-export",
"current-status-unassessed",
"source-full-text-not-present"
] | [] | {
"created_at": "2026-09-08T03:32:15.935Z",
"updated_at": "2026-09-08T03:32:15.935Z",
"published_at": "2026-09-08T03:33:03.542Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"6472b375-ccde-4b9e-a026-c089d3a074f1","format_issues":null,"number":6117} |
1.0.0 | problem:theoremdb-p9238 | problem | theoremdb-p9238 | Power-free values of irreducible polynomials with non-power-of-two degree | {
"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 \in \mathbb{Z}[X]$ be an irreducible polynomial with positive leading coefficient. Let $k$ denote the degree of $f$. Suppose that $k > 3$, that $k$ is not a power of $2$, and that for every prime $p$ there exists a natural number $n$ such that $p^{k-2}$ does not divide $f(n)$. Does it follow that there are infin... | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P9238",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P9238",
"citation": null,
"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_978.parts.ii : answer(sorry) ↔
∀ {f : ℤ[X]}, Irreducible f → f.natDegree > 3 →
(¬ ∃ l : ℕ, f.natDegree = 2 ^ l) → 0 < f.leadingCoeff →
(∀ (p : ℕ), p.Prime → ∃ n : ℕ, ¬ (p : ℤ) ^ (f.natDegree - 2) ∣ f.eval (n : ℤ)) →
{n : ℕ | Powerfree (f.natDegree - 2) (f.eval (n : ℤ))}.Infinite | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem erdos_978.parts.ii : answer(sorry) ↔\n ∀ {f : ℤ[X]}, Irreducible f → f.natDegree > 3 →\n (¬ ∃ l : ℕ, f.natDegree = 2 ^ l) → 0 < f.leadingCoeff →\n (∀ (p : ℕ), p.Prime → ∃ n : ℕ, ¬ (p : ℤ) ^ (f.natDegree - 2) ∣ f.eval (n : ℤ)) →\n {n : ℕ | Powerfree (f.nat... | [] | TheoremDB P9238 · Candidate awaiting source review | null | Source-reported assessment, retrieved 8 September 2026. Frontier Science has not independently reviewed this status.
Open. No signed Lean verification is attached to this problem or its selected resolution.
This record is in the source backlog; its novelty and unresolved status have not passed source review. | null | published | incremental | hard | [
"source-candidate",
"theoremdb"
] | null | null | null | null | [
{
"kind": "website",
"citation": "TheoremDB P9238",
"url": "https://theoremdb.org/statements/P9238",
"note": null,
"sort_order": 0
},
{
"kind": "formalization",
"citation": "Source formal statement (not locally verified)",
"url": "https://theoremdb.org/statements/P9238",
"not... | {
"source_record": "P9238",
"imported_at": "2026-09-08",
"status_as_of": "2026-09-08",
"evidence_label": "Source candidate · not reviewed",
"note": "Full public directory statement. Source lane: backlog; source resolution: open. Source verification claims are attributed only; no local Lean verification has be... | {"created_at":"2026-08-17T19:50:16+00:00","display_standard":{"formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":9238,"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": "2026-09-08T03:33:04.994Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"22c818af-7de0-4059-9e8a-333576709694","format_issues":null,"number":6430} |
1.0.0 | problem:theoremdb-p9240 | problem | theoremdb-p9240 | Upper bound for prime chains with consecutive congruence condition | {
"main": "math",
"sub": "number theory",
"slug": "number-theory",
"name": "Number theory",
"original_discipline": "mathematics"
} | [] | [
{
"code": "11",
"basis": "community-map",
"evidence": "Inferred from community field: number-theory"
}
] | A sequence of primes $q_1 < q_2 < \cdots$ is called a prime chain with consecutive congruence condition if $q_{i+1} \equiv 1 \pmod{q_i}$ for all $i \geq 1$. Determine whether there exists such a prime chain and a function $o : \mathbb{N} \to \mathbb{R}$ with $o(k) = o(1)$ (that is, $o(k) \to 0$ as $k \to \infty$) such ... | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P9240",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P9240",
"citation": null,
"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_695.variants.upperBound : answer(sorry) ↔
∃ q : ℕ → ℕ,
StrictMono q ∧
(∀ i, (q i).Prime) ∧
(∀ i, q (i + 1) % q i = 1) ∧
∃ o : ℕ → ℝ,
(o =o[atTop] (1 : ℕ → ℝ)) ∧
-- We use `(k + 1)` here as the informal statement is 1-indexed.
∀ k, q k ≤ exp ((k + 1) * lo... | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem erdos_695.variants.upperBound : answer(sorry) ↔\n ∃ q : ℕ → ℕ,\n StrictMono q ∧\n (∀ i, (q i).Prime) ∧\n (∀ i, q (i + 1) % q i = 1) ∧\n ∃ o : ℕ → ℝ,\n (o =o[atTop] (1 : ℕ → ℝ)) ∧\n -- We use `(k + 1)` here as the informal statemen... | [] | TheoremDB P9240 · Candidate awaiting source review | null | Source-reported assessment, retrieved 8 September 2026. Frontier Science has not independently reviewed this status.
Open. No signed Lean verification is attached to this problem or its selected resolution.
This record is in the source backlog; its novelty and unresolved status have not passed source review. | null | published | incremental | hard | [
"source-candidate",
"theoremdb"
] | null | null | null | null | [
{
"kind": "website",
"citation": "TheoremDB P9240",
"url": "https://theoremdb.org/statements/P9240",
"note": null,
"sort_order": 0
},
{
"kind": "formalization",
"citation": "Source formal statement (not locally verified)",
"url": "https://theoremdb.org/statements/P9240",
"not... | {
"source_record": "P9240",
"imported_at": "2026-09-08",
"status_as_of": "2026-09-08",
"evidence_label": "Source candidate · not reviewed",
"note": "Full public directory statement. Source lane: backlog; source resolution: open. Source verification claims are attributed only; no local Lean verification has be... | {"created_at":"2026-08-17T19:50:16+00:00","display_standard":{"editorial_review":"submitter_supplied","formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":9240,"public_status":{"evidence":{"hasFormalization":false,"hasInformalProof":false,"hasRecordedWork":false,"solutionRecord":null},"lean":{... | [
"formalization-contains-placeholder",
"formalization-not-compiled-during-export",
"current-status-unassessed",
"source-full-text-not-present"
] | [] | {
"created_at": "2026-09-08T03:32:15.935Z",
"updated_at": "2026-09-08T03:32:15.935Z",
"published_at": "2026-09-08T03:33:07.324Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"b8644ecc-3861-4473-be94-34bfecfccfc7","format_issues":null,"number":6935} |
1.0.0 | problem:theoremdb-p9242 | problem | theoremdb-p9242 | Minimum size of a universal arithmetic progression cover | {
"main": "math",
"sub": "combinatorics",
"slug": "combinatorics",
"name": "Combinatorics",
"original_discipline": "mathematics"
} | [] | [
{
"code": "05",
"basis": "community-map",
"evidence": "Inferred from community field: combinatorics"
}
] | For natural numbers $N$ and $k$, let a set $A \subseteq \mathbb{N}$ be called an \emph{arithmetic progression cover} for $(N, k)$ if for every common difference $d$ with $1 \leq d \leq N$, the set $A$ contains an arithmetic progression of length $k$ with common difference $d$. That is, for each such $d$ there exist int... | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P9242",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P9242",
"citation": null,
"doi": null,
"arxiv_id": null,
"license": "Selected packets label their content CC0-1.0; original references retain their own terms",
"notes": "Selected problem... | theorem green_37 (N k : ℕ) :
IsLeast { m | ∃ A : Finset ℕ, A.card = m ∧ IsAPCover (A : Set ℕ) N k } (answer(sorry)) | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem green_37 (N k : ℕ) :\n IsLeast { m | ∃ A : Finset ℕ, A.card = m ∧ IsAPCover (A : Set ℕ) N k } (answer(sorry))",
"target": null,
"source_url": "https://theoremdb.org/statements/P9242",
"format": "declaration-or-fragment",
"sha256": "a38e52b05b72618a... | [] | TheoremDB P9242 · Candidate awaiting source review | null | Source-reported assessment, retrieved 8 September 2026. Frontier Science has not independently reviewed this status.
Open. No signed Lean verification is attached to this problem or its selected resolution.
This record is in the source backlog; its novelty and unresolved status have not passed source review. | null | published | incremental | hard | [
"source-candidate",
"theoremdb"
] | null | null | null | null | [
{
"kind": "website",
"citation": "TheoremDB P9242",
"url": "https://theoremdb.org/statements/P9242",
"note": null,
"sort_order": 0
},
{
"kind": "formalization",
"citation": "Source formal statement (not locally verified)",
"url": "https://theoremdb.org/statements/P9242",
"not... | {
"source_record": "P9242",
"imported_at": "2026-09-08",
"status_as_of": "2026-09-08",
"evidence_label": "Source candidate · not reviewed",
"note": "Full public directory statement. Source lane: backlog; source resolution: open. Source verification claims are attributed only; no local Lean verification has be... | {"created_at":"2026-08-17T19:50:17+00:00","display_standard":{"editorial_review":"submitter_supplied","formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":9242,"public_status":{"evidence":{"hasFormalization":false,"hasInformalProof":false,"hasRecordedWork":false,"solutionRecord":null},"lean":{... | [
"formalization-contains-placeholder",
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"current-status-unassessed",
"source-full-text-not-present"
] | [] | {
"created_at": "2026-09-08T03:32:15.935Z",
"updated_at": "2026-09-08T03:32:15.935Z",
"published_at": "2026-09-08T03:33:04.036Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"542b204e-5d62-4c6e-a8e8-d0464d2c6a56","format_issues":null,"number":6225} |
1.0.0 | problem:theoremdb-p9244 | problem | theoremdb-p9244 | Erdős's problem on large prime factors 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$ and $k$ with $0 < k < n$, let $P(n,k)$ denote the largest prime factor of the binomial coefficient $\binom{n}{k}$. Does there exist a real number $c > 0$ such that $P(n,k) > \min\{n - k + 1, k^{1+c}\}$ holds for all positive integers $n$ and $k$ with $0 < k < n$? | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P9244",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P9244",
"citation": null,
"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 : answer(sorry) ↔
(∃ c > (0 : ℝ), ∀ n k : ℕ, 0 < k ∧ k < n → P n k > min (n - k + 1 : ℝ) (k ^ (1 + c))) | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem erdos_683 : answer(sorry) ↔\n (∃ c > (0 : ℝ), ∀ n k : ℕ, 0 < k ∧ k < n → P n k > min (n - k + 1 : ℝ) (k ^ (1 + c)))",
"target": null,
"source_url": "https://theoremdb.org/statements/P9244",
"format": "declaration-or-fragment",
"sha256": "c79d764e43... | [] | TheoremDB P9244 · Candidate awaiting source review | null | Source-reported assessment, retrieved 8 September 2026. Frontier Science has not independently reviewed this status.
Open. No signed Lean verification is attached to this problem or its selected resolution.
This record is in the source backlog; its novelty and unresolved status have not passed source review. | null | published | incremental | hard | [
"source-candidate",
"theoremdb"
] | null | null | null | null | [
{
"kind": "website",
"citation": "TheoremDB P9244",
"url": "https://theoremdb.org/statements/P9244",
"note": null,
"sort_order": 0
},
{
"kind": "formalization",
"citation": "Source formal statement (not locally verified)",
"url": "https://theoremdb.org/statements/P9244",
"not... | {
"source_record": "P9244",
"imported_at": "2026-09-08",
"status_as_of": "2026-09-08",
"evidence_label": "Source candidate · not reviewed",
"note": "Full public directory statement. Source lane: backlog; source resolution: open. Source verification claims are attributed only; no local Lean verification has be... | {"created_at":"2026-08-17T19:50:18+00:00","display_standard":{"editorial_review":"submitter_supplied","formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":9244,"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": "2026-09-08T03:32:56.500Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"fe38ff92-3329-437a-a6fc-3dbcfa1a4e65","format_issues":null,"number":4754} |
1.0.0 | problem:theoremdb-p9246 | problem | theoremdb-p9246 | The Banach–Mazur Rotation Problem | {
"main": "math",
"sub": "analysis",
"slug": "analysis",
"name": "Analysis",
"original_discipline": "mathematics"
} | [] | [] | Let $E$ be a separable real Banach space. Suppose the group of linear isometric automorphisms of $E$ acts transitively on the unit sphere $\{x \in E : \|x\| = 1\}$. Must $E$ be linearly isometric to a real Hilbert space? Equivalently, does there exist a real Hilbert space $H$ and a linear isometric bijection from $E$ o... | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P9246",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P9246",
"citation": null,
"doi": null,
"arxiv_id": null,
"license": "Selected packets label their content CC0-1.0; original references retain their own terms",
"notes": "Selected problem... | theorem banach_mazur_rotation_problem : answer(sorry) ↔
∀ (E : Type*) [NormedAddCommGroup E] [NormedSpace ℝ E] [CompleteSpace E] [SeparableSpace E]
[IsPretransitive (E ≃ₗᵢ[ℝ] E) (sphere (0 : E) 1)], ∃ (H : Type*) (_ : NormedAddCommGroup H)
(_ : InnerProductSpace ℝ H),
Nonempty (E ≃ₗᵢ[ℝ] H) | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem banach_mazur_rotation_problem : answer(sorry) ↔\n ∀ (E : Type*) [NormedAddCommGroup E] [NormedSpace ℝ E] [CompleteSpace E] [SeparableSpace E]\n [IsPretransitive (E ≃ₗᵢ[ℝ] E) (sphere (0 : E) 1)], ∃ (H : Type*) (_ : NormedAddCommGroup H)\n (_ : InnerProduc... | [] | TheoremDB P9246 · Candidate awaiting source review | null | Source-reported assessment, retrieved 8 September 2026. Frontier Science has not independently reviewed this status.
Open. No signed Lean verification is attached to this problem or its selected resolution.
This record is in the source backlog; its novelty and unresolved status have not passed source review. | null | published | incremental | hard | [
"source-candidate",
"theoremdb"
] | null | null | null | null | [
{
"kind": "website",
"citation": "TheoremDB P9246",
"url": "https://theoremdb.org/statements/P9246",
"note": null,
"sort_order": 0
},
{
"kind": "formalization",
"citation": "Source formal statement (not locally verified)",
"url": "https://theoremdb.org/statements/P9246",
"not... | {
"source_record": "P9246",
"imported_at": "2026-09-08",
"status_as_of": "2026-09-08",
"evidence_label": "Source candidate · not reviewed",
"note": "Full public directory statement. Source lane: backlog; source resolution: open. Source verification claims are attributed only; no local Lean verification has be... | {"created_at":"2026-08-17T19:50:21+00:00","display_standard":{"formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":9246,"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": "2026-09-08T03:33:06.370Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"b19d0873-656d-45bd-be74-a2e5c10ebadd","format_issues":null,"number":6728} |
1.0.0 | problem:theoremdb-p9248 | problem | theoremdb-p9248 | Selfridge's primality-test conjecture | {
"main": "math",
"sub": "number theory",
"slug": "number-theory",
"name": "Number theory",
"original_discipline": "mathematics"
} | [] | [
{
"code": "11",
"basis": "community-map",
"evidence": "Inferred from community field: number-theory"
}
] | Let $p$ be an odd natural number satisfying the following four conditions:
\begin{enumerate}
\item $p$ is odd,
\item $p \equiv 2 \pmod{5}$ or $p \equiv 3 \pmod{5}$,
\item $2^{p-1} \equiv 1 \pmod{p}$,
\item $F_{p+1} \equiv 0 \pmod{p}$, where $F_n$ denotes the $n$-th Fibonacci number defined by $F_0 = 0$, $F_1 = 1$, and ... | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P9248",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P9248",
"citation": null,
"doi": null,
"arxiv_id": null,
"license": "Selected packets label their content CC0-1.0; original references retain their own terms",
"notes": "Selected problem... | theorem selfridge_conjecture (p : ℕ) (hp : IsSelfridge p) : p.Prime | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem selfridge_conjecture (p : ℕ) (hp : IsSelfridge p) : p.Prime",
"target": null,
"source_url": "https://theoremdb.org/statements/P9248",
"format": "declaration-or-fragment",
"sha256": "8d581bdaa1ffe222ef78e5a4aa3994115eb99cfd3981ef61259272b353ee1323",
... | [] | TheoremDB P9248 · Candidate awaiting source review | null | Source-reported assessment, retrieved 8 September 2026. Frontier Science has not independently reviewed this status.
Open. No signed Lean verification is attached to this problem or its selected resolution.
This record is in the source backlog; its novelty and unresolved status have not passed source review. | null | published | incremental | hard | [
"source-candidate",
"theoremdb"
] | null | null | null | null | [
{
"kind": "website",
"citation": "TheoremDB P9248",
"url": "https://theoremdb.org/statements/P9248",
"note": null,
"sort_order": 0
},
{
"kind": "formalization",
"citation": "Source formal statement (not locally verified)",
"url": "https://theoremdb.org/statements/P9248",
"not... | {
"source_record": "P9248",
"imported_at": "2026-09-08",
"status_as_of": "2026-09-08",
"evidence_label": "Source candidate · not reviewed",
"note": "Full public directory statement. Source lane: backlog; source resolution: open. Source verification claims are attributed only; no local Lean verification has be... | {"created_at":"2026-08-17T19:50:23+00:00","display_standard":{"editorial_review":"submitter_supplied","formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":9248,"public_status":{"evidence":{"hasFormalization":false,"hasInformalProof":false,"hasRecordedWork":false,"solutionRecord":null},"lean":{... | [
"formalization-not-compiled-during-export",
"current-status-unassessed",
"source-full-text-not-present"
] | [] | {
"created_at": "2026-09-08T03:32:15.935Z",
"updated_at": "2026-09-08T03:32:15.935Z",
"published_at": "2026-09-08T03:33:05.770Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"091cecd6-3e71-4cfa-a109-3586eb6461a1","format_issues":null,"number":6599} |
1.0.0 | problem:theoremdb-p9250 | problem | theoremdb-p9250 | Asymptotic Constant for Factorial Product 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 integer $m$ such that $n!$ can be written as a product of distinct integers all greater than $n$, where $m$ is the largest of these integers. Equivalently, $f(n)$ is the infimum of the set of natural numbers $m$ for which there exist a positive integer $k$ a... | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P9250",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P9250",
"citation": null,
"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_390 :
answer(sorry) ↔ ∃ c,
(fun n => f n - 2 * n : ℕ → ℝ) ~[atTop] (fun n => c * n / log (n : ℝ)) | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem erdos_390 :\n answer(sorry) ↔ ∃ c,\n (fun n => f n - 2 * n : ℕ → ℝ) ~[atTop] (fun n => c * n / log (n : ℝ))",
"target": null,
"source_url": "https://theoremdb.org/statements/P9250",
"format": "declaration-or-fragment",
"sha256": "315fb4d80e22c... | [] | TheoremDB P9250 · Candidate awaiting source review | null | Source-reported assessment, retrieved 8 September 2026. Frontier Science has not independently reviewed this status.
Open. No signed Lean verification is attached to this problem or its selected resolution.
This record is in the source backlog; its novelty and unresolved status have not passed source review. | null | published | incremental | hard | [
"source-candidate",
"theoremdb"
] | null | null | null | null | [
{
"kind": "website",
"citation": "TheoremDB P9250",
"url": "https://theoremdb.org/statements/P9250",
"note": null,
"sort_order": 0
},
{
"kind": "formalization",
"citation": "Source formal statement (not locally verified)",
"url": "https://theoremdb.org/statements/P9250",
"not... | {
"source_record": "P9250",
"imported_at": "2026-09-08",
"status_as_of": "2026-09-08",
"evidence_label": "Source candidate · not reviewed",
"note": "Full public directory statement. Source lane: backlog; source resolution: open. Source verification claims are attributed only; no local Lean verification has be... | {"created_at":"2026-08-17T19:50:24+00:00","display_standard":{"editorial_review":"submitter_supplied","formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":9250,"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": "2026-09-08T03:32:49.947Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"b4e69eed-e332-4463-a2e5-61a23be1ac18","format_issues":null,"number":3319} |
1.0.0 | problem:theoremdb-p9252 | problem | theoremdb-p9252 | Products of Two Primes Modulo a Prime | {
"main": "math",
"sub": "number theory",
"slug": "number-theory",
"name": "Number theory",
"original_discipline": "mathematics"
} | [] | [
{
"code": "11",
"basis": "community-map",
"evidence": "Inferred from community field: number-theory"
}
] | Let $p$ be a prime. Let $A_p$ denote the set of all prime numbers that are strictly less than $p$. For each natural number $x$ with $1 \leq x < p$, we consider the residue class of $x$ in the finite field $\mathbb{Z}/p\mathbb{Z}$ with $p$ elements. Determine whether, for all sufficiently large primes $p$, every such re... | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P9252",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P9252",
"citation": null,
"doi": null,
"arxiv_id": null,
"license": "Selected packets label their content CC0-1.0; original references retain their own terms",
"notes": "Selected problem... | theorem green_62 :
answer(sorry) ↔
∀ᶠ p in atTop, p.Prime →
let A := (Finset.range p).filter Nat.Prime
∀ x : ℕ, 1 ≤ x ∧ x < p →
∃ a₁ ∈ A, ∃ a₂ ∈ A, (x : ZMod p) = (a₁ * a₂ : ZMod p) | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem green_62 :\n answer(sorry) ↔\n ∀ᶠ p in atTop, p.Prime →\n let A := (Finset.range p).filter Nat.Prime\n ∀ x : ℕ, 1 ≤ x ∧ x < p →\n ∃ a₁ ∈ A, ∃ a₂ ∈ A, (x : ZMod p) = (a₁ * a₂ : ZMod p)",
"target": null,
"source_url": "https://the... | [] | TheoremDB P9252 · Candidate awaiting source review | null | Source-reported assessment, retrieved 8 September 2026. Frontier Science has not independently reviewed this status.
Open. No signed Lean verification is attached to this problem or its selected resolution.
This record is in the source backlog; its novelty and unresolved status have not passed source review. | null | published | incremental | hard | [
"source-candidate",
"theoremdb"
] | null | null | null | null | [
{
"kind": "website",
"citation": "TheoremDB P9252",
"url": "https://theoremdb.org/statements/P9252",
"note": null,
"sort_order": 0
},
{
"kind": "formalization",
"citation": "Source formal statement (not locally verified)",
"url": "https://theoremdb.org/statements/P9252",
"not... | {
"source_record": "P9252",
"imported_at": "2026-09-08",
"status_as_of": "2026-09-08",
"evidence_label": "Source candidate · not reviewed",
"note": "Full public directory statement. Source lane: backlog; source resolution: open. Source verification claims are attributed only; no local Lean verification has be... | {"created_at":"2026-08-17T19:50:24+00:00","display_standard":{"formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":9252,"public_status":{"evidence":{"hasFormalization":false,"hasInformalProof":false,"hasRecordedWork":false,"solutionRecord":null},"lean":{"kernelAccepted":false,"label":"","reaso... | [
"formalization-contains-placeholder",
"formalization-not-compiled-during-export",
"current-status-unassessed",
"source-full-text-not-present"
] | [] | {
"created_at": "2026-09-08T03:32:15.935Z",
"updated_at": "2026-09-08T03:32:15.935Z",
"published_at": "2026-09-08T03:33:05.307Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"148211ef-b764-4435-bcce-63b3a3924551","format_issues":null,"number":6500} |
1.0.0 | problem:theoremdb-p9254 | problem | theoremdb-p9254 | The Tenth Dedekind Number | {
"main": "math",
"sub": "combinatorics",
"slug": "combinatorics",
"name": "Combinatorics",
"original_discipline": "mathematics"
} | [] | [
{
"code": "05",
"basis": "community-map",
"evidence": "Inferred from community field: combinatorics"
}
] | For a nonnegative integer $n$, let $M(n)$ denote the number of monotone Boolean functions on $n$ variables, or equivalently, the number of antichains (also called Sperner families) in the Boolean lattice of subsets of $\{1, 2, \ldots, n\}$. The known values include $M(0) = 2$, $M(1) = 3$, $M(2) = 6$, $M(3) = 20$, $M(4)... | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P9254",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P9254",
"citation": null,
"doi": null,
"arxiv_id": null,
"license": "Selected packets label their content CC0-1.0; original references retain their own terms",
"notes": "Selected problem... | theorem Dedekind_10 : M 10 = answer(sorry) | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem Dedekind_10 : M 10 = answer(sorry)",
"target": null,
"source_url": "https://theoremdb.org/statements/P9254",
"format": "declaration-or-fragment",
"sha256": "47429f78f2c3b66d39c59993f947ede3105344fcdd86939f82b27fab1b99e4d2",
"license": "Selected pa... | [] | TheoremDB P9254 · Candidate awaiting source review | null | Source-reported assessment, retrieved 8 September 2026. Frontier Science has not independently reviewed this status.
Open. No signed Lean verification is attached to this problem or its selected resolution.
This record is in the source backlog; its novelty and unresolved status have not passed source review. | null | published | incremental | hard | [
"source-candidate",
"theoremdb"
] | null | null | null | null | [
{
"kind": "website",
"citation": "TheoremDB P9254",
"url": "https://theoremdb.org/statements/P9254",
"note": null,
"sort_order": 0
},
{
"kind": "formalization",
"citation": "Source formal statement (not locally verified)",
"url": "https://theoremdb.org/statements/P9254",
"not... | {
"source_record": "P9254",
"imported_at": "2026-09-08",
"status_as_of": "2026-09-08",
"evidence_label": "Source candidate · not reviewed",
"note": "Full public directory statement. Source lane: backlog; source resolution: open. Source verification claims are attributed only; no local Lean verification has be... | {"created_at":"2026-08-17T19:50:25+00:00","display_standard":{"editorial_review":"submitter_supplied","formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":9254,"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": "2026-09-08T03:33:06.803Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"c9be3eed-ada5-4ce0-a6f1-adce41c13860","format_issues":null,"number":6821} |
1.0.0 | problem:theoremdb-p9256 | problem | theoremdb-p9256 | Irrationality of the Catalan constant | {
"main": "math",
"sub": "number theory",
"slug": "number-theory",
"name": "Number theory",
"original_discipline": "mathematics"
} | [] | [
{
"code": "11",
"basis": "community-map",
"evidence": "Inferred from community field: number-theory"
}
] | The Catalan constant $G$ is defined by the convergent series
$$
G \;=\; \sum_{n=0}^{\infty} \frac{(-1)^{n}}{(2n+1)^{2}}
\;\approx\; 0.915965\ldots
$$
Determine whether $G$ is irrational. | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P9256",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P9256",
"citation": null,
"doi": null,
"arxiv_id": null,
"license": "Selected packets label their content CC0-1.0; original references retain their own terms",
"notes": "Selected problem... | theorem irrational_catalanConstant :
answer(sorry) ↔ Irrational catalanConstant | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem irrational_catalanConstant :\n answer(sorry) ↔ Irrational catalanConstant",
"target": null,
"source_url": "https://theoremdb.org/statements/P9256",
"format": "declaration-or-fragment",
"sha256": "a5a40598b59b3f5e2cdeafc9f69306776a7355a3741c6a243d47... | [] | TheoremDB P9256 · Candidate awaiting source review | null | Source-reported assessment, retrieved 8 September 2026. Frontier Science has not independently reviewed this status.
Open. No signed Lean verification is attached to this problem or its selected resolution.
This record is in the source backlog; its novelty and unresolved status have not passed source review. | null | published | incremental | hard | [
"source-candidate",
"theoremdb"
] | null | null | null | null | [
{
"kind": "website",
"citation": "TheoremDB P9256",
"url": "https://theoremdb.org/statements/P9256",
"note": null,
"sort_order": 0
},
{
"kind": "formalization",
"citation": "Source formal statement (not locally verified)",
"url": "https://theoremdb.org/statements/P9256",
"not... | {
"source_record": "P9256",
"imported_at": "2026-09-08",
"status_as_of": "2026-09-08",
"evidence_label": "Source candidate · not reviewed",
"note": "Full public directory statement. Source lane: backlog; source resolution: open. Source verification claims are attributed only; no local Lean verification has be... | {"created_at":"2026-08-17T19:50:26+00:00","display_standard":{"editorial_review":"submitter_supplied","formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":9256,"public_status":{"evidence":{"hasFormalization":false,"hasInformalProof":false,"hasRecordedWork":false,"solutionRecord":null},"lean":{... | [
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"updated_at": "2026-09-08T03:32:15.935Z",
"published_at": "2026-09-08T03:33:01.742Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"05abf7e9-dd12-4c38-bc18-bd4b4b329c5f","format_issues":null,"number":5725} |
1.0.0 | problem:theoremdb-p9258 | problem | theoremdb-p9258 | Dirac's conjecture on 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 $k$ be an integer with $k \geq 4$. Does there exist a finite simple graph $G$ with chromatic number $k$ such that every vertex of $G$ is critical, yet every critical set of edges of $G$ has size greater than $1$? | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P9258",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P9258",
"citation": null,
"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 :
answer(sorry) ↔ ∀ k ≥ 4, ∃ (V : Type u) (G : SimpleGraph V), G.IsErdos944 k 1 | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem erdos_944.variants.dirac_conjecture :\n answer(sorry) ↔ ∀ k ≥ 4, ∃ (V : Type u) (G : SimpleGraph V), G.IsErdos944 k 1",
"target": null,
"source_url": "https://theoremdb.org/statements/P9258",
"format": "declaration-or-fragment",
"sha256": "07e5c229... | [] | TheoremDB P9258 · Candidate awaiting source review | null | Source-reported assessment, retrieved 8 September 2026. Frontier Science has not independently reviewed this status.
Open. No signed Lean verification is attached to this problem or its selected resolution.
This record is in the source backlog; its novelty and unresolved status have not passed source review. | null | published | incremental | hard | [
"source-candidate",
"theoremdb"
] | null | null | null | null | [
{
"kind": "website",
"citation": "TheoremDB P9258",
"url": "https://theoremdb.org/statements/P9258",
"note": null,
"sort_order": 0
},
{
"kind": "formalization",
"citation": "Source formal statement (not locally verified)",
"url": "https://theoremdb.org/statements/P9258",
"not... | {
"source_record": "P9258",
"imported_at": "2026-09-08",
"status_as_of": "2026-09-08",
"evidence_label": "Source candidate · not reviewed",
"note": "Full public directory statement. Source lane: backlog; source resolution: open. Source verification claims are attributed only; no local Lean verification has be... | {"created_at":"2026-08-17T19:50:27+00:00","display_standard":{"editorial_review":"submitter_supplied","formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":9258,"public_status":{"evidence":{"hasFormalization":false,"hasInformalProof":false,"hasRecordedWork":false,"solutionRecord":null},"lean":{... | [
"formalization-contains-placeholder",
"formalization-not-compiled-during-export",
"current-status-unassessed",
"source-full-text-not-present"
] | [] | {
"created_at": "2026-09-08T03:32:15.935Z",
"updated_at": "2026-09-08T03:32:15.935Z",
"published_at": "2026-09-08T03:32:53.177Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"23ddb698-9a1f-4755-a9b6-1c0d6ddd1647","format_issues":null,"number":4020} |
1.0.0 | problem:theoremdb-p9260 | problem | theoremdb-p9260 | Breakdown scenario for the Navier–Stokes equations on \(\mathbb{R}^3\) | {
"main": "math",
"sub": "partial differential equations",
"slug": "partial-differential-equations",
"name": "Partial differential equations",
"original_discipline": "mathematics"
} | [] | [
{
"code": "35",
"basis": "community-map",
"evidence": "Inferred from community field: partial-differential-equations"
}
] | Let $\nu > 0$ be a fixed viscosity constant. A vector field $u_0 : \mathbb{R}^3 \to \mathbb{R}^3$ satisfies the decaying initial velocity condition if it is smooth ($C^\infty$), divergence-free ($\nabla \cdot u_0 = 0$), and all derivatives of $u_0$ decay faster than any polynomial at infinity: for every derivative orde... | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P9260",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P9260",
"citation": null,
"doi": null,
"arxiv_id": null,
"license": "Selected packets label their content CC0-1.0; original references retain their own terms",
"notes": "Selected problem... | theorem navier_stokes_breakdown_R3 (nu : ℝ) (hnu : nu > 0) :
∃ (u₀ : ℝ³ → ℝ³) (f : ℝ³ → ℝ → ℝ³),
InitialVelocityConditionDecay u₀ ∧ ForceConditionDecay f ∧
¬ (∃ v p, NavierStokesExistenceAndSmoothnessRn nu u₀ f v p) | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem navier_stokes_breakdown_R3 (nu : ℝ) (hnu : nu > 0) :\n ∃ (u₀ : ℝ³ → ℝ³) (f : ℝ³ → ℝ → ℝ³),\n InitialVelocityConditionDecay u₀ ∧ ForceConditionDecay f ∧\n ¬ (∃ v p, NavierStokesExistenceAndSmoothnessRn nu u₀ f v p)",
"target": null,
"source_url": "htt... | [] | TheoremDB P9260 · Candidate awaiting source review | null | Source-reported assessment, retrieved 8 September 2026. Frontier Science has not independently reviewed this status.
Open. No signed Lean verification is attached to this problem or its selected resolution.
This record is in the source backlog; its novelty and unresolved status have not passed source review. | null | published | incremental | hard | [
"source-candidate",
"theoremdb"
] | null | null | null | null | [
{
"kind": "website",
"citation": "TheoremDB P9260",
"url": "https://theoremdb.org/statements/P9260",
"note": null,
"sort_order": 0
},
{
"kind": "formalization",
"citation": "Source formal statement (not locally verified)",
"url": "https://theoremdb.org/statements/P9260",
"not... | {
"source_record": "P9260",
"imported_at": "2026-09-08",
"status_as_of": "2026-09-08",
"evidence_label": "Source candidate · not reviewed",
"note": "Full public directory statement. Source lane: backlog; source resolution: open. Source verification claims are attributed only; no local Lean verification has be... | {"created_at":"2026-08-17T19:50:28+00:00","display_standard":{"editorial_review":"submitter_supplied","formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":9260,"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": "2026-09-08T03:32:50.767Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"1c9baa1e-4c03-4396-9d44-485c235ac4eb","format_issues":null,"number":3488} |
1.0.0 | problem:theoremdb-p9262 | problem | theoremdb-p9262 | Positive Density of Differences in 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 $A = \{a_1, a_2, a_3, \ldots\}$ be the greedy Sidon sequence defined as follows: begin with $a_1 = 1$, and for each $n \geq 2$, let $a_n$ be the smallest integer greater than $a_{n-1}$ such that the set $\{a_1, \ldots, a_n\}$ is a Sidon set. A set $S$ of integers is called a Sidon set if all pairwise sums $a + b$ w... | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P9262",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P9262",
"citation": null,
"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.sub_hasPosDensity :
Set.HasPosDensity (Set.range greedySidon - Set.range greedySidon) | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem erdos_340.variants.sub_hasPosDensity :\n Set.HasPosDensity (Set.range greedySidon - Set.range greedySidon)",
"target": null,
"source_url": "https://theoremdb.org/statements/P9262",
"format": "declaration-or-fragment",
"sha256": "aeddaf3c7f0a1fe46f2... | [] | TheoremDB P9262 · Candidate awaiting source review | null | Source-reported assessment, retrieved 8 September 2026. Frontier Science has not independently reviewed this status.
Open. No signed Lean verification is attached to this problem or its selected resolution.
This record is in the source backlog; its novelty and unresolved status have not passed source review. | null | published | incremental | hard | [
"source-candidate",
"theoremdb"
] | null | null | null | null | [
{
"kind": "website",
"citation": "TheoremDB P9262",
"url": "https://theoremdb.org/statements/P9262",
"note": null,
"sort_order": 0
},
{
"kind": "formalization",
"citation": "Source formal statement (not locally verified)",
"url": "https://theoremdb.org/statements/P9262",
"not... | {
"source_record": "P9262",
"imported_at": "2026-09-08",
"status_as_of": "2026-09-08",
"evidence_label": "Source candidate · not reviewed",
"note": "Full public directory statement. Source lane: backlog; source resolution: open. Source verification claims are attributed only; no local Lean verification has be... | {"created_at":"2026-08-17T19:50:29+00:00","display_standard":{"formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":9262,"public_status":{"evidence":{"hasFormalization":false,"hasInformalProof":false,"hasRecordedWork":false,"solutionRecord":null},"lean":{"kernelAccepted":false,"label":"","reaso... | [
"formalization-not-compiled-during-export",
"current-status-unassessed",
"source-full-text-not-present"
] | [] | {
"created_at": "2026-09-08T03:32:15.935Z",
"updated_at": "2026-09-08T03:32:15.935Z",
"published_at": "2026-09-08T03:33:04.935Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"f4103b3c-59dd-4c6e-9a58-59a070609eda","format_issues":null,"number":6418} |
1.0.0 | problem:theoremdb-p9264 | problem | theoremdb-p9264 | Distinctness of sums of fifth powers for unequal nonnegative integers | {
"main": "math",
"sub": "number theory",
"slug": "number-theory",
"name": "Number theory",
"original_discipline": "mathematics"
} | [] | [
{
"code": "11",
"basis": "community-map",
"evidence": "Inferred from community field: number-theory"
}
] | Let $S$ denote the set of ordered pairs $(a,b)$ of nonnegative integers with $a < b$. Consider the function $F: S \to \mathbb{N}$ defined by $F(a,b) = a^5 + b^5$. Determine whether $F$ is injective on $S$, i.e., whether for all $(a,b), (c,d) \in S$, the equality $a^5 + b^5 = c^5 + d^5$ implies $(a,b) = (c,d)$. | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P9264",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P9264",
"citation": null,
"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_324.variants.quintic : {(a, b) : ℕ × ℕ | a < b}.InjOn fun (a, b) => a ^ 5 + b ^ 5 | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem erdos_324.variants.quintic : {(a, b) : ℕ × ℕ | a < b}.InjOn fun (a, b) => a ^ 5 + b ^ 5",
"target": null,
"source_url": "https://theoremdb.org/statements/P9264",
"format": "declaration-or-fragment",
"sha256": "6b556daa0de8df10b5b87463c5effdf41121e8e0c... | [] | TheoremDB P9264 · Candidate awaiting source review | null | Source-reported assessment, retrieved 8 September 2026. Frontier Science has not independently reviewed this status.
Open. No signed Lean verification is attached to this problem or its selected resolution.
This record is in the source backlog; its novelty and unresolved status have not passed source review. | null | published | incremental | hard | [
"source-candidate",
"theoremdb"
] | null | null | null | null | [
{
"kind": "website",
"citation": "TheoremDB P9264",
"url": "https://theoremdb.org/statements/P9264",
"note": null,
"sort_order": 0
},
{
"kind": "formalization",
"citation": "Source formal statement (not locally verified)",
"url": "https://theoremdb.org/statements/P9264",
"not... | {
"source_record": "P9264",
"imported_at": "2026-09-08",
"status_as_of": "2026-09-08",
"evidence_label": "Source candidate · not reviewed",
"note": "Full public directory statement. Source lane: backlog; source resolution: open. Source verification claims are attributed only; no local Lean verification has be... | {"created_at":"2026-08-17T19:50:29+00:00","display_standard":{"editorial_review":"submitter_supplied","formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":9264,"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": "2026-09-08T03:32:53.262Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"5443a7bf-1d05-4a02-b735-c0b9b1003ae2","format_issues":null,"number":4038} |
1.0.0 | problem:theoremdb-p9266 | problem | theoremdb-p9266 | Lower bound for the 2-color van der Waerden number W(3, 21) | {
"main": "math",
"sub": "combinatorics",
"slug": "combinatorics",
"name": "Combinatorics",
"original_discipline": "mathematics"
} | [] | [
{
"code": "05",
"basis": "community-map",
"evidence": "Inferred from community field: combinatorics"
}
] | For positive integers $k$ and $r$, the 2-color van der Waerden number $W(k, r)$ is defined as the least positive integer $N$ such that every coloring of $\{1, 2, \ldots, N\}$ with two colors contains either a monochromatic arithmetic progression of length $k$ in the first color or a monochromatic arithmetic progression... | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P9266",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P9266",
"citation": null,
"doi": null,
"arxiv_id": null,
"license": "Selected packets label their content CC0-1.0; original references retain their own terms",
"notes": "Selected problem... | theorem W_3_21_lower : answer(sorry) ↔ W 3 21 ≥ 416 | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem W_3_21_lower : answer(sorry) ↔ W 3 21 ≥ 416",
"target": null,
"source_url": "https://theoremdb.org/statements/P9266",
"format": "declaration-or-fragment",
"sha256": "e2a76e957228f39b10bccf15a68b8ed5902bdce07c275ea804d86d8066ad080a",
"license": "Se... | [] | TheoremDB P9266 · Candidate awaiting source review | null | Source-reported assessment, retrieved 8 September 2026. Frontier Science has not independently reviewed this status.
Open. No signed Lean verification is attached to this problem or its selected resolution.
This record is in the source backlog; its novelty and unresolved status have not passed source review. | null | published | incremental | hard | [
"source-candidate",
"theoremdb"
] | null | null | null | null | [
{
"kind": "website",
"citation": "TheoremDB P9266",
"url": "https://theoremdb.org/statements/P9266",
"note": null,
"sort_order": 0
},
{
"kind": "formalization",
"citation": "Source formal statement (not locally verified)",
"url": "https://theoremdb.org/statements/P9266",
"not... | {
"source_record": "P9266",
"imported_at": "2026-09-08",
"status_as_of": "2026-09-08",
"evidence_label": "Source candidate · not reviewed",
"note": "Full public directory statement. Source lane: backlog; source resolution: open. Source verification claims are attributed only; no local Lean verification has be... | {"created_at":"2026-08-17T19:50:30+00:00","display_standard":{"editorial_review":"submitter_supplied","formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":9266,"public_status":{"evidence":{"hasFormalization":false,"hasInformalProof":false,"hasRecordedWork":false,"solutionRecord":null},"lean":{... | [
"formalization-contains-placeholder",
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"current-status-unassessed",
"source-full-text-not-present"
] | [] | {
"created_at": "2026-09-08T03:32:15.935Z",
"updated_at": "2026-09-08T03:32:15.935Z",
"published_at": "2026-09-08T03:33:03.424Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"c8d1220f-4792-4d66-a502-cadb158e11e7","format_issues":null,"number":6093} |
1.0.0 | problem:theoremdb-p9268 | problem | theoremdb-p9268 | Local connectivity of Multibrot sets | {
"main": "math",
"sub": "dynamical systems",
"slug": "dynamical-systems",
"name": "Dynamical systems",
"original_discipline": "mathematics"
} | [] | [
{
"code": "37",
"basis": "community-map",
"evidence": "Inferred from community field: dynamical-systems"
}
] | For every natural number $n$, the multibrot set of power $n$ is locally connected. The multibrot set of power $n$ is defined as the set of all complex parameters $c$ for which the orbit of $0$ under iteration of the map $z \mapsto z^n + c$ does not escape to infinity; equivalently, it is the set of all $c \in \mathbb{C... | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P9268",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P9268",
"citation": null,
"doi": null,
"arxiv_id": null,
"license": "Selected packets label their content CC0-1.0; original references retain their own terms",
"notes": "Selected problem... | theorem MLC_general_exponent (n : ℕ) : LocallyConnectedSpace (multibrotSet n) | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem MLC_general_exponent (n : ℕ) : LocallyConnectedSpace (multibrotSet n)",
"target": null,
"source_url": "https://theoremdb.org/statements/P9268",
"format": "declaration-or-fragment",
"sha256": "334282d6b455c5d32a1ee2c3b3feca1586ad01bcd146a7ced02843c0e5b... | [] | TheoremDB P9268 · Candidate awaiting source review | null | Source-reported assessment, retrieved 8 September 2026. Frontier Science has not independently reviewed this status.
Open. No signed Lean verification is attached to this problem or its selected resolution.
This record is in the source backlog; its novelty and unresolved status have not passed source review. | null | published | incremental | hard | [
"source-candidate",
"theoremdb"
] | null | null | null | null | [
{
"kind": "website",
"citation": "TheoremDB P9268",
"url": "https://theoremdb.org/statements/P9268",
"note": null,
"sort_order": 0
},
{
"kind": "formalization",
"citation": "Source formal statement (not locally verified)",
"url": "https://theoremdb.org/statements/P9268",
"not... | {
"source_record": "P9268",
"imported_at": "2026-09-08",
"status_as_of": "2026-09-08",
"evidence_label": "Source candidate · not reviewed",
"note": "Full public directory statement. Source lane: backlog; source resolution: open. Source verification claims are attributed only; no local Lean verification has be... | {"created_at":"2026-08-17T19:50:30+00:00","display_standard":{"editorial_review":"submitter_supplied","formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":9268,"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": "2026-09-08T03:33:03.292Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"128b49fa-699e-44f1-9c22-77f29921ab91","format_issues":null,"number":6064} |
1.0.0 | problem:theoremdb-p9270 | problem | theoremdb-p9270 | Least square side length for packing 21 unit circles | {
"main": "math",
"sub": "geometry and topology",
"slug": "geometry-and-topology",
"name": "Geometry and topology",
"original_discipline": "mathematics"
} | [] | [] | Let a unit circle be the set of points in the Euclidean plane at distance strictly less than 1 from the origin. For a real number $x > 0$, let the square of side length $x$ be the set of points $(a,b) \in \mathbb{R}^2$ satisfying $0 < a < x$ and $0 < b < x$. A packing of $n$ copies of a set $s$ inside a set $S$ consist... | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P9270",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P9270",
"citation": null,
"doi": null,
"arxiv_id": null,
"license": "Selected packets label their content CC0-1.0; original references retain their own terms",
"notes": "Selected problem... | theorem least_twenty_one_circle_packing_in_square :
IsLeast {x : ℝ | Nonempty (Packing 21 UnitCircle (Square x))} answer(sorry) | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem least_twenty_one_circle_packing_in_square :\n IsLeast {x : ℝ | Nonempty (Packing 21 UnitCircle (Square x))} answer(sorry)",
"target": null,
"source_url": "https://theoremdb.org/statements/P9270",
"format": "declaration-or-fragment",
"sha256": "abd2... | [] | TheoremDB P9270 · Candidate awaiting source review | null | Source-reported assessment, retrieved 8 September 2026. Frontier Science has not independently reviewed this status.
Open. No signed Lean verification is attached to this problem or its selected resolution.
This record is in the source backlog; its novelty and unresolved status have not passed source review. | null | published | incremental | hard | [
"source-candidate",
"theoremdb"
] | null | null | null | null | [
{
"kind": "website",
"citation": "TheoremDB P9270",
"url": "https://theoremdb.org/statements/P9270",
"note": null,
"sort_order": 0
},
{
"kind": "formalization",
"citation": "Source formal statement (not locally verified)",
"url": "https://theoremdb.org/statements/P9270",
"not... | {
"source_record": "P9270",
"imported_at": "2026-09-08",
"status_as_of": "2026-09-08",
"evidence_label": "Source candidate · not reviewed",
"note": "Full public directory statement. Source lane: backlog; source resolution: open. Source verification claims are attributed only; no local Lean verification has be... | {"created_at":"2026-08-17T19:50:31+00:00","display_standard":{"editorial_review":"submitter_supplied","formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":9270,"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": "2026-09-08T03:33:03.126Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"3edb51ef-2bc0-4f43-b02e-14dab4bba81f","format_issues":null,"number":6028} |
1.0.0 | problem:theoremdb-p9272 | problem | theoremdb-p9272 | Logarithmic density of integers avoiding prescribed residue classes | {
"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 $X_n$ be a subset of the ring $\mathbb{Z}/n\mathbb{Z}$ of integers modulo $n$. Define the set
$$
B = \{m \in \mathbb{N} : \text{for every positive integer } n, \text{ the residue class of } m \text{ modulo } n \text{ does not belong to } X_n\}.
$$
Must $B$ have a logarithmic densi... | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P9272",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P9272",
"citation": null,
"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_486 : answer(sorry) ↔
∀ X : (n : ℕ) → Set (ZMod n), ∃ d, {m : ℕ | ∀ n, (m : ZMod n) ∉ X n}.HasLogDensity d | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem erdos_486 : answer(sorry) ↔\n ∀ X : (n : ℕ) → Set (ZMod n), ∃ d, {m : ℕ | ∀ n, (m : ZMod n) ∉ X n}.HasLogDensity d",
"target": null,
"source_url": "https://theoremdb.org/statements/P9272",
"format": "declaration-or-fragment",
"sha256": "50c5c3cb315... | [] | TheoremDB P9272 · Candidate awaiting source review | null | Source-reported assessment, retrieved 8 September 2026. Frontier Science has not independently reviewed this status.
Open. No signed Lean verification is attached to this problem or its selected resolution.
This record is in the source backlog; its novelty and unresolved status have not passed source review. | null | published | incremental | hard | [
"source-candidate",
"theoremdb"
] | null | null | null | null | [
{
"kind": "website",
"citation": "TheoremDB P9272",
"url": "https://theoremdb.org/statements/P9272",
"note": null,
"sort_order": 0
},
{
"kind": "formalization",
"citation": "Source formal statement (not locally verified)",
"url": "https://theoremdb.org/statements/P9272",
"not... | {
"source_record": "P9272",
"imported_at": "2026-09-08",
"status_as_of": "2026-09-08",
"evidence_label": "Source candidate · not reviewed",
"note": "Full public directory statement. Source lane: backlog; source resolution: open. Source verification claims are attributed only; no local Lean verification has be... | {"created_at":"2026-08-17T19:50:31+00:00","display_standard":{"editorial_review":"submitter_supplied","formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":9272,"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": "2026-09-08T03:33:03.322Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"4648543c-2dbe-460a-8040-0d431bddea3c","format_issues":null,"number":6070} |
1.0.0 | problem:theoremdb-p9274 | problem | theoremdb-p9274 | Transcendence of the product of e and π | {
"main": "math",
"sub": "number theory",
"slug": "number-theory",
"name": "Number theory",
"original_discipline": "mathematics"
} | [] | [
{
"code": "11",
"basis": "community-map",
"evidence": "Inferred from community field: number-theory"
}
] | A real number is called transcendental if it is not a root of any nonzero polynomial with rational coefficients. Let $e = \exp(1)$ denote the base of the natural logarithm, and let $\pi$ denote the ratio of the circumference of a circle to its diameter. Prove that the product $e \pi$ is transcendental. | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P9274",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P9274",
"citation": null,
"doi": null,
"arxiv_id": null,
"license": "Selected packets label their content CC0-1.0; original references retain their own terms",
"notes": "Selected problem... | theorem exp_mul_pi_transcendental : Transcendental ℚ (exp 1 * π) | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem exp_mul_pi_transcendental : Transcendental ℚ (exp 1 * π)",
"target": null,
"source_url": "https://theoremdb.org/statements/P9274",
"format": "declaration-or-fragment",
"sha256": "c7d94cd0c53da26c44282e8a75e0c04c95e5feb7859682c18ea3c9553fbf1363",
"... | [] | TheoremDB P9274 · Candidate awaiting source review | null | Source-reported assessment, retrieved 8 September 2026. Frontier Science has not independently reviewed this status.
Open. No signed Lean verification is attached to this problem or its selected resolution.
This record is in the source backlog; its novelty and unresolved status have not passed source review. | null | published | incremental | hard | [
"source-candidate",
"theoremdb"
] | null | null | null | null | [
{
"kind": "website",
"citation": "TheoremDB P9274",
"url": "https://theoremdb.org/statements/P9274",
"note": null,
"sort_order": 0
},
{
"kind": "formalization",
"citation": "Source formal statement (not locally verified)",
"url": "https://theoremdb.org/statements/P9274",
"not... | {
"source_record": "P9274",
"imported_at": "2026-09-08",
"status_as_of": "2026-09-08",
"evidence_label": "Source candidate · not reviewed",
"note": "Full public directory statement. Source lane: backlog; source resolution: open. Source verification claims are attributed only; no local Lean verification has be... | {"created_at":"2026-08-17T19:50:33+00:00","display_standard":{"editorial_review":"submitter_supplied","formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":9274,"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": "2026-09-08T03:33:07.021Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"6f48ca41-b860-449b-b0a1-42804f1965a0","format_issues":null,"number":6868} |
1.0.0 | problem:theoremdb-p9276 | problem | theoremdb-p9276 | Infinitely many factorial indices with strict descending gaps in consecutive covering moduli | {
"main": "math",
"sub": "number theory",
"slug": "number-theory",
"name": "Number theory",
"original_discipline": "mathematics"
} | [] | [
{
"code": "11",
"basis": "community-map",
"evidence": "Inferred from community field: number-theory"
}
] | For positive integers $k$ and $n$, let $t_k(n)$ denote the least positive integer $m$ such that $n$ divides the product $m(m+1)(m+2)\cdots(m+k-1)$. Determine whether there exist infinitely many positive integers $n$ such that for every integer $k$ with $2 \leq k < n$, we have $t_k(n!) < t_{k-1}(n!) - 1$. | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P9276",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P9276",
"citation": null,
"doi": null,
"arxiv_id": null,
"license": "Selected packets label their content CC0-1.0; original references retain their own terms",
"notes": "Selected problem... | theorem erdos_394.variants.factorial_gap_conjecture :
answer(sorry) ↔
Set.Infinite { n : ℕ | ∀ k, 2 ≤ k → k < n →
t k (n !) < t (k - 1) (n !) - 1 } | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem erdos_394.variants.factorial_gap_conjecture :\n answer(sorry) ↔\n Set.Infinite { n : ℕ | ∀ k, 2 ≤ k → k < n →\n t k (n !) < t (k - 1) (n !) - 1 }",
"target": null,
"source_url": "https://theoremdb.org/statements/P9276",
"format": "declaration... | [] | TheoremDB P9276 · Candidate awaiting source review | null | Source-reported assessment, retrieved 8 September 2026. Frontier Science has not independently reviewed this status.
Open. No signed Lean verification is attached to this problem or its selected resolution.
This record is in the source backlog; its novelty and unresolved status have not passed source review. | null | published | incremental | hard | [
"source-candidate",
"theoremdb"
] | null | null | null | null | [
{
"kind": "website",
"citation": "TheoremDB P9276",
"url": "https://theoremdb.org/statements/P9276",
"note": null,
"sort_order": 0
},
{
"kind": "formalization",
"citation": "Source formal statement (not locally verified)",
"url": "https://theoremdb.org/statements/P9276",
"not... | {
"source_record": "P9276",
"imported_at": "2026-09-08",
"status_as_of": "2026-09-08",
"evidence_label": "Source candidate · not reviewed",
"note": "Full public directory statement. Source lane: backlog; source resolution: open. Source verification claims are attributed only; no local Lean verification has be... | {"created_at":"2026-08-17T19:50:34+00:00","display_standard":{"editorial_review":"submitter_supplied","formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":9276,"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": "2026-09-08T03:33:00.891Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"a295d887-714c-4c14-9a0f-2edae0dfb721","format_issues":null,"number":5612} |
1.0.0 | problem:theoremdb-p9278 | problem | theoremdb-p9278 | Existence of an Infinite Set of Totient Values with Superlinear Minimal Preimages | {
"main": "math",
"sub": "number theory",
"slug": "number-theory",
"name": "Number theory",
"original_discipline": "mathematics"
} | [] | [
{
"code": "11",
"basis": "community-map",
"evidence": "Inferred from community field: number-theory"
}
] | Let $\varphi$ denote Euler's totient function, which counts the number of integers in $\{1, 2, \ldots, n\}$ that are coprime to $n$. For a positive integer $a$, let $\varphi^{-1}(\{a\})$ denote the set of all positive integers $n$ such that $\varphi(n) = a$, and when this set is nonempty, let $n_a$ denote its least ele... | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P9278",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P9278",
"citation": null,
"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_51 : answer(sorry) ↔ ∃ A : Set ℕ, ∃ n : A → ℕ,
A.Infinite ∧
(∀ a : A, IsLeast (φ ⁻¹' {(a : ℕ)}) (n a)) ∧
Tendsto (fun a : A => (n a : ℝ) / (a : ℝ)) atTop atTop | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem erdos_51 : answer(sorry) ↔ ∃ A : Set ℕ, ∃ n : A → ℕ,\n A.Infinite ∧\n (∀ a : A, IsLeast (φ ⁻¹' {(a : ℕ)}) (n a)) ∧\n Tendsto (fun a : A => (n a : ℝ) / (a : ℝ)) atTop atTop",
"target": null,
"source_url": "https://theoremdb.org/statements/P9278"... | [] | TheoremDB P9278 · Candidate awaiting source review | null | Source-reported assessment, retrieved 8 September 2026. Frontier Science has not independently reviewed this status.
Open. No signed Lean verification is attached to this problem or its selected resolution.
This record is in the source backlog; its novelty and unresolved status have not passed source review. | null | published | incremental | hard | [
"source-candidate",
"theoremdb"
] | null | null | null | null | [
{
"kind": "website",
"citation": "TheoremDB P9278",
"url": "https://theoremdb.org/statements/P9278",
"note": null,
"sort_order": 0
},
{
"kind": "formalization",
"citation": "Source formal statement (not locally verified)",
"url": "https://theoremdb.org/statements/P9278",
"not... | {
"source_record": "P9278",
"imported_at": "2026-09-08",
"status_as_of": "2026-09-08",
"evidence_label": "Source candidate · not reviewed",
"note": "Full public directory statement. Source lane: backlog; source resolution: open. Source verification claims are attributed only; no local Lean verification has be... | {"created_at":"2026-08-17T19:50:34+00:00","display_standard":{"editorial_review":"submitter_supplied","formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":9278,"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": "2026-09-08T03:32:58.692Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"fb81c464-5959-4ae4-9bfd-fe1d9c6f0b98","format_issues":null,"number":5235} |
1.0.0 | problem:theoremdb-p9280 | problem | theoremdb-p9280 | Asymptotic order of the cofactor threshold function | {
"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 the following holds: for every set $A$ of $n$ non-negative integers, the set
$$
\left\{ \frac{a}{(a,b)} : a, b \in A \right\}
$$
has cardinality at least $m$, where $(a,b)$ denotes the greatest common divisor of $a$ and $b$. Determine wh... | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P9280",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P9280",
"citation": null,
"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_539.variants.sq :
(fun n ↦ (cofactorThreshold n : ℝ)) =Θ[atTop] fun n ↦ √n | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem erdos_539.variants.sq :\n (fun n ↦ (cofactorThreshold n : ℝ)) =Θ[atTop] fun n ↦ √n",
"target": null,
"source_url": "https://theoremdb.org/statements/P9280",
"format": "declaration-or-fragment",
"sha256": "0e3db6ee8f72402a35f881419f4f4c931c94a713801... | [] | TheoremDB P9280 · Candidate awaiting source review | null | Source-reported assessment, retrieved 8 September 2026. Frontier Science has not independently reviewed this status.
Open. No signed Lean verification is attached to this problem or its selected resolution.
This record is in the source backlog; its novelty and unresolved status have not passed source review. | null | published | incremental | hard | [
"source-candidate",
"theoremdb"
] | null | null | null | null | [
{
"kind": "website",
"citation": "TheoremDB P9280",
"url": "https://theoremdb.org/statements/P9280",
"note": null,
"sort_order": 0
},
{
"kind": "formalization",
"citation": "Source formal statement (not locally verified)",
"url": "https://theoremdb.org/statements/P9280",
"not... | {
"source_record": "P9280",
"imported_at": "2026-09-08",
"status_as_of": "2026-09-08",
"evidence_label": "Source candidate · not reviewed",
"note": "Full public directory statement. Source lane: backlog; source resolution: open. Source verification claims are attributed only; no local Lean verification has be... | {"created_at":"2026-08-17T19:50:35+00:00","display_standard":{"editorial_review":"submitter_supplied","formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":9280,"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": "2026-09-08T03:32:50.354Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"e7995ce9-a119-47be-8cc1-e39820228e34","format_issues":null,"number":3402} |
1.0.0 | problem:theoremdb-p9282 | problem | theoremdb-p9282 | The Twin Prime Conjecture | {
"main": "math",
"sub": "number theory",
"slug": "number-theory",
"name": "Number theory",
"original_discipline": "mathematics"
} | [] | [
{
"code": "11",
"basis": "community-map",
"evidence": "Inferred from community field: number-theory"
}
] | A pair of twin primes is a pair of prime numbers that differ by 2; that is, primes of the form $p$ and $p + 2$ for some natural number $p$. Determine whether the set of twin primes is infinite. | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P9282",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P9282",
"citation": null,
"doi": null,
"arxiv_id": null,
"license": "Selected packets label their content CC0-1.0; original references retain their own terms",
"notes": "Selected problem... | theorem twin_primes :
answer(sorry) ↔ {p : ℕ | Prime p ∧ Prime (p + 2)}.Infinite | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem twin_primes :\n answer(sorry) ↔ {p : ℕ | Prime p ∧ Prime (p + 2)}.Infinite",
"target": null,
"source_url": "https://theoremdb.org/statements/P9282",
"format": "declaration-or-fragment",
"sha256": "269692ba5132b91923b7015a960a0a85576df9fec915b705b7d... | [] | TheoremDB P9282 · Candidate awaiting source review | null | Source-reported assessment, retrieved 8 September 2026. Frontier Science has not independently reviewed this status.
Open. No signed Lean verification is attached to this problem or its selected resolution.
This record is in the source backlog; its novelty and unresolved status have not passed source review. | null | published | incremental | hard | [
"source-candidate",
"theoremdb"
] | null | null | null | null | [
{
"kind": "website",
"citation": "TheoremDB P9282",
"url": "https://theoremdb.org/statements/P9282",
"note": null,
"sort_order": 0
},
{
"kind": "formalization",
"citation": "Source formal statement (not locally verified)",
"url": "https://theoremdb.org/statements/P9282",
"not... | {
"source_record": "P9282",
"imported_at": "2026-09-08",
"status_as_of": "2026-09-08",
"evidence_label": "Source candidate · not reviewed",
"note": "Full public directory statement. Source lane: backlog; source resolution: open. Source verification claims are attributed only; no local Lean verification has be... | {"created_at":"2026-08-17T19:50:36+00:00","display_standard":{"editorial_review":"submitter_supplied","formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":9282,"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": "2026-09-08T03:33:06.827Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"1f6cf9b6-7154-4129-8aea-07121c22dba9","format_issues":null,"number":6826} |
1.0.0 | problem:theoremdb-p9284 | problem | theoremdb-p9284 | Lower bound for the two-color van der Waerden number $W(3,27)$ | {
"main": "math",
"sub": "combinatorics",
"slug": "combinatorics",
"name": "Combinatorics",
"original_discipline": "mathematics"
} | [] | [
{
"code": "05",
"basis": "community-map",
"evidence": "Inferred from community field: combinatorics"
}
] | For positive integers $k$ and $r$, the two-color van der Waerden number $W(k,r)$ is defined as the least positive integer $N$ such that every coloring of $\{1, 2, \ldots, N\}$ with two colors contains either a monochromatic arithmetic progression of length $k$ in the first color or a monochromatic arithmetic progressio... | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P9284",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P9284",
"citation": null,
"doi": null,
"arxiv_id": null,
"license": "Selected packets label their content CC0-1.0; original references retain their own terms",
"notes": "Selected problem... | theorem W_3_27_lower : answer(sorry) ↔ W 3 27 ≥ 770 | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem W_3_27_lower : answer(sorry) ↔ W 3 27 ≥ 770",
"target": null,
"source_url": "https://theoremdb.org/statements/P9284",
"format": "declaration-or-fragment",
"sha256": "84a55e8c33edc0e5b3294537064f9eaee727ec8c423815a6ddf05960e25aef7b",
"license": "Se... | [] | TheoremDB P9284 · Candidate awaiting source review | null | Source-reported assessment, retrieved 8 September 2026. Frontier Science has not independently reviewed this status.
Open. No signed Lean verification is attached to this problem or its selected resolution.
This record is in the source backlog; its novelty and unresolved status have not passed source review. | null | published | incremental | hard | [
"source-candidate",
"theoremdb"
] | null | null | null | null | [
{
"kind": "website",
"citation": "TheoremDB P9284",
"url": "https://theoremdb.org/statements/P9284",
"note": null,
"sort_order": 0
},
{
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} | 2026-10-10T16:07:49.246Z | {"database_id":"d38e7731-247c-4631-80b0-7b1a79ceba1d","format_issues":null,"number":6111} |
1.0.0 | problem:theoremdb-p9286 | problem | theoremdb-p9286 | Odd Perfect Number Conjecture | {
"main": "math",
"sub": "number theory",
"slug": "number-theory",
"name": "Number theory",
"original_discipline": "mathematics"
} | [] | [
{
"code": "11",
"basis": "community-map",
"evidence": "Inferred from community field: number-theory"
}
] | A natural number $n$ is called perfect if it equals the sum of its proper positive divisors (that is, all positive divisors of $n$ other than $n$ itself). Prove that every perfect number is even. Equivalently, prove that no odd perfect number exists. | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P9286",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P9286",
"citation": null,
"doi": null,
"arxiv_id": null,
"license": "Selected packets label their content CC0-1.0; original references retain their own terms",
"notes": "Selected problem... | theorem odd_perfect_number_conjecture (n : ℕ) (hn : Perfect n) : Even n | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem odd_perfect_number_conjecture (n : ℕ) (hn : Perfect n) : Even n",
"target": null,
"source_url": "https://theoremdb.org/statements/P9286",
"format": "declaration-or-fragment",
"sha256": "cb60a153c08e2dec054c74617b7d2c0b7a6d783c335fcda0811a53c8d3c95b2a"... | [] | TheoremDB P9286 · Candidate awaiting source review | null | Source-reported assessment, retrieved 8 September 2026. Frontier Science has not independently reviewed this status.
Open. No signed Lean verification is attached to this problem or its selected resolution.
This record is in 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 P9286",
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"url": "https://theoremdb.org/statements/P9286",
"not... | {
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"imported_at": "2026-09-08",
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"evidence_label": "Source candidate · not reviewed",
"note": "Full public directory statement. Source lane: backlog; source resolution: open. Source verification claims are attributed only; no local Lean verification has be... | {"created_at":"2026-08-17T19:50:38+00:00","display_standard":{"editorial_review":"submitter_supplied","formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":9286,"public_status":{"evidence":{"hasFormalization":false,"hasInformalProof":false,"hasRecordedWork":false,"solutionRecord":null},"lean":{... | [
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"updated_at": "2026-09-08T03:32:15.935Z",
"published_at": "2026-09-08T03:33:04.613Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"0bbe55b6-1c32-4202-b891-6241f1987237","format_issues":null,"number":6348} |
1.0.0 | problem:theoremdb-p9288 | problem | theoremdb-p9288 | Asymptotic covering of cyclic groups by random translates | {
"main": "math",
"sub": "combinatorics",
"slug": "combinatorics",
"name": "Combinatorics",
"original_discipline": "mathematics"
} | [] | [
{
"code": "05",
"basis": "community-map",
"evidence": "Inferred from community field: combinatorics"
}
] | For a prime $p$ and positive integers $k$ and $c$, let the coverable proportion $\rho(p,k,c)$ be defined as follows. Consider the cyclic group $\mathbb{Z}/p\mathbb{Z}$. Let $\mathcal{S}$ be the collection of all $k$-element subsets of $\mathbb{Z}/p\mathbb{Z}$. A subset $A \in \mathcal{S}$ is called coverable with $c$ t... | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P9288",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P9288",
"citation": null,
"doi": null,
"arxiv_id": null,
"license": "Selected packets label their content CC0-1.0; original references retain their own terms",
"notes": "Selected problem... | theorem green_39.variant_theta : answer(sorry) ↔
∀ (θ : ℝ), 0 < θ → θ ≤ 1/2 →
∃ C > 1, Tendsto
(fun p : {q : ℕ // q.Prime} ↦
let k := ⌊(p : ℝ) ^ θ⌋₊
let c := ⌊C * (p : ℝ) ^ (1 - θ)⌋₊
(proportionCoverable p k c : ℝ))
atTop (𝓝 1) | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem green_39.variant_theta : answer(sorry) ↔\n ∀ (θ : ℝ), 0 < θ → θ ≤ 1/2 →\n ∃ C > 1, Tendsto\n (fun p : {q : ℕ // q.Prime} ↦\n let k := ⌊(p : ℝ) ^ θ⌋₊\n let c := ⌊C * (p : ℝ) ^ (1 - θ)⌋₊\n (proportionCoverable p k c : ℝ))\n atTop (�... | [] | TheoremDB P9288 · Candidate awaiting source review | null | Source-reported assessment, retrieved 8 September 2026. Frontier Science has not independently reviewed this status.
Open. No signed Lean verification is attached to this problem or its selected resolution.
This record is in the source backlog; its novelty and unresolved status have not passed source review. | null | published | incremental | hard | [
"source-candidate",
"theoremdb"
] | null | null | null | null | [
{
"kind": "website",
"citation": "TheoremDB P9288",
"url": "https://theoremdb.org/statements/P9288",
"note": null,
"sort_order": 0
},
{
"kind": "formalization",
"citation": "Source formal statement (not locally verified)",
"url": "https://theoremdb.org/statements/P9288",
"not... | {
"source_record": "P9288",
"imported_at": "2026-09-08",
"status_as_of": "2026-09-08",
"evidence_label": "Source candidate · not reviewed",
"note": "Full public directory statement. Source lane: backlog; source resolution: open. Source verification claims are attributed only; no local Lean verification has be... | {"created_at":"2026-08-17T19:50:38+00:00","display_standard":{"editorial_review":"submitter_supplied","formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":9288,"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": "2026-09-08T03:32:49.987Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"26d92a39-f3e4-43fe-a900-a73f2844bc15","format_issues":null,"number":3327} |
1.0.0 | problem:theoremdb-p9290 | problem | theoremdb-p9290 | Infinitude of Ulam Pairs Differing by Two | {
"main": "math",
"sub": "combinatorics",
"slug": "combinatorics",
"name": "Combinatorics",
"original_discipline": "mathematics"
} | [] | [
{
"code": "05",
"basis": "community-map",
"evidence": "Inferred from community field: combinatorics"
}
] | The Ulam sequence is the unique sequence of positive integers $(a(n))_{n \geq 0}$ defined by $a(0) = 1$, $a(1) = 2$, and for each $n \geq 2$, $a(n)$ is the least integer greater than $a(n-1)$ that can be expressed as $a(i) + a(j)$ with $i < j < n$ in exactly one way. Determine whether there exist infinitely many indice... | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P9290",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P9290",
"citation": null,
"doi": null,
"arxiv_id": null,
"license": "Selected packets label their content CC0-1.0; original references retain their own terms",
"notes": "Selected problem... | theorem erdos_342.parts.i :
answer(sorry) ↔
∀ a : ℕ → ℕ, IsUlamSequence a →
Set.Infinite {n : ℕ | ∃ m, a m = a n + 2} | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem erdos_342.parts.i :\n answer(sorry) ↔\n ∀ a : ℕ → ℕ, IsUlamSequence a →\n Set.Infinite {n : ℕ | ∃ m, a m = a n + 2}",
"target": null,
"source_url": "https://theoremdb.org/statements/P9290",
"format": "declaration-or-fragment",
"sha256":... | [] | TheoremDB P9290 · Candidate awaiting source review | null | Source-reported assessment, retrieved 8 September 2026. Frontier Science has not independently reviewed this status.
Open. No signed Lean verification is attached to this problem or its selected resolution.
This record is in the source backlog; its novelty and unresolved status have not passed source review. | null | published | incremental | hard | [
"source-candidate",
"theoremdb"
] | null | null | null | null | [
{
"kind": "website",
"citation": "TheoremDB P9290",
"url": "https://theoremdb.org/statements/P9290",
"note": null,
"sort_order": 0
},
{
"kind": "formalization",
"citation": "Source formal statement (not locally verified)",
"url": "https://theoremdb.org/statements/P9290",
"not... | {
"source_record": "P9290",
"imported_at": "2026-09-08",
"status_as_of": "2026-09-08",
"evidence_label": "Source candidate · not reviewed",
"note": "Full public directory statement. Source lane: backlog; source resolution: open. Source verification claims are attributed only; no local Lean verification has be... | {"created_at":"2026-08-17T19:50:39+00:00","display_standard":{"editorial_review":"submitter_supplied","formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":9290,"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": "2026-09-08T03:33:01.271Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"e5716c59-b320-46cc-aa70-bddb89d2a0df","format_issues":null,"number":5694} |
1.0.0 | problem:theoremdb-p9292 | problem | theoremdb-p9292 | The ordinal partition relation for triples at the continuum with double omega and four | {
"main": "math",
"sub": "logic and foundations",
"slug": "logic-and-foundations",
"name": "Logic and foundations",
"original_discipline": "mathematics"
} | [] | [
{
"code": "03",
"basis": "community-map",
"evidence": "Inferred from community field: logic-and-foundations"
}
] | Let $\mathfrak{c}$ denote the cardinality of the continuum. For ordinals $\alpha, \beta$ and a cardinal $c$, the 3-uniform ordinal Ramsey property $\alpha \to (\beta, c)^3_2$ is defined as follows: for any 2-coloring of all 3-element subsets of the ordinal type $\alpha$ (where the coloring is given by a symmetric predi... | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P9292",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P9292",
"citation": null,
"doi": null,
"arxiv_id": null,
"license": "Selected packets label their content CC0-1.0; original references retain their own terms",
"notes": "Selected problem... | theorem omega_times_two_four :
answer(sorry) ↔ OrdinalCardinalRamsey3 (𝔠).ord (ω * 2) 4 | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem omega_times_two_four :\n answer(sorry) ↔ OrdinalCardinalRamsey3 (𝔠).ord (ω * 2) 4",
"target": null,
"source_url": "https://theoremdb.org/statements/P9292",
"format": "declaration-or-fragment",
"sha256": "bd91d6132f28142d23e8ce8ebfedd66b1eb98b2d683... | [] | TheoremDB P9292 · Candidate awaiting source review | null | Source-reported assessment, retrieved 8 September 2026. Frontier Science has not independently reviewed this status.
Open. No signed Lean verification is attached to this problem or its selected resolution.
This record is in the source backlog; its novelty and unresolved status have not passed source review. | null | published | incremental | hard | [
"source-candidate",
"theoremdb"
] | null | null | null | null | [
{
"kind": "website",
"citation": "TheoremDB P9292",
"url": "https://theoremdb.org/statements/P9292",
"note": null,
"sort_order": 0
},
{
"kind": "formalization",
"citation": "Source formal statement (not locally verified)",
"url": "https://theoremdb.org/statements/P9292",
"not... | {
"source_record": "P9292",
"imported_at": "2026-09-08",
"status_as_of": "2026-09-08",
"evidence_label": "Source candidate · not reviewed",
"note": "Full public directory statement. Source lane: backlog; source resolution: open. Source verification claims are attributed only; no local Lean verification has be... | {"created_at":"2026-08-17T19:50:40+00:00","display_standard":{"formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":9292,"public_status":{"evidence":{"hasFormalization":false,"hasInformalProof":false,"hasRecordedWork":false,"solutionRecord":null},"lean":{"kernelAccepted":false,"label":"","reaso... | [
"formalization-contains-placeholder",
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"current-status-unassessed",
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] | [] | {
"created_at": "2026-09-08T03:32:15.935Z",
"updated_at": "2026-09-08T03:32:15.935Z",
"published_at": "2026-09-08T03:33:06.685Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"eff74d8f-d9c6-4c3a-8e5e-fa82bbf8fd75","format_issues":null,"number":6796} |
1.0.0 | problem:theoremdb-p9294 | problem | theoremdb-p9294 | Asymptotic size of subsets with distinct r-fold sums | {
"main": "math",
"sub": "combinatorics",
"slug": "combinatorics",
"name": "Combinatorics",
"original_discipline": "mathematics"
} | [] | [
{
"code": "05",
"basis": "community-map",
"evidence": "Inferred from community field: combinatorics"
}
] | For integers $r \geq 2$ and $N \geq 1$, let $f(N,r)$ denote the maximum cardinality of a subset $A \subseteq \{1, \ldots, N\}$ such that all unordered $r$-fold sums of elements from $A$ are distinct, apart from the trivial coincidences arising from permutations of the summands. More precisely, if $m_1$ and $m_2$ are mu... | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P9294",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P9294",
"citation": null,
"doi": null,
"arxiv_id": null,
"license": "Selected packets label their content CC0-1.0; original references retain their own terms",
"notes": "Selected problem... | theorem erdos_241.variants.generalization (r : ℕ) (hr : r ≥ 2) : BoseChowlaConjecture r | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem erdos_241.variants.generalization (r : ℕ) (hr : r ≥ 2) : BoseChowlaConjecture r",
"target": null,
"source_url": "https://theoremdb.org/statements/P9294",
"format": "declaration-or-fragment",
"sha256": "05a120fd50e432d32baca21c329524b7c55f2e9bbe3eb513... | [] | TheoremDB P9294 · Candidate awaiting source review | null | Source-reported assessment, retrieved 8 September 2026. Frontier Science has not independently reviewed this status.
Open. No signed Lean verification is attached to this problem or its selected resolution.
This record is in the source backlog; its novelty and unresolved status have not passed source review. | null | published | incremental | hard | [
"source-candidate",
"theoremdb"
] | null | null | null | null | [
{
"kind": "website",
"citation": "TheoremDB P9294",
"url": "https://theoremdb.org/statements/P9294",
"note": null,
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},
{
"kind": "formalization",
"citation": "Source formal statement (not locally verified)",
"url": "https://theoremdb.org/statements/P9294",
"not... | {
"source_record": "P9294",
"imported_at": "2026-09-08",
"status_as_of": "2026-09-08",
"evidence_label": "Source candidate · not reviewed",
"note": "Full public directory statement. Source lane: backlog; source resolution: open. Source verification claims are attributed only; no local Lean verification has be... | {"created_at":"2026-08-17T19:50:41+00:00","display_standard":{"editorial_review":"submitter_supplied","formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":9294,"public_status":{"evidence":{"hasFormalization":false,"hasInformalProof":false,"hasRecordedWork":false,"solutionRecord":null},"lean":{... | [
"formalization-not-compiled-during-export",
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"created_at": "2026-09-08T03:32:15.935Z",
"updated_at": "2026-09-08T03:32:15.935Z",
"published_at": "2026-09-08T03:32:50.391Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"62f63e04-9eb4-4037-b581-9bc2c18ee950","format_issues":null,"number":3410} |
1.0.0 | problem:theoremdb-p9296 | problem | theoremdb-p9296 | Transcendence of the sine of Euler's number | {
"main": "math",
"sub": "number theory",
"slug": "number-theory",
"name": "Number theory",
"original_discipline": "mathematics"
} | [] | [
{
"code": "11",
"basis": "community-map",
"evidence": "Inferred from community field: number-theory"
}
] | Let $e$ denote the base of the natural logarithm, and let $\sin$ denote the sine function on the real numbers. Determine whether the real number $\sin(e)$ is transcendental over the rational numbers. | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P9296",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P9296",
"citation": null,
"doi": null,
"arxiv_id": null,
"license": "Selected packets label their content CC0-1.0; original references retain their own terms",
"notes": "Selected problem... | theorem sin_exp_transcendental : Transcendental ℚ (Real.sin (exp 1)) | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem sin_exp_transcendental : Transcendental ℚ (Real.sin (exp 1))",
"target": null,
"source_url": "https://theoremdb.org/statements/P9296",
"format": "declaration-or-fragment",
"sha256": "d9055d18597912b77e12f92f1848fb5cf72536885f0b71658a9accb775746aa7",
... | [] | TheoremDB P9296 · Candidate awaiting source review | null | Source-reported assessment, retrieved 8 September 2026. Frontier Science has not independently reviewed this status.
Open. No signed Lean verification is attached to this problem or its selected resolution.
This record is in the source backlog; its novelty and unresolved status have not passed source review. | null | published | incremental | hard | [
"source-candidate",
"theoremdb"
] | null | null | null | null | [
{
"kind": "website",
"citation": "TheoremDB P9296",
"url": "https://theoremdb.org/statements/P9296",
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"citation": "Source formal statement (not locally verified)",
"url": "https://theoremdb.org/statements/P9296",
"not... | {
"source_record": "P9296",
"imported_at": "2026-09-08",
"status_as_of": "2026-09-08",
"evidence_label": "Source candidate · not reviewed",
"note": "Full public directory statement. Source lane: backlog; source resolution: open. Source verification claims are attributed only; no local Lean verification has be... | {"created_at":"2026-08-17T19:50:41+00:00","display_standard":{"editorial_review":"submitter_supplied","formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":9296,"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": "2026-09-08T03:33:07.026Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"3a4809ed-1759-48ad-9d6a-df8c44349e52","format_issues":null,"number":6869} |
1.0.0 | problem:theoremdb-p9298 | problem | theoremdb-p9298 | Granville–Soundararajan conjecture on sums of a prime and 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"
}
] | For a natural number $k$, let $\Sigma_k$ denote the set of natural numbers that can be written as $p + \sum_{i=1}^{m} 2^{e_i}$ where $p$ is a prime number, $m \leq k$, and $e_1, \ldots, e_m$ are natural numbers (not necessarily distinct). Prove that every odd integer $n > 1$ belongs to $\Sigma_3$ and every even integer... | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P9298",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P9298",
"citation": null,
"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_10.variants.granville_soundararajan_odd :
{n : ℕ | Odd n ∧ 1 < n} ⊆ sumPrimeAndTwoPows 3 ∧
{n : ℕ | Even n ∧ n ≠ 0} ⊆ sumPrimeAndTwoPows 4 | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem erdos_10.variants.granville_soundararajan_odd :\n {n : ℕ | Odd n ∧ 1 < n} ⊆ sumPrimeAndTwoPows 3 ∧\n {n : ℕ | Even n ∧ n ≠ 0} ⊆ sumPrimeAndTwoPows 4",
"target": null,
"source_url": "https://theoremdb.org/statements/P9298",
"format": "declaration-o... | [] | TheoremDB P9298 · Candidate awaiting source review | null | Source-reported assessment, retrieved 8 September 2026. Frontier Science has not independently reviewed this status.
Open. No signed Lean verification is attached to this problem or its selected resolution.
This record is in the source backlog; its novelty and unresolved status have not passed source review. | null | published | incremental | hard | [
"source-candidate",
"theoremdb"
] | null | null | null | null | [
{
"kind": "website",
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"citation": "Source formal statement (not locally verified)",
"url": "https://theoremdb.org/statements/P9298",
"not... | {
"source_record": "P9298",
"imported_at": "2026-09-08",
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"evidence_label": "Source candidate · not reviewed",
"note": "Full public directory statement. Source lane: backlog; source resolution: open. Source verification claims are attributed only; no local Lean verification has be... | {"created_at":"2026-08-17T19:50:43+00:00","display_standard":{"editorial_review":"submitter_supplied","formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":9298,"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": "2026-09-08T03:32:59.394Z",
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} | 2026-10-10T16:07:49.246Z | {"database_id":"1ed5ed8e-c2ed-4f0c-8d4f-c15ae3956f03","format_issues":null,"number":5386} |
1.0.0 | problem:theoremdb-p9300 | problem | theoremdb-p9300 | Largest Product-Free Subsets of Alternating Groups | {
"main": "math",
"sub": "algebra",
"slug": "algebra",
"name": "Algebra",
"original_discipline": "mathematics"
} | [] | [] | For each natural number $n$, let $A_n$ denote the alternating group on the finite set $\{0, 1, \ldots, n-1\}$, consisting of all even permutations. A subset $S \subseteq A_n$ is called product-free if there do not exist elements $x, y, z \in S$ such that $x \cdot y = z$, where $\cdot$ denotes the group multiplication. ... | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P9300",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P9300",
"citation": null,
"doi": null,
"arxiv_id": null,
"license": "Selected packets label their content CC0-1.0; original references retain their own terms",
"notes": "Selected problem... | theorem green_4 (n : ℕ) :
let S : ∀ n, Set (alternatingGroup <| Fin n) := answer(sorry)
MaximalFor (ProdFree (M := alternatingGroup <| Fin n)) Set.ncard (S n) | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem green_4 (n : ℕ) :\n let S : ∀ n, Set (alternatingGroup <| Fin n) := answer(sorry)\n MaximalFor (ProdFree (M := alternatingGroup <| Fin n)) Set.ncard (S n)",
"target": null,
"source_url": "https://theoremdb.org/statements/P9300",
"format": "declarati... | [] | TheoremDB P9300 · Candidate awaiting source review | null | Source-reported assessment, retrieved 8 September 2026. Frontier Science has not independently reviewed this status.
Open. No signed Lean verification is attached to this problem or its selected resolution.
This record is in the source backlog; its novelty and unresolved status have not passed source review. | null | published | incremental | hard | [
"source-candidate",
"theoremdb"
] | null | null | null | null | [
{
"kind": "website",
"citation": "TheoremDB P9300",
"url": "https://theoremdb.org/statements/P9300",
"note": null,
"sort_order": 0
},
{
"kind": "formalization",
"citation": "Source formal statement (not locally verified)",
"url": "https://theoremdb.org/statements/P9300",
"not... | {
"source_record": "P9300",
"imported_at": "2026-09-08",
"status_as_of": "2026-09-08",
"evidence_label": "Source candidate · not reviewed",
"note": "Full public directory statement. Source lane: backlog; source resolution: open. Source verification claims are attributed only; no local Lean verification has be... | {"created_at":"2026-08-17T19:50:43+00:00","display_standard":{"editorial_review":"submitter_supplied","formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":9300,"public_status":{"evidence":{"hasFormalization":false,"hasInformalProof":false,"hasRecordedWork":false,"solutionRecord":null},"lean":{... | [
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] | [] | {
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"updated_at": "2026-09-08T03:32:15.935Z",
"published_at": "2026-09-08T03:33:03.013Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"62849aba-b83a-4ee3-b361-ec55eef6474d","format_issues":null,"number":6004} |
1.0.0 | problem:theoremdb-p9302 | problem | theoremdb-p9302 | The Erdős–Hajnal Conjecture on Polynomial Ramsey-Type Bounds | {
"main": "math",
"sub": "combinatorics",
"slug": "combinatorics",
"name": "Combinatorics",
"original_discipline": "mathematics"
} | [] | [
{
"code": "05",
"basis": "community-map",
"evidence": "Inferred from community field: combinatorics"
}
] | For a finite simple graph $H$, say that a simple graph $G$ is $H$\textit{-free} if $H$ does not appear as an induced subgraph of $G$. For a graph $G$, let $\alpha(G)$ denote the independence number of $G$ (the size of a largest independent set), and let $\omega(G)$ denote the clique number of $G$ (the size of a largest... | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P9302",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P9302",
"citation": null,
"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_61 :
answer(sorry) ↔ ∀ {α : Type*} [Fintype α] [DecidableEq α] (H : SimpleGraph α),
∃ c > (0 : ℝ), IsErdosHajnalLowerBound H (fun n : ℕ => (n : ℝ) ^ c) | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem erdos_61 :\n answer(sorry) ↔ ∀ {α : Type*} [Fintype α] [DecidableEq α] (H : SimpleGraph α),\n ∃ c > (0 : ℝ), IsErdosHajnalLowerBound H (fun n : ℕ => (n : ℝ) ^ c)",
"target": null,
"source_url": "https://theoremdb.org/statements/P9302",
"format": "... | [] | TheoremDB P9302 · Candidate awaiting source review | null | Source-reported assessment, retrieved 8 September 2026. Frontier Science has not independently reviewed this status.
Open. No signed Lean verification is attached to this problem or its selected resolution.
This record is in the source backlog; its novelty and unresolved status have not passed source review. | null | published | incremental | hard | [
"source-candidate",
"theoremdb"
] | null | null | null | null | [
{
"kind": "website",
"citation": "TheoremDB P9302",
"url": "https://theoremdb.org/statements/P9302",
"note": null,
"sort_order": 0
},
{
"kind": "formalization",
"citation": "Source formal statement (not locally verified)",
"url": "https://theoremdb.org/statements/P9302",
"not... | {
"source_record": "P9302",
"imported_at": "2026-09-08",
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"evidence_label": "Source candidate · not reviewed",
"note": "Full public directory statement. Source lane: backlog; source resolution: open. Source verification claims are attributed only; no local Lean verification has be... | {"created_at":"2026-08-17T19:50:44+00:00","display_standard":{"formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":9302,"public_status":{"evidence":{"hasFormalization":false,"hasInformalProof":false,"hasRecordedWork":false,"solutionRecord":null},"lean":{"kernelAccepted":false,"label":"","reaso... | [
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"updated_at": "2026-09-08T03:32:15.935Z",
"published_at": "2026-09-08T03:33:06.463Z",
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} | 2026-10-10T16:07:49.246Z | {"database_id":"256a574b-8c43-46f2-8118-135b83d8262b","format_issues":null,"number":6748} |
1.0.0 | problem:theoremdb-p9304 | problem | theoremdb-p9304 | The lonely runner conjecture | {
"main": "math",
"sub": "number theory",
"slug": "number-theory",
"name": "Number theory",
"original_discipline": "mathematics"
} | [] | [
{
"code": "11",
"basis": "community-map",
"evidence": "Inferred from community field: number-theory"
}
] | Consider $n$ runners on a circular track of unit length. At time $t = 0$, all runners start from the same position and run with constant speeds. The speeds are pairwise distinct real numbers and may be negative. For a runner $r$ and time $t \geq 0$, we say that runner $r$ is \textit{lonely} at time $t$ if the distance ... | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P9304",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P9304",
"citation": null,
"doi": null,
"arxiv_id": null,
"license": "Selected packets label their content CC0-1.0; original references retain their own terms",
"notes": "Selected problem... | theorem lonely_runner_conjecture (n : ℕ)
(speed : Fin n ↪ ℝ) (lonely : Fin n → ℝ → Prop)
(lonely_def :
∀ r t, lonely r t ↔
∀ r2 : Fin n, r2 ≠ r →
dist (t * speed r : UnitAddCircle) (t * speed r2) ≥ 1 / n)
(r : Fin n) : ∃ t ≥ 0, lonely r t | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem lonely_runner_conjecture (n : ℕ)\n (speed : Fin n ↪ ℝ) (lonely : Fin n → ℝ → Prop)\n (lonely_def :\n ∀ r t, lonely r t ↔\n ∀ r2 : Fin n, r2 ≠ r →\n dist (t * speed r : UnitAddCircle) (t * speed r2) ≥ 1 / n)\n (r : Fin n) : ∃ t ≥ 0, lonely ... | [] | TheoremDB P9304 · Candidate awaiting source review | null | Source-reported assessment, retrieved 8 September 2026. Frontier Science has not independently reviewed this status.
Open. No signed Lean verification is attached to this problem or its selected resolution.
This record is in the source backlog; its novelty and unresolved status have not passed source review. | null | published | incremental | hard | [
"source-candidate",
"theoremdb"
] | null | null | null | null | [
{
"kind": "website",
"citation": "TheoremDB P9304",
"url": "https://theoremdb.org/statements/P9304",
"note": null,
"sort_order": 0
},
{
"kind": "formalization",
"citation": "Source formal statement (not locally verified)",
"url": "https://theoremdb.org/statements/P9304",
"not... | {
"source_record": "P9304",
"imported_at": "2026-09-08",
"status_as_of": "2026-09-08",
"evidence_label": "Source candidate · not reviewed",
"note": "Full public directory statement. Source lane: backlog; source resolution: open. Source verification claims are attributed only; no local Lean verification has be... | {"created_at":"2026-08-17T19:50:44+00:00","display_standard":{"formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":9304,"public_status":{"evidence":{"hasFormalization":false,"hasInformalProof":false,"hasRecordedWork":false,"solutionRecord":null},"lean":{"kernelAccepted":false,"label":"","reaso... | [
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"created_at": "2026-09-08T03:32:15.935Z",
"updated_at": "2026-09-08T03:32:15.935Z",
"published_at": "2026-09-08T03:33:06.639Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"1b1cbe8c-d48c-404f-b9a3-92658b97197b","format_issues":null,"number":6786} |
1.0.0 | problem:theoremdb-p9306 | problem | theoremdb-p9306 | Upper bound on the maximal number of distinct prime factors of pairwise sums | {
"main": "math",
"sub": "number theory",
"slug": "number-theory",
"name": "Number theory",
"original_discipline": "mathematics"
} | [] | [
{
"code": "11",
"basis": "community-map",
"evidence": "Inferred from community field: number-theory"
}
] | For each positive integer $n$, let $f(n)$ denote the largest integer $m$ such that for every set $A \subseteq \mathbb{N}$ with $|A| = n$, the product $\prod_{\substack{a, b \in A \\ a \neq b}} (a + b)$ has at least $m$ distinct prime factors. Determine whether $f(n) = o\!\left(\frac{n}{\log n}\right)$, that is, whether... | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P9306",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P9306",
"citation": null,
"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_126.variants.isLittleO
(f : ℕ → ℕ)
(hf : IsMaximalAddFactorsCard f) :
(fun (n : ℕ) => (f n : ℝ)) =o[atTop] (fun (n : ℕ) => n / Real.log n) | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem erdos_126.variants.isLittleO\n (f : ℕ → ℕ)\n (hf : IsMaximalAddFactorsCard f) :\n (fun (n : ℕ) => (f n : ℝ)) =o[atTop] (fun (n : ℕ) => n / Real.log n)",
"target": null,
"source_url": "https://theoremdb.org/statements/P9306",
"format": "declaratio... | [] | TheoremDB P9306 · Candidate awaiting source review | null | Source-reported assessment, retrieved 8 September 2026. Frontier Science has not independently reviewed this status.
Open. No signed Lean verification is attached to this problem or its selected resolution.
This record is in the source backlog; its novelty and unresolved status have not passed source review. | null | published | incremental | hard | [
"source-candidate",
"theoremdb"
] | null | null | null | null | [
{
"kind": "website",
"citation": "TheoremDB P9306",
"url": "https://theoremdb.org/statements/P9306",
"note": null,
"sort_order": 0
},
{
"kind": "formalization",
"citation": "Source formal statement (not locally verified)",
"url": "https://theoremdb.org/statements/P9306",
"not... | {
"source_record": "P9306",
"imported_at": "2026-09-08",
"status_as_of": "2026-09-08",
"evidence_label": "Source candidate · not reviewed",
"note": "Full public directory statement. Source lane: backlog; source resolution: open. Source verification claims are attributed only; no local Lean verification has be... | {"created_at":"2026-08-17T19:50:46+00:00","display_standard":{"editorial_review":"submitter_supplied","formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":9306,"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": "2026-09-08T03:33:07.399Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"e20b471c-a293-432c-9517-e5ff3eb4253c","format_issues":null,"number":6951} |
1.0.0 | problem:theoremdb-p9308 | problem | theoremdb-p9308 | Termination of the Odd-Denominator Greedy Unit Fraction Algorithm | {
"main": "math",
"sub": "combinatorics",
"slug": "combinatorics",
"name": "Combinatorics",
"original_discipline": "mathematics"
} | [] | [
{
"code": "05",
"basis": "community-map",
"evidence": "Inferred from community field: combinatorics"
}
] | Let $x$ be a rational number in the open interval $(0,1)$ whose denominator (in lowest terms) is odd. Consider the following greedy algorithm for representing $x$ as a sum of distinct unit fractions with odd denominators: at each step, subtract from the current remainder the largest unit fraction $\frac{1}{n}$ such tha... | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P9308",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P9308",
"citation": null,
"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 {x : ℚ} (hx : x ∈ Set.Ioo 0 1) (hx_den : Odd x.den) :
greedyUnitFractionRem { n | Odd n } x =ᶠ[atTop] 0 | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem erdos_282 {x : ℚ} (hx : x ∈ Set.Ioo 0 1) (hx_den : Odd x.den) :\n greedyUnitFractionRem { n | Odd n } x =ᶠ[atTop] 0",
"target": null,
"source_url": "https://theoremdb.org/statements/P9308",
"format": "declaration-or-fragment",
"sha256": "60ad8e443c... | [] | TheoremDB P9308 · Candidate awaiting source review | null | Source-reported assessment, retrieved 8 September 2026. Frontier Science has not independently reviewed this status.
Open. No signed Lean verification is attached to this problem or its selected resolution.
This record is in the source backlog; its novelty and unresolved status have not passed source review. | null | published | incremental | hard | [
"source-candidate",
"theoremdb"
] | null | null | null | null | [
{
"kind": "website",
"citation": "TheoremDB P9308",
"url": "https://theoremdb.org/statements/P9308",
"note": null,
"sort_order": 0
},
{
"kind": "formalization",
"citation": "Source formal statement (not locally verified)",
"url": "https://theoremdb.org/statements/P9308",
"not... | {
"source_record": "P9308",
"imported_at": "2026-09-08",
"status_as_of": "2026-09-08",
"evidence_label": "Source candidate · not reviewed",
"note": "Full public directory statement. Source lane: backlog; source resolution: open. Source verification claims are attributed only; no local Lean verification has be... | {"created_at":"2026-08-17T19:50:48+00:00","display_standard":{"formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":9308,"public_status":{"evidence":{"hasFormalization":false,"hasInformalProof":false,"hasRecordedWork":false,"solutionRecord":null},"lean":{"kernelAccepted":false,"label":"","reaso... | [
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"created_at": "2026-09-08T03:32:15.935Z",
"updated_at": "2026-09-08T03:32:15.935Z",
"published_at": "2026-09-08T03:33:06.303Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"aa24e91b-623f-4048-a2d2-4f900bf63ef5","format_issues":null,"number":6714} |
1.0.0 | problem:theoremdb-p9310 | problem | theoremdb-p9310 | Global Existence and Smoothness of Navier–Stokes Solutions on ℝ³ | {
"main": "math",
"sub": "partial differential equations",
"slug": "partial-differential-equations",
"name": "Partial differential equations",
"original_discipline": "mathematics"
} | [] | [
{
"code": "35",
"basis": "community-map",
"evidence": "Inferred from community field: partial-differential-equations"
}
] | Let $\nu > 0$ be a real number (the viscosity), and let $u_0 : \mathbb{R}^3 \to \mathbb{R}^3$ be a smooth vector field satisfying the following conditions: $u_0$ is divergence-free, meaning $\nabla \cdot u_0 = 0$ everywhere; and $u_0$ and all of its derivatives decay faster than any polynomial at infinity, meaning for ... | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P9310",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P9310",
"citation": null,
"doi": null,
"arxiv_id": null,
"license": "Selected packets label their content CC0-1.0; original references retain their own terms",
"notes": "Selected problem... | theorem navier_stokes_existence_and_smoothness_R3 (nu : ℝ) (hnu : nu > 0)
(u₀ : ℝ³ → ℝ³) (hu₀ : InitialVelocityConditionDecay u₀) :
∃ v p, NavierStokesExistenceAndSmoothnessRn nu u₀ (f := 0) v p | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem navier_stokes_existence_and_smoothness_R3 (nu : ℝ) (hnu : nu > 0)\n (u₀ : ℝ³ → ℝ³) (hu₀ : InitialVelocityConditionDecay u₀) :\n ∃ v p, NavierStokesExistenceAndSmoothnessRn nu u₀ (f := 0) v p",
"target": null,
"source_url": "https://theoremdb.org/stateme... | [] | TheoremDB P9310 · Candidate awaiting source review | null | Source-reported assessment, retrieved 8 September 2026. Frontier Science has not independently reviewed this status.
Open. No signed Lean verification is attached to this problem or its selected resolution.
This record is in the source backlog; its novelty and unresolved status have not passed source review. | null | published | incremental | hard | [
"source-candidate",
"theoremdb"
] | null | null | null | null | [
{
"kind": "website",
"citation": "TheoremDB P9310",
"url": "https://theoremdb.org/statements/P9310",
"note": null,
"sort_order": 0
},
{
"kind": "formalization",
"citation": "Source formal statement (not locally verified)",
"url": "https://theoremdb.org/statements/P9310",
"not... | {
"source_record": "P9310",
"imported_at": "2026-09-08",
"status_as_of": "2026-09-08",
"evidence_label": "Source candidate · not reviewed",
"note": "Full public directory statement. Source lane: backlog; source resolution: open. Source verification claims are attributed only; no local Lean verification has be... | {"created_at":"2026-08-17T19:50:48+00:00","display_standard":{"editorial_review":"submitter_supplied","formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":9310,"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": "2026-09-08T03:32:59.336Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"a4e26fab-050d-4e65-91cf-77225ff215fb","format_issues":null,"number":5373} |
1.0.0 | problem:theoremdb-p9312 | problem | theoremdb-p9312 | Existence of a SIC-POVM in dimension 71 | {
"main": "math",
"sub": "algebra",
"slug": "algebra",
"name": "Algebra",
"original_discipline": "mathematics"
} | [] | [] | Let $d$ be a positive integer. A state vector in $\mathbb{C}^d$ is a vector of $d$ complex coordinates with the standard Hermitian inner product. A state vector is normalized if its $L^2$ norm equals $1$. For two state vectors $\phi$ and $\psi$ in $\mathbb{C}^d$, their squared overlap is the squared magnitude of their ... | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P9312",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P9312",
"citation": null,
"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_71 : answer(sorry) ↔ HasSICPOVM 71 | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem hasSICPOVM_71 : answer(sorry) ↔ HasSICPOVM 71",
"target": null,
"source_url": "https://theoremdb.org/statements/P9312",
"format": "declaration-or-fragment",
"sha256": "29209c6db6af5c282aad9d3eadd5efb18b8ac340e4ea190f5ed768385f7a6303",
"license": "... | [] | TheoremDB P9312 · Candidate awaiting source review | null | Source-reported assessment, retrieved 8 September 2026. Frontier Science has not independently reviewed this status.
Open. No signed Lean verification is attached to this problem or its selected resolution.
This record is in the source backlog; its novelty and unresolved status have not passed source review. | null | published | incremental | hard | [
"source-candidate",
"theoremdb"
] | null | null | null | null | [
{
"kind": "website",
"citation": "TheoremDB P9312",
"url": "https://theoremdb.org/statements/P9312",
"note": null,
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},
{
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"citation": "Source formal statement (not locally verified)",
"url": "https://theoremdb.org/statements/P9312",
"not... | {
"source_record": "P9312",
"imported_at": "2026-09-08",
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"evidence_label": "Source candidate · not reviewed",
"note": "Full public directory statement. Source lane: backlog; source resolution: open. Source verification claims are attributed only; no local Lean verification has be... | {"created_at":"2026-08-17T19:50:50+00:00","display_standard":{"editorial_review":"submitter_supplied","formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":9312,"public_status":{"evidence":{"hasFormalization":false,"hasInformalProof":false,"hasRecordedWork":false,"solutionRecord":null},"lean":{... | [
"formalization-contains-placeholder",
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"source-full-text-not-present"
] | [] | {
"created_at": "2026-09-08T03:32:15.935Z",
"updated_at": "2026-09-08T03:32:15.935Z",
"published_at": "2026-09-08T03:32:58.527Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"36ffe476-2481-4a15-adb8-a40125776ce7","format_issues":null,"number":5201} |
1.0.0 | problem:theoremdb-p9314 | problem | theoremdb-p9314 | Equidistribution Modulo One of the Sequence $(3/2)^n$ | {
"main": "math",
"sub": "number theory",
"slug": "number-theory",
"name": "Number theory",
"original_discipline": "mathematics"
} | [] | [
{
"code": "11",
"basis": "community-map",
"evidence": "Inferred from community field: number-theory"
}
] | For each real number $x$ and each positive integer $n$, let $\{x\}$ denote the fractional part of $x$, defined as $\{x\} = x - \lfloor x \rfloor$ where $\lfloor x \rfloor$ is the greatest integer less than or equal to $x$. A sequence $(a_n)_{n=1}^{\infty}$ of real numbers is said to be equidistributed modulo $1$ if for... | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P9314",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P9314",
"citation": null,
"doi": null,
"arxiv_id": null,
"license": "Selected packets label their content CC0-1.0; original references retain their own terms",
"notes": "Selected problem... | theorem isEquidistributedModuloOne_three_halves_pow :
IsEquidistributedModuloOne (fun n => (3 / 2 : ℝ)^n) | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem isEquidistributedModuloOne_three_halves_pow :\n IsEquidistributedModuloOne (fun n => (3 / 2 : ℝ)^n)",
"target": null,
"source_url": "https://theoremdb.org/statements/P9314",
"format": "declaration-or-fragment",
"sha256": "393106a460aaafc3bd14f15132... | [] | TheoremDB P9314 · Candidate awaiting source review | null | Source-reported assessment, retrieved 8 September 2026. Frontier Science has not independently reviewed this status.
Open. No signed Lean verification is attached to this problem or its selected resolution.
This record is in the source backlog; its novelty and unresolved status have not passed source review. | null | published | incremental | hard | [
"source-candidate",
"theoremdb"
] | null | null | null | null | [
{
"kind": "website",
"citation": "TheoremDB P9314",
"url": "https://theoremdb.org/statements/P9314",
"note": null,
"sort_order": 0
},
{
"kind": "formalization",
"citation": "Source formal statement (not locally verified)",
"url": "https://theoremdb.org/statements/P9314",
"not... | {
"source_record": "P9314",
"imported_at": "2026-09-08",
"status_as_of": "2026-09-08",
"evidence_label": "Source candidate · not reviewed",
"note": "Full public directory statement. Source lane: backlog; source resolution: open. Source verification claims are attributed only; no local Lean verification has be... | {"created_at":"2026-08-17T19:50:51+00:00","display_standard":{"formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":9314,"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": "2026-09-08T03:32:54.772Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"ec7abc1a-1feb-4715-80e3-35ccb02dfd8d","format_issues":null,"number":4375} |
1.0.0 | problem:theoremdb-p9316 | problem | theoremdb-p9316 | Infinitude of Composite 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 $F_n = 2^{2^n} + 1$. Determine whether there are infinitely many natural numbers $n$ such that $F_n$ is not prime. | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P9316",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P9316",
"citation": null,
"doi": null,
"arxiv_id": null,
"license": "Selected packets label their content CC0-1.0; original references retain their own terms",
"notes": "Selected problem... | theorem infinite_fermat_composite : answer(sorry) ↔ Infinite {n : ℕ | ¬Prime n.fermatNumber} | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem infinite_fermat_composite : answer(sorry) ↔ Infinite {n : ℕ | ¬Prime n.fermatNumber}",
"target": null,
"source_url": "https://theoremdb.org/statements/P9316",
"format": "declaration-or-fragment",
"sha256": "cf3554f66d51f76d1fa3c9529a81f2cd1baeff2d4255... | [] | TheoremDB P9316 · Candidate awaiting source review | null | Source-reported assessment, retrieved 8 September 2026. Frontier Science has not independently reviewed this status.
Open. No signed Lean verification is attached to this problem or its selected resolution.
This record is in the source backlog; its novelty and unresolved status have not passed source review. | null | published | incremental | hard | [
"source-candidate",
"theoremdb"
] | null | null | null | null | [
{
"kind": "website",
"citation": "TheoremDB P9316",
"url": "https://theoremdb.org/statements/P9316",
"note": null,
"sort_order": 0
},
{
"kind": "formalization",
"citation": "Source formal statement (not locally verified)",
"url": "https://theoremdb.org/statements/P9316",
"not... | {
"source_record": "P9316",
"imported_at": "2026-09-08",
"status_as_of": "2026-09-08",
"evidence_label": "Source candidate · not reviewed",
"note": "Full public directory statement. Source lane: backlog; source resolution: open. Source verification claims are attributed only; no local Lean verification has be... | {"created_at":"2026-08-17T19:50:51+00:00","display_standard":{"editorial_review":"submitter_supplied","formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":9316,"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": "2026-09-08T03:33:01.117Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"1e8c0893-b4b5-477f-a8f0-aad84b6e87d2","format_issues":null,"number":5661} |
1.0.0 | problem:theoremdb-p9318 | problem | theoremdb-p9318 | Erdős's problem on integers with small prime factors between consecutive primes | {
"main": "math",
"sub": "number theory",
"slug": "number-theory",
"name": "Number theory",
"original_discipline": "mathematics"
} | [] | [
{
"code": "11",
"basis": "community-map",
"evidence": "Inferred from community field: number-theory"
}
] | Let $p_k$ denote the $k$th prime number. For a positive integer $m$, let $P(m)$ denote the largest prime factor of $m$. For each positive integer $r$, consider the set of integers $m$ strictly between $p_r$ and $p_{r+1}$ such that $P(m) < p_{r+1} - p_r$. Prove that the set of positive integers $r$ for which this set co... | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P9318",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P9318",
"citation": null,
"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_932 :
{ r : ℕ | 2 ≤ (Finset.Ioo (r.nth Nat.Prime) (r.succ.nth Nat.Prime) |>.filter
(fun m => m.maxPrimeFac < r.succ.nth Nat.Prime - r.nth Nat.Prime)).card }.Infinite | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem erdos_932 :\n { r : ℕ | 2 ≤ (Finset.Ioo (r.nth Nat.Prime) (r.succ.nth Nat.Prime) |>.filter\n (fun m => m.maxPrimeFac < r.succ.nth Nat.Prime - r.nth Nat.Prime)).card }.Infinite",
"target": null,
"source_url": "https://theoremdb.org/statements/P9318",
... | [] | TheoremDB P9318 · Candidate awaiting source review | null | Source-reported assessment, retrieved 8 September 2026. Frontier Science has not independently reviewed this status.
Open. No signed Lean verification is attached to this problem or its selected resolution.
This record is in the source backlog; its novelty and unresolved status have not passed source review. | null | published | incremental | hard | [
"source-candidate",
"theoremdb"
] | null | null | null | null | [
{
"kind": "website",
"citation": "TheoremDB P9318",
"url": "https://theoremdb.org/statements/P9318",
"note": null,
"sort_order": 0
},
{
"kind": "formalization",
"citation": "Source formal statement (not locally verified)",
"url": "https://theoremdb.org/statements/P9318",
"not... | {
"source_record": "P9318",
"imported_at": "2026-09-08",
"status_as_of": "2026-09-08",
"evidence_label": "Source candidate · not reviewed",
"note": "Full public directory statement. Source lane: backlog; source resolution: open. Source verification claims are attributed only; no local Lean verification has be... | {"created_at":"2026-08-17T19:50:51+00:00","display_standard":{"editorial_review":"submitter_supplied","formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":9318,"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": "2026-09-08T03:32:56.492Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"b20adc55-ea07-48b0-aac5-8492a4c07705","format_issues":null,"number":4752} |
1.0.0 | problem:theoremdb-p9320 | problem | theoremdb-p9320 | Complex manifold structures on the 6-sphere | {
"main": "math",
"sub": "mathematics general",
"slug": "mathematics-general",
"name": "Mathematics · general",
"original_discipline": "mathematics"
} | [] | [] | Let $S^6$ denote the unit $6$-sphere, defined as the set of points at distance $1$ from the origin in the $7$-dimensional real Euclidean space $\mathbb{R}^7$. Determine whether there exists an atlas of charts on $S^6$ with values in $\mathbb{C}^3$ such that $S^6$ becomes a complex manifold of complex dimension $3$. Equ... | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P9320",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P9320",
"citation": null,
"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_1973 :
answer(sorry) ↔ ∃ atlas : ChartedSpace (EuclideanSpace ℂ (Fin 3)) (unitSphere 6),
IsManifold 𝓘(ℂ, EuclideanSpace ℂ (Fin 3)) 1 (unitSphere 6) | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem mathoverflow_1973 :\n answer(sorry) ↔ ∃ atlas : ChartedSpace (EuclideanSpace ℂ (Fin 3)) (unitSphere 6),\n IsManifold 𝓘(ℂ, EuclideanSpace ℂ (Fin 3)) 1 (unitSphere 6)",
"target": null,
"source_url": "https://theoremdb.org/statements/P9320",
"format... | [] | TheoremDB P9320 · Candidate awaiting source review | null | Source-reported assessment, retrieved 8 September 2026. Frontier Science has not independently reviewed this status.
Open. No signed Lean verification is attached to this problem or its selected resolution.
This record is in the source backlog; its novelty and unresolved status have not passed source review. | null | published | incremental | hard | [
"source-candidate",
"theoremdb"
] | null | null | null | null | [
{
"kind": "website",
"citation": "TheoremDB P9320",
"url": "https://theoremdb.org/statements/P9320",
"note": null,
"sort_order": 0
},
{
"kind": "formalization",
"citation": "Source formal statement (not locally verified)",
"url": "https://theoremdb.org/statements/P9320",
"not... | {
"source_record": "P9320",
"imported_at": "2026-09-08",
"status_as_of": "2026-09-08",
"evidence_label": "Source candidate · not reviewed",
"note": "Full public directory statement. Source lane: backlog; source resolution: open. Source verification claims are attributed only; no local Lean verification has be... | {"created_at":"2026-08-17T19:50:52+00:00","display_standard":{"editorial_review":"submitter_supplied","formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":9320,"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": "2026-09-08T03:32:52.519Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"ad79fa52-777e-42c3-9fce-650f2b3a3bbd","format_issues":null,"number":3878} |
1.0.0 | problem:theoremdb-p9322 | problem | theoremdb-p9322 | Babai–Seress Conjecture on Diameters of Non-Abelian Finite Simple Groups | {
"main": "math",
"sub": "combinatorics",
"slug": "combinatorics",
"name": "Combinatorics",
"original_discipline": "mathematics"
} | [] | [
{
"code": "05",
"basis": "community-map",
"evidence": "Inferred from community field: combinatorics"
}
] | Does there exist an absolute constant $C \in \mathbb{N}$ such that for every non-abelian finite simple group $G$, the group diameter of $G$ satisfies $\operatorname{diam}(G) \leq (\log |G|)^C$? | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P9322",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P9322",
"citation": null,
"doi": null,
"arxiv_id": null,
"license": "Selected packets label their content CC0-1.0; original references retain their own terms",
"notes": "Selected problem... | theorem babai_seress_conjecture :
∃ C : ℕ,
∀ (G : Type) [Group G] [Fintype G] [IsSimpleGroup G],
(∃ a b : G, a * b ≠ b * a) →
(groupDiam G : ℝ) ≤ (Real.log (Fintype.card G : ℝ)) ^ C | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem babai_seress_conjecture :\n ∃ C : ℕ,\n ∀ (G : Type) [Group G] [Fintype G] [IsSimpleGroup G],\n (∃ a b : G, a * b ≠ b * a) →\n (groupDiam G : ℝ) ≤ (Real.log (Fintype.card G : ℝ)) ^ C",
"target": null,
"source_url": "https://theoremdb.org/statements... | [] | TheoremDB P9322 · Candidate awaiting source review | null | Source-reported assessment, retrieved 8 September 2026. Frontier Science has not independently reviewed this status.
Open. No signed Lean verification is attached to this problem or its selected resolution.
This record is in the source backlog; its novelty and unresolved status have not passed source review. | null | published | incremental | hard | [
"source-candidate",
"theoremdb"
] | null | null | null | null | [
{
"kind": "website",
"citation": "TheoremDB P9322",
"url": "https://theoremdb.org/statements/P9322",
"note": null,
"sort_order": 0
},
{
"kind": "formalization",
"citation": "Source formal statement (not locally verified)",
"url": "https://theoremdb.org/statements/P9322",
"not... | {
"source_record": "P9322",
"imported_at": "2026-09-08",
"status_as_of": "2026-09-08",
"evidence_label": "Source candidate · not reviewed",
"note": "Full public directory statement. Source lane: backlog; source resolution: open. Source verification claims are attributed only; no local Lean verification has be... | {"created_at":"2026-08-17T19:50:53+00:00","display_standard":{"editorial_review":"submitter_supplied","formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":9322,"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": "2026-09-08T03:32:50.562Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"c22623c8-59fb-4486-9ca3-a0b97a0300a1","format_issues":null,"number":3443} |
1.0.0 | problem:theoremdb-p9324 | problem | theoremdb-p9324 | Density of natural numbers iterating to a fixed prime under Euler's totient plus one | {
"main": "math",
"sub": "number theory",
"slug": "number-theory",
"name": "Number theory",
"original_discipline": "mathematics"
} | [] | [
{
"code": "11",
"basis": "community-map",
"evidence": "Inferred from community field: number-theory"
}
] | For a positive integer $n$, let $\phi(n)$ denote Euler's totient function, which counts the number of integers in $\{1, 2, \ldots, n\}$ that are relatively prime to $n$. Consider the iteration $n \mapsto \phi(n) + 1$. For a fixed prime number $p$, let $S_p$ denote the set of positive integers $n$ such that some iterate... | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P9324",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P9324",
"citation": null,
"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.iii (p : ℕ) (h : p.Prime) (α : ℝ)
(hα : { n | ∃ i, (φ · + 1)^[i] n = p }.HasDensity α) :
α = answer(sorry) | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem erdos_409.parts.iii (p : ℕ) (h : p.Prime) (α : ℝ)\n (hα : { n | ∃ i, (φ · + 1)^[i] n = p }.HasDensity α) :\n α = answer(sorry)",
"target": null,
"source_url": "https://theoremdb.org/statements/P9324",
"format": "declaration-or-fragment",
"sha256... | [] | TheoremDB P9324 · Candidate awaiting source review | null | Source-reported assessment, retrieved 8 September 2026. Frontier Science has not independently reviewed this status.
Open. No signed Lean verification is attached to this problem or its selected resolution.
This record is in the source backlog; its novelty and unresolved status have not passed source review. | null | published | incremental | hard | [
"source-candidate",
"theoremdb"
] | null | null | null | null | [
{
"kind": "website",
"citation": "TheoremDB P9324",
"url": "https://theoremdb.org/statements/P9324",
"note": null,
"sort_order": 0
},
{
"kind": "formalization",
"citation": "Source formal statement (not locally verified)",
"url": "https://theoremdb.org/statements/P9324",
"not... | {
"source_record": "P9324",
"imported_at": "2026-09-08",
"status_as_of": "2026-09-08",
"evidence_label": "Source candidate · not reviewed",
"note": "Full public directory statement. Source lane: backlog; source resolution: open. Source verification claims are attributed only; no local Lean verification has be... | {"created_at":"2026-08-17T19:50:54+00:00","display_standard":{"editorial_review":"submitter_supplied","formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":9324,"public_status":{"evidence":{"hasFormalization":false,"hasInformalProof":false,"hasRecordedWork":false,"solutionRecord":null},"lean":{... | [
"formalization-contains-placeholder",
"formalization-not-compiled-during-export",
"current-status-unassessed",
"source-full-text-not-present"
] | [] | {
"created_at": "2026-09-08T03:32:15.935Z",
"updated_at": "2026-09-08T03:32:15.935Z",
"published_at": "2026-09-08T03:32:53.098Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"26717638-163f-4b04-9bf2-02304ba3233e","format_issues":null,"number":4003} |
1.0.0 | problem:theoremdb-p9326 | problem | theoremdb-p9326 | Waldschmidt's Conjecture on the Distance from $e^n$ to the Nearest Integer | {
"main": "math",
"sub": "number theory",
"slug": "number-theory",
"name": "Number theory",
"original_discipline": "mathematics"
} | [] | [
{
"code": "11",
"basis": "community-map",
"evidence": "Inferred from community field: number-theory"
}
] | For a real number $x$, let $\lVert x \rVert$ denote the distance from $x$ to the nearest integer, defined as $\lVert x \rVert = \min_{m \in \mathbb{Z}} |x - m|$. Prove that there exists a positive real number $c$ such that for every integer $n \geq 2$, the inequality $\lVert e^n \rVert > n^{-c}$ holds. | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P9326",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P9326",
"citation": null,
"doi": null,
"arxiv_id": null,
"license": "Selected packets label their content CC0-1.0; original references retain their own terms",
"notes": "Selected problem... | theorem waldschmidt :
∃ c : ℝ, 0 < c ∧ ∀ n : ℕ, 2 ≤ n → (n : ℝ) ^ (-c) < distToNearestInt (Real.exp n) | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem waldschmidt :\n ∃ c : ℝ, 0 < c ∧ ∀ n : ℕ, 2 ≤ n → (n : ℝ) ^ (-c) < distToNearestInt (Real.exp n)",
"target": null,
"source_url": "https://theoremdb.org/statements/P9326",
"format": "declaration-or-fragment",
"sha256": "0a2148291f2401c972f0bb43f520d... | [] | TheoremDB P9326 · Candidate awaiting source review | null | Source-reported assessment, retrieved 8 September 2026. Frontier Science has not independently reviewed this status.
Open. No signed Lean verification is attached to this problem or its selected resolution.
This record is in the source backlog; its novelty and unresolved status have not passed source review. | null | published | incremental | hard | [
"source-candidate",
"theoremdb"
] | null | null | null | null | [
{
"kind": "website",
"citation": "TheoremDB P9326",
"url": "https://theoremdb.org/statements/P9326",
"note": null,
"sort_order": 0
},
{
"kind": "formalization",
"citation": "Source formal statement (not locally verified)",
"url": "https://theoremdb.org/statements/P9326",
"not... | {
"source_record": "P9326",
"imported_at": "2026-09-08",
"status_as_of": "2026-09-08",
"evidence_label": "Source candidate · not reviewed",
"note": "Full public directory statement. Source lane: backlog; source resolution: open. Source verification claims are attributed only; no local Lean verification has be... | {"created_at":"2026-08-17T19:50:54+00:00","display_standard":{"editorial_review":"submitter_supplied","formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":9326,"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": "2026-09-08T03:33:07.446Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"3f8b8ca2-78cd-4967-9224-664ae5a3f7a7","format_issues":null,"number":6961} |
1.0.0 | problem:theoremdb-p9328 | problem | theoremdb-p9328 | Nonexistence of a Complex-Weighted Monochromatic Quantum Graph with Six Vertices and Three Colours | {
"main": "math",
"sub": "combinatorics",
"slug": "combinatorics",
"name": "Combinatorics",
"original_discipline": "mathematics"
} | [] | [
{
"code": "05",
"basis": "community-map",
"evidence": "Inferred from community field: combinatorics"
}
] | Let $N$ be a positive integer. Define $V_N = \{0, 1, \ldots, N-1\}$. An edge of the complete graph on $V_N$ is a 2-element subset $\{i,j\} \subseteq V_N$ with $i < j$. Let $D$ be a positive integer. A weight function on edges with $D$ colours and coefficients in a ring $\alpha$ is a function $W : \{\{i,j\} : 0 \leq i <... | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P9328",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P9328",
"citation": null,
"doi": null,
"arxiv_id": null,
"license": "Selected packets label their content CC0-1.0; original references retain their own terms",
"notes": "Selected problem... | theorem eqSystem6_no_solution_d3 :
answer(sorry) ↔
¬ ∃ W : WeightsN 6 3 ℂ, EqSystemN 6 3 W | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem eqSystem6_no_solution_d3 :\n answer(sorry) ↔\n ¬ ∃ W : WeightsN 6 3 ℂ, EqSystemN 6 3 W",
"target": null,
"source_url": "https://theoremdb.org/statements/P9328",
"format": "declaration-or-fragment",
"sha256": "26de66d6adcafb36c4f35b66eee8cef5dc... | [] | TheoremDB P9328 · Candidate awaiting source review | null | Source-reported assessment, retrieved 8 September 2026. Frontier Science has not independently reviewed this status.
Open. No signed Lean verification is attached to this problem or its selected resolution.
This record is in the source backlog; its novelty and unresolved status have not passed source review. | null | published | incremental | hard | [
"source-candidate",
"theoremdb"
] | null | null | null | null | [
{
"kind": "website",
"citation": "TheoremDB P9328",
"url": "https://theoremdb.org/statements/P9328",
"note": null,
"sort_order": 0
},
{
"kind": "formalization",
"citation": "Source formal statement (not locally verified)",
"url": "https://theoremdb.org/statements/P9328",
"not... | {
"source_record": "P9328",
"imported_at": "2026-09-08",
"status_as_of": "2026-09-08",
"evidence_label": "Source candidate · not reviewed",
"note": "Full public directory statement. Source lane: backlog; source resolution: open. Source verification claims are attributed only; no local Lean verification has be... | {"created_at":"2026-08-17T19:50:55+00:00","display_standard":{"editorial_review":"submitter_supplied","formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":9328,"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": "2026-09-08T03:33:04.470Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"30831ed4-5674-4ee4-831a-7201573030ba","format_issues":null,"number":6316} |
1.0.0 | problem:theoremdb-p9330 | problem | theoremdb-p9330 | Asymptotic covering probability for random subsets of prime cyclic groups with 1.01√p translates | {
"main": "math",
"sub": "combinatorics",
"slug": "combinatorics",
"name": "Combinatorics",
"original_discipline": "mathematics"
} | [] | [
{
"code": "05",
"basis": "community-map",
"evidence": "Inferred from community field: combinatorics"
}
] | Let $p$ be a prime number, let $k = \lfloor\sqrt{p}\rfloor$, and let $c = \lfloor 1.01\sqrt{p}\rfloor$. For a subset $A \subseteq \mathbb{Z}/p\mathbb{Z}$ with $|A| = k$, we say that $A$ is coverable with $c$ translates if there exists a subset $T \subseteq \mathbb{Z}/p\mathbb{Z}$ with $|T| \leq c$ such that $A + T = \m... | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P9330",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P9330",
"citation": null,
"doi": null,
"arxiv_id": null,
"license": "Selected packets label their content CC0-1.0; original references retain their own terms",
"notes": "Selected problem... | theorem green_39.variant_101 : answer(sorry) ↔
Tendsto
(fun p : {q : ℕ // q.Prime} ↦
let k := Nat.sqrt p
let c := ⌊1.01 * (k : ℝ)⌋₊
(proportionCoverable p k c : ℝ))
atTop (𝓝 1) | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem green_39.variant_101 : answer(sorry) ↔\n Tendsto\n (fun p : {q : ℕ // q.Prime} ↦\n let k := Nat.sqrt p\n let c := ⌊1.01 * (k : ℝ)⌋₊\n (proportionCoverable p k c : ℝ))\n atTop (𝓝 1)",
"target": null,
"source_url": "https://th... | [] | TheoremDB P9330 · Candidate awaiting source review | null | Source-reported assessment, retrieved 8 September 2026. Frontier Science has not independently reviewed this status.
Open. No signed Lean verification is attached to this problem or its selected resolution.
This record is in the source backlog; its novelty and unresolved status have not passed source review. | null | published | incremental | hard | [
"source-candidate",
"theoremdb"
] | null | null | null | null | [
{
"kind": "website",
"citation": "TheoremDB P9330",
"url": "https://theoremdb.org/statements/P9330",
"note": null,
"sort_order": 0
},
{
"kind": "formalization",
"citation": "Source formal statement (not locally verified)",
"url": "https://theoremdb.org/statements/P9330",
"not... | {
"source_record": "P9330",
"imported_at": "2026-09-08",
"status_as_of": "2026-09-08",
"evidence_label": "Source candidate · not reviewed",
"note": "Full public directory statement. Source lane: backlog; source resolution: open. Source verification claims are attributed only; no local Lean verification has be... | {"created_at":"2026-08-17T19:50:56+00:00","display_standard":{"editorial_review":"submitter_supplied","formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":9330,"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": "2026-09-08T03:32:49.992Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"8218d000-66a0-4700-bbda-88ed859aa736","format_issues":null,"number":3328} |
1.0.0 | problem:theoremdb-p9332 | problem | theoremdb-p9332 | Erdős problem 891 for the case k=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 $\omega(m)$ denote the number of distinct prime factors of a positive integer $m$. Is it true that for all sufficiently large integers $n$, there exists an integer $m$ in the interval $[n, n+6)$ such that $\omega(m) \geq 3$? | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P9332",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P9332",
"citation": null,
"doi": null,
"arxiv_id": null,
"license": "Selected packets label their content CC0-1.0; original references retain their own terms",
"notes": "Selected problem... | theorem erdos_891.variants.case_k_2 :
answer(sorry) ↔ ∀ᶠ n in atTop,
∃ m ∈ Ico n (n + 6), 3 ≤ ω m | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem erdos_891.variants.case_k_2 :\n answer(sorry) ↔ ∀ᶠ n in atTop,\n ∃ m ∈ Ico n (n + 6), 3 ≤ ω m",
"target": null,
"source_url": "https://theoremdb.org/statements/P9332",
"format": "declaration-or-fragment",
"sha256": "923d6d17e03636e2b7830cdef32... | [] | TheoremDB P9332 · Candidate awaiting source review | null | Source-reported assessment, retrieved 8 September 2026. Frontier Science has not independently reviewed this status.
Open. No signed Lean verification is attached to this problem or its selected resolution.
This record is in the source backlog; its novelty and unresolved status have not passed source review. | null | published | incremental | hard | [
"source-candidate",
"theoremdb"
] | null | null | null | null | [
{
"kind": "website",
"citation": "TheoremDB P9332",
"url": "https://theoremdb.org/statements/P9332",
"note": null,
"sort_order": 0
},
{
"kind": "formalization",
"citation": "Source formal statement (not locally verified)",
"url": "https://theoremdb.org/statements/P9332",
"not... | {
"source_record": "P9332",
"imported_at": "2026-09-08",
"status_as_of": "2026-09-08",
"evidence_label": "Source candidate · not reviewed",
"note": "Full public directory statement. Source lane: backlog; source resolution: open. Source verification claims are attributed only; no local Lean verification has be... | {"created_at":"2026-08-17T19:50:57+00:00","display_standard":{"formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":9332,"public_status":{"evidence":{"hasFormalization":false,"hasInformalProof":false,"hasRecordedWork":false,"solutionRecord":null},"lean":{"kernelAccepted":false,"label":"","reaso... | [
"formalization-contains-placeholder",
"formalization-not-compiled-during-export",
"current-status-unassessed",
"source-full-text-not-present"
] | [] | {
"created_at": "2026-09-08T03:32:15.935Z",
"updated_at": "2026-09-08T03:32:15.935Z",
"published_at": "2026-09-08T03:32:56.193Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"a45afa7c-8a4e-4de9-8203-bdc376538e7a","format_issues":null,"number":4687} |
1.0.0 | problem:theoremdb-p9334 | problem | theoremdb-p9334 | Erdős–Eggleton–Selfridge lower bound for maximal composite sum with least 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 $F(n)$ to be the maximum value of $m + p(m)$ over all composite integers $m$ with $m < n$, where $p(m)$ denotes the least prime divisor of $m$. Determine whether there exists a function $e: \mathbb{N} \to \mathbb{R}$ such that $e(n) = o(1)$ as $n \to \infty$ and $n + (1 - e(n))\sqr... | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P9334",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P9334",
"citation": null,
"doi": null,
"arxiv_id": null,
"license": "Selected packets label their content CC0-1.0; original references retain their own terms",
"notes": "Selected problem... | theorem erdos_385.variants.lb : answer(sorry) ↔ ∃ (e : ℕ → ℝ) (he : e =o[atTop] (1 : ℕ → ℝ)),
∀ n, n + (1 - e n) * √n ≤ F n | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem erdos_385.variants.lb : answer(sorry) ↔ ∃ (e : ℕ → ℝ) (he : e =o[atTop] (1 : ℕ → ℝ)),\n ∀ n, n + (1 - e n) * √n ≤ F n",
"target": null,
"source_url": "https://theoremdb.org/statements/P9334",
"format": "declaration-or-fragment",
"sha256": "929326e7... | [] | TheoremDB P9334 · Candidate awaiting source review | null | Source-reported assessment, retrieved 8 September 2026. Frontier Science has not independently reviewed this status.
Open. No signed Lean verification is attached to this problem or its selected resolution.
This record is in the source backlog; its novelty and unresolved status have not passed source review. | null | published | incremental | hard | [
"source-candidate",
"theoremdb"
] | null | null | null | null | [
{
"kind": "website",
"citation": "TheoremDB P9334",
"url": "https://theoremdb.org/statements/P9334",
"note": null,
"sort_order": 0
},
{
"kind": "formalization",
"citation": "Source formal statement (not locally verified)",
"url": "https://theoremdb.org/statements/P9334",
"not... | {
"source_record": "P9334",
"imported_at": "2026-09-08",
"status_as_of": "2026-09-08",
"evidence_label": "Source candidate · not reviewed",
"note": "Full public directory statement. Source lane: backlog; source resolution: open. Source verification claims are attributed only; no local Lean verification has be... | {"created_at":"2026-08-17T19:50:58+00:00","display_standard":{"editorial_review":"submitter_supplied","formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":9334,"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": "2026-09-08T03:32:56.659Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"1e5ad7bf-a57c-4e28-8ac6-a2d3d657cf3d","format_issues":null,"number":4787} |
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