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1.0.0 | problem:theoremdb-p9536 | problem | theoremdb-p9536 | Erdős Problem 143, Part (i) | {
"main": "math",
"sub": "number theory",
"slug": "number-theory",
"name": "Number theory",
"original_discipline": "mathematics"
} | [] | [
{
"code": "11",
"basis": "community-map",
"evidence": "Inferred from community field: number-theory"
}
] | A subset $A$ of the real numbers is called a well-separated set if $A \subseteq (1, \infty)$, the set $A$ is countably infinite, and for all distinct elements $x, y \in A$ and every integer $k \geq 1$, we have $|kx - y| \geq 1$.
Does every well-separated set $A \subseteq \mathbb{R}$ satisfy
$$
\liminf_{x \to \infty}... | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P9536",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P9536",
"citation": null,
"doi": null,
"arxiv_id": null,
"license": "Selected packets label their content CC0-1.0; original references retain their own terms",
"notes": "Selected problem... | theorem erdos_143.parts.i : answer(sorry) ↔ ∀ (A : Set ℝ), WellSeparatedSet A →
liminf (fun x => (A ∩ (Set.Icc 1 x)).ncard / x) atTop = 0 | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem erdos_143.parts.i : answer(sorry) ↔ ∀ (A : Set ℝ), WellSeparatedSet A →\n liminf (fun x => (A ∩ (Set.Icc 1 x)).ncard / x) atTop = 0",
"target": null,
"source_url": "https://theoremdb.org/statements/P9536",
"format": "declaration-or-fragment",
"sha2... | [] | TheoremDB P9536 · Candidate awaiting source review | null | Source-reported assessment, retrieved 8 September 2026. Frontier Science has not independently reviewed this status.
Open. No signed Lean verification is attached to this problem or its selected resolution.
This record is in the source backlog; its novelty and unresolved status have not passed source review. | null | published | incremental | hard | [
"source-candidate",
"theoremdb"
] | null | null | null | null | [
{
"kind": "website",
"citation": "TheoremDB P9536",
"url": "https://theoremdb.org/statements/P9536",
"note": null,
"sort_order": 0
},
{
"kind": "formalization",
"citation": "Source formal statement (not locally verified)",
"url": "https://theoremdb.org/statements/P9536",
"not... | {
"source_record": "P9536",
"imported_at": "2026-09-08",
"status_as_of": "2026-09-08",
"evidence_label": "Source candidate · not reviewed",
"note": "Full public directory statement. Source lane: backlog; source resolution: open. Source verification claims are attributed only; no local Lean verification has be... | {"created_at":"2026-08-17T19:54:46+00:00","display_standard":{"formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":9536,"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": "2026-09-08T03:32:55.152Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"dceb0676-cea5-4e41-a156-c17c9acf4055","format_issues":null,"number":4458} |
1.0.0 | problem:theoremdb-p9538 | problem | theoremdb-p9538 | Optimal constant for infinitely many integers with many divisors near their square root | {
"main": "math",
"sub": "number theory",
"slug": "number-theory",
"name": "Number theory",
"original_discipline": "mathematics"
} | [] | [
{
"code": "11",
"basis": "community-map",
"evidence": "Inferred from community field: number-theory"
}
] | For a positive integer $n$ and real numbers $C > 0$, let $D(n, C)$ denote the set of divisors $d$ of $n$ satisfying $\sqrt{n} < d < \sqrt{n} + C n^{1/4}$. For each positive integer $K$, say that $K$ is \emph{attainable} if there exists some $C > 0$ such that infinitely many positive integers $n$ have at least $K$ divis... | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P9538",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P9538",
"citation": null,
"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.variants.rosenfeld_4 :
IsGreatest {K | ∃ C > (0 : ℝ),
Infinite {n : ℕ | K ≤ #{ d ∈ Ioo ⌊√n⌋₊ ⌈√n + C * n^((1 : ℝ) / 4)⌉₊ | d ∣ n }}} 4 | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem erdos_887.variants.rosenfeld_4 :\n IsGreatest {K | ∃ C > (0 : ℝ),\n Infinite {n : ℕ | K ≤ #{ d ∈ Ioo ⌊√n⌋₊ ⌈√n + C * n^((1 : ℝ) / 4)⌉₊ | d ∣ n }}} 4",
"target": null,
"source_url": "https://theoremdb.org/statements/P9538",
"format": "declaration-o... | [] | TheoremDB P9538 · Candidate awaiting source review | null | Source-reported assessment, retrieved 8 September 2026. Frontier Science has not independently reviewed this status.
Open. No signed Lean verification is attached to this problem or its selected resolution.
This record is in the source backlog; its novelty and unresolved status have not passed source review. | null | published | incremental | hard | [
"source-candidate",
"theoremdb"
] | null | null | null | null | [
{
"kind": "website",
"citation": "TheoremDB P9538",
"url": "https://theoremdb.org/statements/P9538",
"note": null,
"sort_order": 0
},
{
"kind": "formalization",
"citation": "Source formal statement (not locally verified)",
"url": "https://theoremdb.org/statements/P9538",
"not... | {
"source_record": "P9538",
"imported_at": "2026-09-08",
"status_as_of": "2026-09-08",
"evidence_label": "Source candidate · not reviewed",
"note": "Full public directory statement. Source lane: backlog; source resolution: open. Source verification claims are attributed only; no local Lean verification has be... | {"created_at":"2026-08-17T19:54:49+00:00","display_standard":{"editorial_review":"submitter_supplied","formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":9538,"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": "2026-09-08T03:33:04.672Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"beb8d65a-bbfa-4d4c-bac6-4c8fa08c0a07","format_issues":null,"number":6361} |
1.0.0 | problem:theoremdb-p9540 | problem | theoremdb-p9540 | The Vaught conjecture on countable models of complete theories | {
"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 $L$ be a first-order language with countably many symbols, and let $T$ be a complete $L$-theory. The number of countable models of $T$ up to isomorphism is the cardinality of the set of isomorphism classes of countable models of $T$. Prove that this number is either at most countably infinite or exactly equal to th... | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P9540",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P9540",
"citation": null,
"doi": null,
"arxiv_id": null,
"license": "Selected packets label their content CC0-1.0; original references retain their own terms",
"notes": "Selected problem... | theorem vaught_conjecture {L : FirstOrder.Language} (hL : Countable L.Symbols)
{T : L.Theory} (hT : T.IsComplete) :
numberOfCountableModels T ≤ Cardinal.aleph0 ∨ numberOfCountableModels T = Cardinal.continuum | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem vaught_conjecture {L : FirstOrder.Language} (hL : Countable L.Symbols)\n {T : L.Theory} (hT : T.IsComplete) :\n numberOfCountableModels T ≤ Cardinal.aleph0 ∨ numberOfCountableModels T = Cardinal.continuum",
"target": null,
"source_ur... | [] | TheoremDB P9540 · Candidate awaiting source review | null | Source-reported assessment, retrieved 8 September 2026. Frontier Science has not independently reviewed this status.
Open. No signed Lean verification is attached to this problem or its selected resolution.
This record is in the source backlog; its novelty and unresolved status have not passed source review. | null | published | incremental | hard | [
"source-candidate",
"theoremdb"
] | null | null | null | null | [
{
"kind": "website",
"citation": "TheoremDB P9540",
"url": "https://theoremdb.org/statements/P9540",
"note": null,
"sort_order": 0
},
{
"kind": "formalization",
"citation": "Source formal statement (not locally verified)",
"url": "https://theoremdb.org/statements/P9540",
"not... | {
"source_record": "P9540",
"imported_at": "2026-09-08",
"status_as_of": "2026-09-08",
"evidence_label": "Source candidate · not reviewed",
"note": "Full public directory statement. Source lane: backlog; source resolution: open. Source verification claims are attributed only; no local Lean verification has be... | {"created_at":"2026-08-17T19:54:52+00:00","display_standard":{"editorial_review":"submitter_supplied","formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":9540,"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": "2026-09-08T03:33:06.854Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"9e4b90ee-32e6-4e65-99be-4e0dd73164cc","format_issues":null,"number":6832} |
1.0.0 | problem:theoremdb-p9542 | problem | theoremdb-p9542 | The New Mersenne Conjecture for Odd Primes | {
"main": "math",
"sub": "number theory",
"slug": "number-theory",
"name": "Number theory",
"original_discipline": "mathematics"
} | [] | [
{
"code": "11",
"basis": "community-map",
"evidence": "Inferred from community field: number-theory"
}
] | Let $p$ be an odd prime number. The Mersenne number with exponent $p$ is $M_p = 2^p - 1$. The number $p$ gives a Wagstaff prime if $(2^p + 1)/3$ is prime. The number $p$ is of special form if there exists a natural number $k$ such that $p = 2^k + 1$, or $p = 2^k - 1$, or $p = 4^k + 3$, or $p = 4^k - 3$. Prove that if a... | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P9542",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P9542",
"citation": null,
"doi": null,
"arxiv_id": null,
"license": "Selected packets label their content CC0-1.0; original references retain their own terms",
"notes": "Selected problem... | theorem new_mersenne_conjecture.variants.prime (p : ℕ) (hp : p.Prime) (h : Odd p) :
NewMersenneConjectureStatement p | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem new_mersenne_conjecture.variants.prime (p : ℕ) (hp : p.Prime) (h : Odd p) :\n NewMersenneConjectureStatement p",
"target": null,
"source_url": "https://theoremdb.org/statements/P9542",
"format": "declaration-or-fragment",
"sha256": "006ebccbf6827b7... | [] | TheoremDB P9542 · Candidate awaiting source review | null | Source-reported assessment, retrieved 8 September 2026. Frontier Science has not independently reviewed this status.
Open. No signed Lean verification is attached to this problem or its selected resolution.
This record is in the source backlog; its novelty and unresolved status have not passed source review. | null | published | incremental | hard | [
"source-candidate",
"theoremdb"
] | null | null | null | null | [
{
"kind": "website",
"citation": "TheoremDB P9542",
"url": "https://theoremdb.org/statements/P9542",
"note": null,
"sort_order": 0
},
{
"kind": "formalization",
"citation": "Source formal statement (not locally verified)",
"url": "https://theoremdb.org/statements/P9542",
"not... | {
"source_record": "P9542",
"imported_at": "2026-09-08",
"status_as_of": "2026-09-08",
"evidence_label": "Source candidate · not reviewed",
"note": "Full public directory statement. Source lane: backlog; source resolution: open. Source verification claims are attributed only; no local Lean verification has be... | {"created_at":"2026-08-17T19:54:54+00:00","display_standard":{"formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":9542,"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": "2026-09-08T03:33:06.667Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"049e57f5-5341-4f30-94e4-7169c9940fb6","format_issues":null,"number":6792} |
1.0.0 | problem:theoremdb-p9544 | problem | theoremdb-p9544 | Erdős Problem 680, Part II | {
"main": "math",
"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 $\epsilon > 0$, does there exist a real number $C > 0$ such that the following statement is false: for all sufficiently large natural numbers $n$, there exists a nonzero integer $k$ such that the least prime factor of $n + k$ exceeds $\exp\big((1 + \epsilon)\sqrt{k}\big) + C$? | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P9544",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P9544",
"citation": null,
"doi": null,
"arxiv_id": null,
"license": "Selected packets label their content CC0-1.0; original references retain their own terms",
"notes": "Selected problem... | theorem erdos_680.parts.ii : answer(sorry) ↔ ∀ ε > 0, ∃ C > 0,
¬ ∀ᶠ (n : ℕ) in Filter.atTop, ∃ k ≠ 0,
Nat.minFac (n + k) > exp ((1 + ε) * √k) + C | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem erdos_680.parts.ii : answer(sorry) ↔ ∀ ε > 0, ∃ C > 0,\n ¬ ∀ᶠ (n : ℕ) in Filter.atTop, ∃ k ≠ 0,\n Nat.minFac (n + k) > exp ((1 + ε) * √k) + C",
"target": null,
"source_url": "https://theoremdb.org/statements/P9544",
"format": "declaration-or-fragmen... | [] | TheoremDB P9544 · Candidate awaiting source review | null | Source-reported assessment, retrieved 8 September 2026. Frontier Science has not independently reviewed this status.
Open. No signed Lean verification is attached to this problem or its selected resolution.
This record is in the source backlog; its novelty and unresolved status have not passed source review. | null | published | incremental | hard | [
"source-candidate",
"theoremdb"
] | null | null | null | null | [
{
"kind": "website",
"citation": "TheoremDB P9544",
"url": "https://theoremdb.org/statements/P9544",
"note": null,
"sort_order": 0
},
{
"kind": "formalization",
"citation": "Source formal statement (not locally verified)",
"url": "https://theoremdb.org/statements/P9544",
"not... | {
"source_record": "P9544",
"imported_at": "2026-09-08",
"status_as_of": "2026-09-08",
"evidence_label": "Source candidate · not reviewed",
"note": "Full public directory statement. Source lane: backlog; source resolution: open. Source verification claims are attributed only; no local Lean verification has be... | {"created_at":"2026-08-17T19:54:56+00:00","display_standard":{"formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":9544,"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": "2026-09-08T03:32:55.888Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"060bb44d-6a34-4daf-8067-e7c00d8cb8c1","format_issues":null,"number":4619} |
1.0.0 | problem:theoremdb-p9546 | problem | theoremdb-p9546 | The Prime Power Conjecture for Finite Projective Planes | {
"main": "math",
"sub": "combinatorics",
"slug": "combinatorics",
"name": "Combinatorics",
"original_discipline": "mathematics"
} | [] | [
{
"code": "05",
"basis": "community-map",
"evidence": "Inferred from community field: combinatorics"
}
] | A finite projective plane is a geometric structure consisting of a finite set of points and a finite set of lines, where each line is a subset of points, satisfying the following axioms: any two distinct points lie on exactly one common line; any two distinct lines intersect in exactly one point; and there exist four p... | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P9546",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P9546",
"citation": null,
"doi": null,
"arxiv_id": null,
"license": "Selected packets label their content CC0-1.0; original references retain their own terms",
"notes": "Selected problem... | theorem erdos_723 :
answer(sorry) ↔ ∀ {P L : Type} (_: Membership P L) (_ : Fintype P) (_ : Fintype L),
∀ pp : ProjectivePlane P L, IsPrimePow pp.order | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem erdos_723 :\n answer(sorry) ↔ ∀ {P L : Type} (_: Membership P L) (_ : Fintype P) (_ : Fintype L),\n ∀ pp : ProjectivePlane P L, IsPrimePow pp.order",
"target": null,
"source_url": "https://theoremdb.org/statements/P9546",
"format": "declaration-or... | [] | TheoremDB P9546 · Candidate awaiting source review | null | Source-reported assessment, retrieved 8 September 2026. Frontier Science has not independently reviewed this status.
Open. No signed Lean verification is attached to this problem or its selected resolution.
This record is in the source backlog; its novelty and unresolved status have not passed source review. | null | published | incremental | hard | [
"source-candidate",
"theoremdb"
] | null | null | null | null | [
{
"kind": "website",
"citation": "TheoremDB P9546",
"url": "https://theoremdb.org/statements/P9546",
"note": null,
"sort_order": 0
},
{
"kind": "formalization",
"citation": "Source formal statement (not locally verified)",
"url": "https://theoremdb.org/statements/P9546",
"not... | {
"source_record": "P9546",
"imported_at": "2026-09-08",
"status_as_of": "2026-09-08",
"evidence_label": "Source candidate · not reviewed",
"note": "Full public directory statement. Source lane: backlog; source resolution: open. Source verification claims are attributed only; no local Lean verification has be... | {"created_at":"2026-08-17T19:54:59+00:00","display_standard":{"editorial_review":"submitter_supplied","formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":9546,"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": "2026-09-08T03:33:06.713Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"64f34ba1-c6bd-4415-bfb8-8e241ad9af53","format_issues":null,"number":6802} |
1.0.0 | problem:theoremdb-p9548 | problem | theoremdb-p9548 | Lower bound for the off-diagonal van der Waerden number W(3,31) | {
"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,\dots,N\}$ with two colors contains either a monochromatic arithmetic progression of length $k$ in the first color or a monochromatic arithmetic progression of l... | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P9548",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P9548",
"citation": null,
"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_31_lower : answer(sorry) ↔ W 3 31 > 930 | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem W_3_31_lower : answer(sorry) ↔ W 3 31 > 930",
"target": null,
"source_url": "https://theoremdb.org/statements/P9548",
"format": "declaration-or-fragment",
"sha256": "993a6bf7972fef2cca715706003b29629cfb30dd345dc3c62db23470c7f6cb16",
"license": "Se... | [] | TheoremDB P9548 · Candidate awaiting source review | null | Source-reported assessment, retrieved 8 September 2026. Frontier Science has not independently reviewed this status.
Open. No signed Lean verification is attached to this problem or its selected resolution.
This record is in the source 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/P9548",
"not... | {
"source_record": "P9548",
"imported_at": "2026-09-08",
"status_as_of": "2026-09-08",
"evidence_label": "Source candidate · not reviewed",
"note": "Full public directory statement. Source lane: backlog; source resolution: open. Source verification claims are attributed only; no local Lean verification has be... | {"created_at":"2026-08-17T19:55:02+00:00","display_standard":{"formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":9548,"public_status":{"evidence":{"hasFormalization":false,"hasInformalProof":false,"hasRecordedWork":false,"solutionRecord":null},"lean":{"kernelAccepted":false,"label":"","reaso... | [
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"created_at": "2026-09-08T03:32:15.935Z",
"updated_at": "2026-09-08T03:32:15.935Z",
"published_at": "2026-09-08T03:33:03.484Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"541b5108-be25-46ab-b265-03c31d027324","format_issues":null,"number":6105} |
1.0.0 | problem:theoremdb-p9550 | problem | theoremdb-p9550 | Existence of dense infinite Sidon sets | {
"main": "math",
"sub": "number theory",
"slug": "number-theory",
"name": "Number theory",
"original_discipline": "mathematics"
} | [] | [
{
"code": "11",
"basis": "community-map",
"evidence": "Inferred from community field: number-theory"
}
] | A set $A$ of natural numbers is called a Sidon set if all pairwise sums of its elements are distinct, i.e., if $a + b = c + d$ with $a, b, c, d \in A$, then $\{a, b\} = \{c, d\}$. Does there exist an infinite Sidon set $A \subseteq \mathbb{N}$ such that for every real number $\varepsilon > 0$, the counting function sat... | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P9550",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P9550",
"citation": null,
"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_39 : answer(sorry) ↔ ∃ (A : Set ℕ), A.Infinite ∧ IsSidon A ∧
∀ᵉ (ε > (0 : ℝ)),
(· ^ (1 / 2 - ε) : ℕ → ℝ) =O[atTop] fun N => (((Set.Icc 1 N) ∩ A).ncard : ℝ) | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem erdos_39 : answer(sorry) ↔ ∃ (A : Set ℕ), A.Infinite ∧ IsSidon A ∧\n ∀ᵉ (ε > (0 : ℝ)),\n (· ^ (1 / 2 - ε) : ℕ → ℝ) =O[atTop] fun N => (((Set.Icc 1 N) ∩ A).ncard : ℝ)",
"target": null,
"source_url": "https://theoremdb.org/statements/P9550",
"format... | [] | TheoremDB P9550 · Candidate awaiting source review | null | Source-reported assessment, retrieved 8 September 2026. Frontier Science has not independently reviewed this status.
Open. No signed Lean verification is attached to this problem or its selected resolution.
This record is in the source backlog; its novelty and unresolved status have not passed source review. | null | published | incremental | hard | [
"source-candidate",
"theoremdb"
] | null | null | null | null | [
{
"kind": "website",
"citation": "TheoremDB P9550",
"url": "https://theoremdb.org/statements/P9550",
"note": null,
"sort_order": 0
},
{
"kind": "formalization",
"citation": "Source formal statement (not locally verified)",
"url": "https://theoremdb.org/statements/P9550",
"not... | {
"source_record": "P9550",
"imported_at": "2026-09-08",
"status_as_of": "2026-09-08",
"evidence_label": "Source candidate · not reviewed",
"note": "Full public directory statement. Source lane: backlog; source resolution: open. Source verification claims are attributed only; no local Lean verification has be... | {"created_at":"2026-08-17T19:55:07+00:00","display_standard":{"editorial_review":"submitter_supplied","formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":9550,"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": "2026-09-08T03:32:58.725Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"5399c75d-28ee-4794-b014-d6d428d236ff","format_issues":null,"number":5242} |
1.0.0 | problem:theoremdb-p9552 | problem | theoremdb-p9552 | Non-existence of monochromatic quantum graphs with 16 vertices and 3 colours over the complex numbers | {
"main": "math",
"sub": "combinatorics",
"slug": "combinatorics",
"name": "Combinatorics",
"original_discipline": "mathematics"
} | [] | [
{
"code": "05",
"basis": "community-map",
"evidence": "Inferred from community field: combinatorics"
}
] | Let $N = 16$ and $D = 3$. A \emph{weight function} on the complete graph $K_N$ with $D$ colours and coefficients in a ring $\alpha$ is a function $W : E_N \times \{1, \dots, D\} \to \alpha$, where $E_N$ denotes the set of unordered pairs of vertices $\{i,j\}$ with $1 \leq i < j \leq N$. For each vertex colouring $\iota... | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P9552",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P9552",
"citation": null,
"doi": null,
"arxiv_id": null,
"license": "Selected packets label their content CC0-1.0; original references retain their own terms",
"notes": "Selected problem... | theorem eqSystem16_no_solution_d3 :
answer(sorry) ↔
¬ ∃ W : WeightsN 16 3 ℂ, EqSystemN 16 3 W | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem eqSystem16_no_solution_d3 :\n answer(sorry) ↔\n ¬ ∃ W : WeightsN 16 3 ℂ, EqSystemN 16 3 W",
"target": null,
"source_url": "https://theoremdb.org/statements/P9552",
"format": "declaration-or-fragment",
"sha256": "3b2da43b3a49793564d41efa25ce038... | [] | TheoremDB P9552 · Candidate awaiting source review | null | Source-reported assessment, retrieved 8 September 2026. Frontier Science has not independently reviewed this status.
Open. No signed Lean verification is attached to this problem or its selected resolution.
This record is in the source backlog; its novelty and unresolved status have not passed source review. | null | published | incremental | hard | [
"source-candidate",
"theoremdb"
] | null | null | null | null | [
{
"kind": "website",
"citation": "TheoremDB P9552",
"url": "https://theoremdb.org/statements/P9552",
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},
{
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"citation": "Source formal statement (not locally verified)",
"url": "https://theoremdb.org/statements/P9552",
"not... | {
"source_record": "P9552",
"imported_at": "2026-09-08",
"status_as_of": "2026-09-08",
"evidence_label": "Source candidate · not reviewed",
"note": "Full public directory statement. Source lane: backlog; source resolution: open. Source verification claims are attributed only; no local Lean verification has be... | {"created_at":"2026-08-17T19:55:11+00:00","display_standard":{"editorial_review":"submitter_supplied","formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":9552,"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": "2026-09-08T03:33:04.299Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"a77acecf-5637-4002-8fe4-cf88508882f6","format_issues":null,"number":6282} |
1.0.0 | problem:theoremdb-p9554 | problem | theoremdb-p9554 | Markel's $S_3$-conjecture on anti-homogeneous groups | {
"main": "math",
"sub": "algebra",
"slug": "algebra",
"name": "Algebra",
"original_discipline": "mathematics"
} | [] | [] | A finite group $G$ is called anti-homogeneous (or an ah-group) if it has the property that distinct conjugacy classes have distinct cardinalities. Equivalently, the map assigning to each conjugacy class its cardinality is injective. Let $G$ be a nontrivial finite group that is anti-homogeneous. Determine whether $G$ is... | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P9554",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P9554",
"citation": null,
"doi": null,
"arxiv_id": null,
"license": "Selected packets label their content CC0-1.0; original references retain their own terms",
"notes": "Selected problem... | theorem conjClassSizes_iff_sym_three
(G : Type) [Group G] [Fintype G] [Nontrivial G]
(h : HasDistinctConjClassSizes (G := G)) :
Nonempty (G ≃* Equiv.Perm (Fin 3)) | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem conjClassSizes_iff_sym_three\n (G : Type) [Group G] [Fintype G] [Nontrivial G]\n (h : HasDistinctConjClassSizes (G := G)) :\n Nonempty (G ≃* Equiv.Perm (Fin 3))",
"target": null,
"source_url": "https://theoremdb.org/statements/P9554",
"format": "... | [] | TheoremDB P9554 · Candidate awaiting source review | null | Source-reported assessment, retrieved 8 September 2026. Frontier Science has not independently reviewed this status.
Open. No signed Lean verification is attached to this problem or its selected resolution.
This record is in the source backlog; its novelty and unresolved status have not passed source review. | null | published | incremental | hard | [
"source-candidate",
"theoremdb"
] | null | null | null | null | [
{
"kind": "website",
"citation": "TheoremDB P9554",
"url": "https://theoremdb.org/statements/P9554",
"note": null,
"sort_order": 0
},
{
"kind": "formalization",
"citation": "Source formal statement (not locally verified)",
"url": "https://theoremdb.org/statements/P9554",
"not... | {
"source_record": "P9554",
"imported_at": "2026-09-08",
"status_as_of": "2026-09-08",
"evidence_label": "Source candidate · not reviewed",
"note": "Full public directory statement. Source lane: backlog; source resolution: open. Source verification claims are attributed only; no local Lean verification has be... | {"created_at":"2026-08-17T19:55:12+00:00","display_standard":{"formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":9554,"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": "2026-09-08T03:33:03.623Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"2b0ec9dd-b55c-4447-b91b-ff11adffc6c4","format_issues":null,"number":6134} |
1.0.0 | problem:theoremdb-p9556 | problem | theoremdb-p9556 | Asymptotic growth rate of transversals in cyclic Latin squares | {
"main": "math",
"sub": "combinatorics",
"slug": "combinatorics",
"name": "Combinatorics",
"original_discipline": "mathematics"
} | [] | [
{
"code": "05",
"basis": "community-map",
"evidence": "Inferred from community field: combinatorics"
}
] | Let $z(n)$ denote the number of transversals in the Cayley table of the cyclic group $\mathbb{Z}_n$, that is, the $n \times n$ Latin square whose entry in row $i$ and column $j$ is $i + j \pmod n$. It is known that $z(n) = 0$ for all even $n > 0$. For odd integers, parametrize $n = 2k + 1$ with $k \in \mathbb{N}$. Does... | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P9556",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P9556",
"citation": null,
"doi": null,
"arxiv_id": null,
"license": "Selected packets label their content CC0-1.0; original references retain their own terms",
"notes": "Selected problem... | theorem growthRateZn : answer(sorry) ↔
Filter.Tendsto (fun k => (1 : ℝ) / (2 * k + 1) *
Real.log (z (2 * k + 1) / (2 * k + 1).factorial)) Filter.atTop
(nhds (-1)) | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem growthRateZn : answer(sorry) ↔\n Filter.Tendsto (fun k => (1 : ℝ) / (2 * k + 1) *\n Real.log (z (2 * k + 1) / (2 * k + 1).factorial)) Filter.atTop\n (nhds (-1))",
"target": null,
"source_url": "https://theoremdb.org/statements/P9556",
"format... | [] | TheoremDB P9556 · Candidate awaiting source review | null | Source-reported assessment, retrieved 8 September 2026. Frontier Science has not independently reviewed this status.
Open. No signed Lean verification is attached to this problem or its selected resolution.
This record is in the source backlog; its novelty and unresolved status have not passed source review. | null | published | incremental | hard | [
"source-candidate",
"theoremdb"
] | null | null | null | null | [
{
"kind": "website",
"citation": "TheoremDB P9556",
"url": "https://theoremdb.org/statements/P9556",
"note": null,
"sort_order": 0
},
{
"kind": "formalization",
"citation": "Source formal statement (not locally verified)",
"url": "https://theoremdb.org/statements/P9556",
"not... | {
"source_record": "P9556",
"imported_at": "2026-09-08",
"status_as_of": "2026-09-08",
"evidence_label": "Source candidate · not reviewed",
"note": "Full public directory statement. Source lane: backlog; source resolution: open. Source verification claims are attributed only; no local Lean verification has be... | {"created_at":"2026-08-17T19:55:14+00:00","display_standard":{"editorial_review":"submitter_supplied","formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":9556,"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": "2026-09-08T03:32:50.292Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"c4ccf0eb-791d-44eb-980b-06a0cf0acbeb","format_issues":null,"number":3390} |
1.0.0 | problem:theoremdb-p9558 | problem | theoremdb-p9558 | Transcendence of the iterated logarithm of 2 | {
"main": "math",
"sub": "number theory",
"slug": "number-theory",
"name": "Number theory",
"original_discipline": "mathematics"
} | [] | [
{
"code": "11",
"basis": "community-map",
"evidence": "Inferred from community field: number-theory"
}
] | A real number is said to be transcendental if it is not algebraic over the rational numbers, meaning it is not a root of any non-zero polynomial with rational coefficients. Let $\log$ denote the natural logarithm. Determine whether $\log(\log 2)$ is transcendental. | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P9558",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P9558",
"citation": null,
"doi": null,
"arxiv_id": null,
"license": "Selected packets label their content CC0-1.0; original references retain their own terms",
"notes": "Selected problem... | theorem rlog_rlog_two_transcendental : Transcendental ℚ ((2 : ℝ).log.log) | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem rlog_rlog_two_transcendental : Transcendental ℚ ((2 : ℝ).log.log)",
"target": null,
"source_url": "https://theoremdb.org/statements/P9558",
"format": "declaration-or-fragment",
"sha256": "2998ca80b0954faf583ce75b054aff2edb49ba8a1c9a6d5851f5a53129aeae1... | [] | TheoremDB P9558 · Candidate awaiting source review | null | Source-reported assessment, retrieved 8 September 2026. Frontier Science has not independently reviewed this status.
Open. No signed Lean verification is attached to this problem or its selected resolution.
This record is in the source backlog; its novelty and unresolved status have not passed source review. | null | published | incremental | hard | [
"source-candidate",
"theoremdb"
] | null | null | null | null | [
{
"kind": "website",
"citation": "TheoremDB P9558",
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"citation": "Source formal statement (not locally verified)",
"url": "https://theoremdb.org/statements/P9558",
"not... | {
"source_record": "P9558",
"imported_at": "2026-09-08",
"status_as_of": "2026-09-08",
"evidence_label": "Source candidate · not reviewed",
"note": "Full public directory statement. Source lane: backlog; source resolution: open. Source verification claims are attributed only; no local Lean verification has be... | {"created_at":"2026-08-17T19:55:16+00:00","display_standard":{"editorial_review":"submitter_supplied","formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":9558,"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": "2026-09-08T03:33:07.012Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"0d5a1a20-dcba-4ff2-a64b-feac219efacb","format_issues":null,"number":6866} |
1.0.0 | problem:theoremdb-p9560 | problem | theoremdb-p9560 | Existence of an absolutely maximally entangled pure state for eight parties of local dimension ten | {
"main": "math",
"sub": "combinatorics",
"slug": "combinatorics",
"name": "Combinatorics",
"original_discipline": "mathematics"
} | [] | [
{
"code": "05",
"basis": "community-map",
"evidence": "Inferred from community field: combinatorics"
}
] | Let $n$ and $d$ be positive integers. An $n$-partite pure quantum state with local dimension $d$ is called absolutely maximally entangled (AME) if, for every subset of at most $\lfloor n/2 \rfloor$ parties, the reduced density matrix of that subset is maximally mixed. Does there exist an absolutely maximally entangled ... | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P9560",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P9560",
"citation": null,
"doi": null,
"arxiv_id": null,
"license": "Selected packets label their content CC0-1.0; original references retain their own terms",
"notes": "Selected problem... | theorem ame_8_10_open :
answer(sorry) ↔ ExistsAME 8 10 | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem ame_8_10_open :\n answer(sorry) ↔ ExistsAME 8 10",
"target": null,
"source_url": "https://theoremdb.org/statements/P9560",
"format": "declaration-or-fragment",
"sha256": "ae5c4d3f96489a73125bebc8f0e516e73df79b4ace797903515c345cdb72d8ce",
"licen... | [] | TheoremDB P9560 · Candidate awaiting source review | null | Source-reported assessment, retrieved 8 September 2026. Frontier Science has not independently reviewed this status.
Open. No signed Lean verification is attached to this problem or its selected resolution.
This record is in the source backlog; its novelty and unresolved status have not passed source review. | null | published | incremental | hard | [
"source-candidate",
"theoremdb"
] | null | null | null | null | [
{
"kind": "website",
"citation": "TheoremDB P9560",
"url": "https://theoremdb.org/statements/P9560",
"note": null,
"sort_order": 0
},
{
"kind": "formalization",
"citation": "Source formal statement (not locally verified)",
"url": "https://theoremdb.org/statements/P9560",
"not... | {
"source_record": "P9560",
"imported_at": "2026-09-08",
"status_as_of": "2026-09-08",
"evidence_label": "Source candidate · not reviewed",
"note": "Full public directory statement. Source lane: backlog; source resolution: open. Source verification claims are attributed only; no local Lean verification has be... | {"created_at":"2026-08-17T19:55:17+00:00","display_standard":{"editorial_review":"submitter_supplied","formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":9560,"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": "2026-09-08T03:32:58.599Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"30303dd1-9b41-41d2-b4d4-d30996cee3a9","format_issues":null,"number":5216} |
1.0.0 | problem:theoremdb-p9562 | problem | theoremdb-p9562 | Perfect Powers Among Partition Numbers | {
"main": "math",
"sub": "number theory",
"slug": "number-theory",
"name": "Number theory",
"original_discipline": "mathematics"
} | [] | [
{
"code": "11",
"basis": "community-map",
"evidence": "Inferred from community field: number-theory"
}
] | Let $a(n)$ denote the number of partitions of $n$, that is, the number of ways to write $n$ as a sum of positive integers where the order of summands does not matter. A positive integer is called a perfect power if it can be expressed as $x^m$ for some integers $x > 1$ and $m > 1$. Determine whether there exists any in... | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P9562",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P9562",
"citation": null,
"doi": null,
"arxiv_id": null,
"license": "Selected packets label their content CC0-1.0; original references retain their own terms",
"notes": "Selected problem... | theorem noPowerPartitionNumber : answer(sorry) ↔ ∀ k, ¬IsPerfectPower (a k) | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem noPowerPartitionNumber : answer(sorry) ↔ ∀ k, ¬IsPerfectPower (a k)",
"target": null,
"source_url": "https://theoremdb.org/statements/P9562",
"format": "declaration-or-fragment",
"sha256": "d3a8eb4cf9d72a3d45b3061d01073b7e012b78bf472e304c35b29b02ed297... | [] | TheoremDB P9562 · Candidate awaiting source review | null | Source-reported assessment, retrieved 8 September 2026. Frontier Science has not independently reviewed this status.
Open. No signed Lean verification is attached to this problem or its selected resolution.
This record is in 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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} | 2026-10-10T16:07:49.246Z | {"database_id":"3426cf50-d1ac-41a5-af67-6566ef6b9064","format_issues":null,"number":6384} |
1.0.0 | problem:theoremdb-p9564 | problem | theoremdb-p9564 | Lower bound for the off-diagonal van der Waerden number W(3, 38) | {
"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 · P9564",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P9564",
"citation": null,
"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_38_lower : answer(sorry) ↔ W 3 38 > 1378 | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem W_3_38_lower : answer(sorry) ↔ W 3 38 > 1378",
"target": null,
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"sha256": "8b41167e299c39d9e146143de42048b3f6d20cfdf9c4a21fc80ad04ee53e4fcf",
"license": "S... | [] | TheoremDB P9564 · Candidate awaiting source review | null | Source-reported assessment, retrieved 8 September 2026. Frontier Science has not independently reviewed this status.
Open. No signed Lean verification is attached to this problem or its selected resolution.
This record is in the source backlog; its novelty and unresolved status have not passed source review. | null | published | incremental | hard | [
"source-candidate",
"theoremdb"
] | null | null | null | null | [
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"url": "https://theoremdb.org/statements/P9564",
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} | 2026-10-10T16:07:49.246Z | {"database_id":"ec41a267-858e-4b0b-9214-3b680a08e9a1","format_issues":null,"number":6103} |
1.0.0 | problem:theoremdb-p9566 | problem | theoremdb-p9566 | Smooth Poincaré Conjecture in Dimension Four | {
"main": "math",
"sub": "geometry and topology",
"slug": "geometry-and-topology",
"name": "Geometry and topology",
"original_discipline": "mathematics"
} | [] | [] | For a natural number $n$, the smooth Poincaré conjecture in dimension $n$ is the statement that every smooth manifold $M$ of dimension $n$ which is homotopy equivalent to the $n$-dimensional unit sphere $\mathbb{S}^n$ is diffeomorphic to $\mathbb{S}^n$. Here $\mathbb{S}^n$ denotes the unit sphere in $(n+1)$-dimensional... | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P9566",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P9566",
"citation": null,
"doi": null,
"arxiv_id": null,
"license": "Selected packets label their content CC0-1.0; original references retain their own terms",
"notes": "Selected problem... | theorem poincare_conjecture.variants.smooth_dimension_four : SmoothConjectureFor 4 | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem poincare_conjecture.variants.smooth_dimension_four : SmoothConjectureFor 4",
"target": null,
"source_url": "https://theoremdb.org/statements/P9566",
"format": "declaration-or-fragment",
"sha256": "011e45465b4a10e1f552923c37033f8d548e496dd909f78b3c6bee... | [] | TheoremDB P9566 · Candidate awaiting source review | null | Source-reported assessment, retrieved 8 September 2026. Frontier Science has not independently reviewed this status.
Open. No signed Lean verification is attached to this problem or its selected resolution.
This record is in 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 P9566",
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"citation": "Source formal statement (not locally verified)",
"url": "https://theoremdb.org/statements/P9566",
"not... | {
"source_record": "P9566",
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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":"2026-08-17T19:55:29+00:00","display_standard":{"editorial_review":"submitter_supplied","formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":9566,"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": "2026-09-08T03:33:05.959Z",
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} | 2026-10-10T16:07:49.246Z | {"database_id":"74924194-c870-4c31-8566-d615fea08d91","format_issues":null,"number":6640} |
1.0.0 | problem:theoremdb-p9568 | problem | theoremdb-p9568 | Converse implication for the maximum normalized upper density of Sidon sets | {
"main": "math",
"sub": "combinatorics",
"slug": "combinatorics",
"name": "Combinatorics",
"original_discipline": "mathematics"
} | [] | [
{
"code": "05",
"basis": "community-map",
"evidence": "Inferred from community field: combinatorics"
}
] | For a set $A \subseteq \mathbb{N}$, define the square-root partial density of $A$ up to $N$ by
$$
\frac{|A \cap \{1, 2, \dots, N\}|}{\sqrt{N}}.
$$
The Sidon upper density of $A$ is the limit superior of these partial densities as $N \to \infty$:
$$
\limsup_{N \to \infty} \frac{|A \cap \{1, 2, \dots, N\}|}{\sqrt{N... | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P9568",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P9568",
"citation": null,
"doi": null,
"arxiv_id": null,
"license": "Selected packets label their content CC0-1.0; original references retain their own terms",
"notes": "Selected problem... | theorem erdos_329.variants.converse_implication :
(sSup {sidonUpperDensity A | (A : Set ℕ) (_ : IsSidon A)} = 1) →
(∀ (A : Finset ℕ), IsSidon (A : Set ℕ) → ∃ (D : Set ℕ) (n : ℕ) (_ : n > 0),
↑A ⊆ D ∧ IsPerfectDifferenceSet D n) | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem erdos_329.variants.converse_implication :\n (sSup {sidonUpperDensity A | (A : Set ℕ) (_ : IsSidon A)} = 1) →\n (∀ (A : Finset ℕ), IsSidon (A : Set ℕ) → ∃ (D : Set ℕ) (n : ℕ) (_ : n > 0),\n ↑A ⊆ D ∧ IsPerfectDifferenceSet D n)",
"target": null,
"sou... | [] | TheoremDB P9568 · Candidate awaiting source review | null | Source-reported assessment, retrieved 8 September 2026. Frontier Science has not independently reviewed this status.
Open. No signed Lean verification is attached to this problem or its selected resolution.
This record is in 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 P9568",
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"url": "https://theoremdb.org/statements/P9568",
"not... | {
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"note": "Full public directory statement. Source lane: backlog; source resolution: open. Source verification claims are attributed only; no local Lean verification has be... | {"created_at":"2026-08-17T19:55:34+00:00","display_standard":{"editorial_review":"submitter_supplied","formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":9568,"public_status":{"evidence":{"hasFormalization":false,"hasInformalProof":false,"hasRecordedWork":false,"solutionRecord":null},"lean":{... | [
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"published_at": "2026-09-08T03:32:52.697Z",
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} | 2026-10-10T16:07:49.246Z | {"database_id":"b10eef66-dc8c-4570-920a-3c8d2fb0bbb9","format_issues":null,"number":3916} |
1.0.0 | problem:theoremdb-p9570 | problem | theoremdb-p9570 | Erdős's Coloring Problem for $k$-Subsets of a $2k$-Set | {
"main": "math",
"sub": "combinatorics",
"slug": "combinatorics",
"name": "Combinatorics",
"original_discipline": "mathematics"
} | [] | [
{
"code": "05",
"basis": "community-map",
"evidence": "Inferred from community field: combinatorics"
}
] | For a positive integer $k$, let $\binom{[2k]}{k}$ denote the collection of all $k$-element subsets of $\{1, 2, \ldots, 2k\}$. We say that $k$ has the coloring property if there exists a coloring $c: \binom{[2k]}{k} \to \{0, 1, \ldots, k\}$ with $k+1$ colors such that for every subset $A \subseteq \{1, 2, \ldots, 2k\}$ ... | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P9570",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P9570",
"citation": null,
"doi": null,
"arxiv_id": null,
"license": "Selected packets label their content CC0-1.0; original references retain their own terms",
"notes": "Selected problem... | theorem erdos_835 : (∃ k > 2, Property k) ↔ answer(sorry) | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem erdos_835 : (∃ k > 2, Property k) ↔ answer(sorry)",
"target": null,
"source_url": "https://theoremdb.org/statements/P9570",
"format": "declaration-or-fragment",
"sha256": "f2ad9e4121ae997d8a0d8755172e8abeebb91da921ea1192b69349e50ed6a190",
"license... | [] | TheoremDB P9570 · Candidate awaiting source review | null | Source-reported assessment, retrieved 8 September 2026. Frontier Science has not independently reviewed this status.
Open. No signed Lean verification is attached to this problem or its selected resolution.
This record is in 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 P9570",
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"citation": "Source formal statement (not locally verified)",
"url": "https://theoremdb.org/statements/P9570",
"not... | {
"source_record": "P9570",
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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":"2026-08-17T19:55:38+00:00","display_standard":{"editorial_review":"submitter_supplied","formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":9570,"public_status":{"evidence":{"hasFormalization":false,"hasInformalProof":false,"hasRecordedWork":false,"solutionRecord":null},"lean":{... | [
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"created_at": "2026-09-08T03:32:15.935Z",
"updated_at": "2026-09-08T03:32:15.935Z",
"published_at": "2026-09-08T03:32:56.329Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"3ec19e5b-2599-4f36-abe1-630b958d1558","format_issues":null,"number":4717} |
1.0.0 | problem:theoremdb-p9572 | problem | theoremdb-p9572 | Existence of a SIC-POVM in dimension 60 | {
"main": "math",
"sub": "algebra",
"slug": "algebra",
"name": "Algebra",
"original_discipline": "mathematics"
} | [] | [] | For a positive integer $d$, let $\mathbb{C}^d$ denote the $d$-dimensional complex Hilbert space with its standard inner product $\langle \phi, \psi \rangle = \sum_{i=0}^{d-1} \overline{\phi_i} \, \psi_i$. A state vector in $\mathbb{C}^d$ is a vector of unit $L^2$ norm, i.e., $\|\psi\| = 1$. For two state vectors $\phi,... | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P9572",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P9572",
"citation": null,
"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_60 : answer(sorry) ↔ HasSICPOVM 60 | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem hasSICPOVM_60 : answer(sorry) ↔ HasSICPOVM 60",
"target": null,
"source_url": "https://theoremdb.org/statements/P9572",
"format": "declaration-or-fragment",
"sha256": "c1e7e02e65b561aa7f23b5a7113b753cca977f5a7a2129ee691bd3e73d4bf76d",
"license": "... | [] | TheoremDB P9572 · Candidate awaiting source review | null | Source-reported assessment, retrieved 8 September 2026. Frontier Science has not independently reviewed this status.
Open. No signed Lean verification is attached to this problem or its selected resolution.
This record is in 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 P9572",
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"url": "https://theoremdb.org/statements/P9572",
"not... | {
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"note": "Full public directory statement. Source lane: backlog; source resolution: open. Source verification claims are attributed only; no local Lean verification has be... | {"created_at":"2026-08-17T19:55:42+00:00","display_standard":{"editorial_review":"submitter_supplied","formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":9572,"public_status":{"evidence":{"hasFormalization":false,"hasInformalProof":false,"hasRecordedWork":false,"solutionRecord":null},"lean":{... | [
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"created_at": "2026-09-08T03:32:15.935Z",
"updated_at": "2026-09-08T03:32:15.935Z",
"published_at": "2026-09-08T03:32:58.502Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"afb683f7-54ef-4986-951a-64ae81849701","format_issues":null,"number":5196} |
1.0.0 | problem:theoremdb-p9574 | problem | theoremdb-p9574 | Infinitely many integers with uniformly bounded shifted divisor sums | {
"main": "math",
"sub": "number theory",
"slug": "number-theory",
"name": "Number theory",
"original_discipline": "mathematics"
} | [] | [
{
"code": "11",
"basis": "community-map",
"evidence": "Inferred from community field: number-theory"
}
] | Let $\tau(m)$ denote the number of positive divisors of a positive integer $m$. Determine whether there exist infinitely many positive integers $n$ such that there exists a real constant $C > 0$ for which $\tau(n + k) \leq C \cdot k$ holds for every integer $k \geq 1$. | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P9574",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P9574",
"citation": null,
"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_826 : answer(sorry) ↔
∃ C > (0 : ℝ), { n | ∀ k ≥ 1, σ 0 (n + k) ≤ C * k }.Infinite | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem erdos_826 : answer(sorry) ↔\n ∃ C > (0 : ℝ), { n | ∀ k ≥ 1, σ 0 (n + k) ≤ C * k }.Infinite",
"target": null,
"source_url": "https://theoremdb.org/statements/P9574",
"format": "declaration-or-fragment",
"sha256": "61342b17cd87a7e399e5294f9a69f39cbd9... | [] | TheoremDB P9574 · Candidate awaiting source review | null | Source-reported assessment, retrieved 8 September 2026. Frontier Science has not independently reviewed this status.
Open. No signed Lean verification is attached to this problem or its selected resolution.
This record is in the source backlog; its novelty and unresolved status have not passed source review. | null | published | incremental | hard | [
"source-candidate",
"theoremdb"
] | null | null | null | null | [
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"url": "https://theoremdb.org/statements/P9574",
"not... | {
"source_record": "P9574",
"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":"2026-08-17T19:55:46+00:00","display_standard":{"editorial_review":"submitter_supplied","formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":9574,"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": "2026-09-08T03:33:00.930Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"88a3fff1-fffd-48c5-a953-59d7ef1046d1","format_issues":null,"number":5621} |
1.0.0 | problem:theoremdb-p9576 | problem | theoremdb-p9576 | Irrationality of the Prime-Weighted Binary 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"
}
] | Let $p_n$ denote the $n$-th prime number, with $p_1 = 2, p_2 = 3, p_3 = 5, \ldots$. Consider the infinite series
$$
S = \sum_{n=1}^{\infty} \frac{p_n}{2^n}.
$$
Determine whether $S$ is irrational. | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P9576",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P9576",
"citation": null,
"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_251 : answer(sorry) ↔ Irrational (∑' n : ℕ, (Nat.nth Nat.Prime n) / (2 ^ n)) | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem erdos_251 : answer(sorry) ↔ Irrational (∑' n : ℕ, (Nat.nth Nat.Prime n) / (2 ^ n))",
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"format": "declaration-or-fragment",
"sha256": "1f9da02dc570ecfb62c17ed19498acaa039c39762e4b63... | [] | TheoremDB P9576 · Candidate awaiting source review | null | Source-reported assessment, retrieved 8 September 2026. Frontier Science has not independently reviewed this status.
Open. No signed Lean verification is attached to this problem or its selected resolution.
This record is in the source backlog; its novelty and unresolved status have not passed source review. | null | published | incremental | hard | [
"source-candidate",
"theoremdb"
] | null | null | null | null | [
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"citation": "Source formal statement (not locally verified)",
"url": "https://theoremdb.org/statements/P9576",
"not... | {
"source_record": "P9576",
"imported_at": "2026-09-08",
"status_as_of": "2026-09-08",
"evidence_label": "Source candidate · not reviewed",
"note": "Full public directory statement. Source lane: backlog; source resolution: open. Source verification claims are attributed only; no local Lean verification has be... | {"created_at":"2026-08-17T19:55:50+00:00","display_standard":{"editorial_review":"submitter_supplied","formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":9576,"public_status":{"evidence":{"hasFormalization":false,"hasInformalProof":false,"hasRecordedWork":false,"solutionRecord":null},"lean":{... | [
"formalization-contains-placeholder",
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"current-status-unassessed",
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"created_at": "2026-09-08T03:32:15.935Z",
"updated_at": "2026-09-08T03:32:15.935Z",
"published_at": "2026-09-08T03:33:01.803Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"340e3971-d654-4bc0-9b7e-d074b243783b","format_issues":null,"number":5738} |
1.0.0 | problem:theoremdb-p9578 | problem | theoremdb-p9578 | First Countability of Homogeneous Countably Monolithic Compact Hausdorff Spaces | {
"main": "math",
"sub": "geometry and topology",
"slug": "geometry-and-topology",
"name": "Geometry and topology",
"original_discipline": "mathematics"
} | [] | [] | A topological space $X$ is called *homogeneous* if for all $x, y \in X$ there exists a homeomorphism $f : X \to X$ with $f(x) = y$. A topological space $X$ is called *countably monolithic* (or $\omega$-monolithic) if the closure of every countable subspace is metrizable. A topological space satisfies the *first countab... | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P9578",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P9578",
"citation": null,
"doi": null,
"arxiv_id": null,
"license": "Selected packets label their content CC0-1.0; original references retain their own terms",
"notes": "Selected problem... | theorem firstCountableTopology_of_countablyMonolithicSpace :
answer(sorry) ↔ ∀ (X : Type) (_ : TopologicalSpace X), T2Space X → CompactSpace X →
HomogeneousSpace X → CountablyMonolithicSpace X → FirstCountableTopology X | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem firstCountableTopology_of_countablyMonolithicSpace :\n answer(sorry) ↔ ∀ (X : Type) (_ : TopologicalSpace X), T2Space X → CompactSpace X →\n HomogeneousSpace X → CountablyMonolithicSpace X → FirstCountableTopology X",
"target": null,
"source_url": "ht... | [] | TheoremDB P9578 · Candidate awaiting source review | null | Source-reported assessment, retrieved 8 September 2026. Frontier Science has not independently reviewed this status.
Open. No signed Lean verification is attached to this problem or its selected resolution.
This record is in the source backlog; its novelty and unresolved status have not passed source review. | null | published | incremental | hard | [
"source-candidate",
"theoremdb"
] | null | null | null | null | [
{
"kind": "website",
"citation": "TheoremDB P9578",
"url": "https://theoremdb.org/statements/P9578",
"note": null,
"sort_order": 0
},
{
"kind": "formalization",
"citation": "Source formal statement (not locally verified)",
"url": "https://theoremdb.org/statements/P9578",
"not... | {
"source_record": "P9578",
"imported_at": "2026-09-08",
"status_as_of": "2026-09-08",
"evidence_label": "Source candidate · not reviewed",
"note": "Full public directory statement. Source lane: backlog; source resolution: open. Source verification claims are attributed only; no local Lean verification has be... | {"created_at":"2026-08-17T19:55:54+00:00","display_standard":{"editorial_review":"submitter_supplied","formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":9578,"public_status":{"evidence":{"hasFormalization":false,"hasInformalProof":false,"hasRecordedWork":false,"solutionRecord":null},"lean":{... | [
"formalization-contains-placeholder",
"formalization-not-compiled-during-export",
"current-status-unassessed",
"source-full-text-not-present"
] | [] | {
"created_at": "2026-09-08T03:32:15.935Z",
"updated_at": "2026-09-08T03:32:15.935Z",
"published_at": "2026-09-08T03:32:59.150Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"b128dcd9-8496-4704-b466-781730ca34df","format_issues":null,"number":5333} |
1.0.0 | problem:theoremdb-p9580 | problem | theoremdb-p9580 | The Černý Conjecture on Synchronizing Words | {
"main": "computer science",
"sub": "algorithms",
"slug": "algorithms",
"name": "Algorithms and data structures",
"original_discipline": "computer-science"
} | [] | [
{
"code": "68W",
"basis": "community-map",
"evidence": "Inferred from community field: algorithms"
}
] | A deterministic finite automaton (DFA) is a tuple $M = (Q, \Sigma, \delta)$ where $Q$ is a finite set of states, $\Sigma$ is an alphabet, and $\delta : Q \times \Sigma \to Q$ is a transition function. A synchronizing word (or reset word) for a DFA is a word $w \in \Sigma^*$ such that there exists a single state $p \in ... | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P9580",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P9580",
"citation": null,
"doi": null,
"arxiv_id": null,
"license": "Selected packets label their content CC0-1.0; original references retain their own terms",
"notes": "Selected problem... | theorem cerny_conjecture :
answer(sorry) ↔ ∀ {α : Type*} {σ : Type*} [Fintype σ] (M : DFA α σ) (hM : M.IsSynchronizing) ,
∃ w : List α, M.IsSynchronizingWord w ∧ w.length ≤ (Fintype.card σ - 1)^2 | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem cerny_conjecture :\n answer(sorry) ↔ ∀ {α : Type*} {σ : Type*} [Fintype σ] (M : DFA α σ) (hM : M.IsSynchronizing) ,\n ∃ w : List α, M.IsSynchronizingWord w ∧ w.length ≤ (Fintype.card σ - 1)^2",
"target": null,
"source_url": "https://theoremdb.org/statem... | [] | TheoremDB P9580 · Candidate awaiting source review | null | Source-reported assessment, retrieved 8 September 2026. Frontier Science has not independently reviewed this status.
Open. No signed Lean verification is attached to this problem or its selected resolution.
This record is in the source backlog; its novelty and unresolved status have not passed source review. | null | published | incremental | hard | [
"source-candidate",
"theoremdb"
] | null | null | null | null | [
{
"kind": "website",
"citation": "TheoremDB P9580",
"url": "https://theoremdb.org/statements/P9580",
"note": null,
"sort_order": 0
},
{
"kind": "formalization",
"citation": "Source formal statement (not locally verified)",
"url": "https://theoremdb.org/statements/P9580",
"not... | {
"source_record": "P9580",
"imported_at": "2026-09-08",
"status_as_of": "2026-09-08",
"evidence_label": "Source candidate · not reviewed",
"note": "Full public directory statement. Source lane: backlog; source resolution: open. Source verification claims are attributed only; no local Lean verification has be... | {"created_at":"2026-08-17T19:55:57+00:00","display_standard":{"editorial_review":"submitter_supplied","formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":9580,"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": "2026-09-08T03:33:06.871Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"4db09494-64f9-449d-b03f-49674dbd25d1","format_issues":null,"number":6835} |
1.0.0 | problem:theoremdb-p9582 | problem | theoremdb-p9582 | Rudin's conjecture on squares in arithmetic progressions | {
"main": "math",
"sub": "number theory",
"slug": "number-theory",
"name": "Number theory",
"original_discipline": "mathematics"
} | [] | [
{
"code": "11",
"basis": "community-map",
"evidence": "Inferred from community field: number-theory"
}
] | For a positive integer $N$ and natural numbers $q, a$, let $Q(N; q, a)$ denote the number of perfect squares among the first $N$ terms of the arithmetic progression $\{qn + a : 0 \le n < N\}$. A pair $(q, a)$ is called non-trivial if $q \ge 1$, $a \ge 1$, $\gcd(q, a) = 1$, and $(q, a) \neq (1, 1)$. Define $Q(N) = \max ... | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P9582",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P9582",
"citation": null,
"doi": null,
"arxiv_id": null,
"license": "Selected packets label their content CC0-1.0; original references retain their own terms",
"notes": "Selected problem... | theorem rudins_conjecture :
(fun N : ℕ => (Qmax N : ℝ)) =O[atTop] fun N : ℕ => Real.sqrt N | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem rudins_conjecture :\n (fun N : ℕ => (Qmax N : ℝ)) =O[atTop] fun N : ℕ => Real.sqrt N",
"target": null,
"source_url": "https://theoremdb.org/statements/P9582",
"format": "declaration-or-fragment",
"sha256": "9a6bb7b377dbc0d6f4cc64779e08e724da050938f... | [] | TheoremDB P9582 · Candidate awaiting source review | null | Source-reported assessment, retrieved 8 September 2026. Frontier Science has not independently reviewed this status.
Open. No signed Lean verification is attached to this problem or its selected resolution.
This record is in the source backlog; its novelty and unresolved status have not passed source review. | null | published | incremental | hard | [
"source-candidate",
"theoremdb"
] | null | null | null | null | [
{
"kind": "website",
"citation": "TheoremDB P9582",
"url": "https://theoremdb.org/statements/P9582",
"note": null,
"sort_order": 0
},
{
"kind": "formalization",
"citation": "Source formal statement (not locally verified)",
"url": "https://theoremdb.org/statements/P9582",
"not... | {
"source_record": "P9582",
"imported_at": "2026-09-08",
"status_as_of": "2026-09-08",
"evidence_label": "Source candidate · not reviewed",
"note": "Full public directory statement. Source lane: backlog; source resolution: open. Source verification claims are attributed only; no local Lean verification has be... | {"created_at":"2026-08-17T19:56:02+00:00","display_standard":{"editorial_review":"submitter_supplied","formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":9582,"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": "2026-09-08T03:33:05.686Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"82ae92c7-fb55-483f-9e0e-3048708b9dce","format_issues":null,"number":6581} |
1.0.0 | problem:theoremdb-p9584 | problem | theoremdb-p9584 | Improved lower bound for the largest admissible cardinality in \(\mathbb{F}_7^n\) | {
"main": "math",
"sub": "combinatorics",
"slug": "combinatorics",
"name": "Combinatorics",
"original_discipline": "mathematics"
} | [] | [
{
"code": "05",
"basis": "community-map",
"evidence": "Inferred from community field: combinatorics"
}
] | For each positive integer $n$, let $\mathbb{F}_7^n$ denote the $n$-dimensional vector space over the field $\mathbb{Z}/7\mathbb{Z}$ of integers modulo 7. A finite subset $A \subseteq \mathbb{F}_7^n$ is called admissible if whenever $a - a' \in \{-1, 0, 1\}^n$ for some $a, a' \in A$, we have $a = a'$; equivalently, the ... | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P9584",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P9584",
"citation": null,
"doi": null,
"arxiv_id": null,
"license": "Selected packets label their content CC0-1.0; original references retain their own terms",
"notes": "Selected problem... | theorem green_38.lower :
let ans := (answer(sorry) : ℕ → ℝ)
ans ≤ᶠ[atTop] LargestAdmissibleCardinality ∧
∃ c > C₁, (fun n ↦ c ^ n) =O[atTop] ans | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem green_38.lower :\n let ans := (answer(sorry) : ℕ → ℝ)\n ans ≤ᶠ[atTop] LargestAdmissibleCardinality ∧\n ∃ c > C₁, (fun n ↦ c ^ n) =O[atTop] ans",
"target": null,
"source_url": "https://theoremdb.org/statements/P9584",
"format": "declaration-or-fra... | [] | TheoremDB P9584 · Candidate awaiting source review | null | Source-reported assessment, retrieved 8 September 2026. Frontier Science has not independently reviewed this status.
Open. No signed Lean verification is attached to this problem or its selected resolution.
This record is in the source backlog; its novelty and unresolved status have not passed source review. | null | published | incremental | hard | [
"source-candidate",
"theoremdb"
] | null | null | null | null | [
{
"kind": "website",
"citation": "TheoremDB P9584",
"url": "https://theoremdb.org/statements/P9584",
"note": null,
"sort_order": 0
},
{
"kind": "formalization",
"citation": "Source formal statement (not locally verified)",
"url": "https://theoremdb.org/statements/P9584",
"not... | {
"source_record": "P9584",
"imported_at": "2026-09-08",
"status_as_of": "2026-09-08",
"evidence_label": "Source candidate · not reviewed",
"note": "Full public directory statement. Source lane: backlog; source resolution: open. Source verification claims are attributed only; no local Lean verification has be... | {"created_at":"2026-08-17T19:56:05+00:00","display_standard":{"editorial_review":"submitter_supplied","formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":9584,"public_status":{"evidence":{"hasFormalization":false,"hasInformalProof":false,"hasRecordedWork":false,"solutionRecord":null},"lean":{... | [
"formalization-contains-placeholder",
"formalization-not-compiled-during-export",
"current-status-unassessed",
"source-full-text-not-present"
] | [] | {
"created_at": "2026-09-08T03:32:15.935Z",
"updated_at": "2026-09-08T03:32:15.935Z",
"published_at": "2026-09-08T03:32:59.656Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"6b0d0078-603f-41d8-b90c-0990b43b6e63","format_issues":null,"number":5442} |
1.0.0 | problem:theoremdb-p9586 | problem | theoremdb-p9586 | Non-existence of monochromatic quantum graphs with fourteen vertices and three colours over the complex numbers | {
"main": "math",
"sub": "combinatorics",
"slug": "combinatorics",
"name": "Combinatorics",
"original_discipline": "mathematics"
} | [] | [
{
"code": "05",
"basis": "community-map",
"evidence": "Inferred from community field: combinatorics"
}
] | Let $N$ and $D$ be positive integers. A quantum graph with $N$ vertices and $D$ colours over a coefficient domain $\alpha$ is specified by a weight function $W$ that assigns a value in $\alpha$ to each edge of the complete graph $K_N$ with a colour from $\{1, \dots, D\}$. For each vertex colouring $\iota : \{1, \dots, ... | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P9586",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P9586",
"citation": null,
"doi": null,
"arxiv_id": null,
"license": "Selected packets label their content CC0-1.0; original references retain their own terms",
"notes": "Selected problem... | theorem eqSystem14_no_solution_d3 :
answer(sorry) ↔
¬ ∃ W : WeightsN 14 3 ℂ, EqSystemN 14 3 W | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem eqSystem14_no_solution_d3 :\n answer(sorry) ↔\n ¬ ∃ W : WeightsN 14 3 ℂ, EqSystemN 14 3 W",
"target": null,
"source_url": "https://theoremdb.org/statements/P9586",
"format": "declaration-or-fragment",
"sha256": "0adc66d529d687f8fabf0216a6393b8... | [] | TheoremDB P9586 · Candidate awaiting source review | null | Source-reported assessment, retrieved 8 September 2026. Frontier Science has not independently reviewed this status.
Open. No signed Lean verification is attached to this problem or its selected resolution.
This record is in the source backlog; its novelty and unresolved status have not passed source review. | null | published | incremental | hard | [
"source-candidate",
"theoremdb"
] | null | null | null | null | [
{
"kind": "website",
"citation": "TheoremDB P9586",
"url": "https://theoremdb.org/statements/P9586",
"note": null,
"sort_order": 0
},
{
"kind": "formalization",
"citation": "Source formal statement (not locally verified)",
"url": "https://theoremdb.org/statements/P9586",
"not... | {
"source_record": "P9586",
"imported_at": "2026-09-08",
"status_as_of": "2026-09-08",
"evidence_label": "Source candidate · not reviewed",
"note": "Full public directory statement. Source lane: backlog; source resolution: open. Source verification claims are attributed only; no local Lean verification has be... | {"created_at":"2026-08-17T19:56:09+00:00","display_standard":{"editorial_review":"submitter_supplied","formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":9586,"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": "2026-09-08T03:33:04.315Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"86be117f-6662-47a0-8d20-33f7f0757457","format_issues":null,"number":6285} |
1.0.0 | problem:theoremdb-p9588 | problem | theoremdb-p9588 | Infinitude of cousin primes | {
"main": "math",
"sub": "number theory",
"slug": "number-theory",
"name": "Number theory",
"original_discipline": "mathematics"
} | [] | [
{
"code": "11",
"basis": "community-map",
"evidence": "Inferred from community field: number-theory"
}
] | A pair of cousin primes is a pair of prime numbers that differ by exactly 4. Determine whether there are infinitely many primes $p$ such that $p + 4$ is also prime. | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P9588",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P9588",
"citation": null,
"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_cousin_primes :
Infinite {p : ℕ | Prime p ∧ Prime (p + 4)} | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem infinite_cousin_primes :\n Infinite {p : ℕ | Prime p ∧ Prime (p + 4)}",
"target": null,
"source_url": "https://theoremdb.org/statements/P9588",
"format": "declaration-or-fragment",
"sha256": "f366811457b6cab8e3cc440681fed964f19a4675245ee13466935977... | [] | TheoremDB P9588 · Candidate awaiting source review | null | Source-reported assessment, retrieved 8 September 2026. Frontier Science has not independently reviewed this status.
Open. No signed Lean verification is attached to this problem or its selected resolution.
This record is in 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 P9588",
"url": "https://theoremdb.org/statements/P9588",
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"url": "https://theoremdb.org/statements/P9588",
"not... | {
"source_record": "P9588",
"imported_at": "2026-09-08",
"status_as_of": "2026-09-08",
"evidence_label": "Source candidate · not reviewed",
"note": "Full public directory statement. Source lane: backlog; source resolution: open. Source verification claims are attributed only; no local Lean verification has be... | {"created_at":"2026-08-17T19:56:13+00:00","display_standard":{"editorial_review":"submitter_supplied","formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":9588,"public_status":{"evidence":{"hasFormalization":false,"hasInformalProof":false,"hasRecordedWork":false,"solutionRecord":null},"lean":{... | [
"formalization-not-compiled-during-export",
"current-status-unassessed",
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] | [] | {
"created_at": "2026-09-08T03:32:15.935Z",
"updated_at": "2026-09-08T03:32:15.935Z",
"published_at": "2026-09-08T03:33:01.146Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"3fcdfef8-848f-4206-9a25-bd09247810a9","format_issues":null,"number":5667} |
1.0.0 | problem:theoremdb-p9590 | problem | theoremdb-p9590 | Infinitely many squarefree values of $n^4 + 2$ | {
"main": "math",
"sub": "number theory",
"slug": "number-theory",
"name": "Number theory",
"original_discipline": "mathematics"
} | [] | [
{
"code": "11",
"basis": "community-map",
"evidence": "Inferred from community field: number-theory"
}
] | A positive integer $m$ is called squarefree if it is not divisible by the square of any prime number; equivalently, in the prime factorization $m = p_1^{e_1} \cdots p_r^{e_r}$, each exponent $e_i$ equals $1$. Determine whether the set $\{n \in \mathbb{N} : n^4 + 2 \text{ is squarefree}\}$ is infinite. | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P9590",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P9590",
"citation": null,
"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.iii : answer(sorry) ↔ {n : ℕ | Squarefree (n ^ 4 + 2)}.Infinite | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem erdos_978.parts.iii : answer(sorry) ↔ {n : ℕ | Squarefree (n ^ 4 + 2)}.Infinite",
"target": null,
"source_url": "https://theoremdb.org/statements/P9590",
"format": "declaration-or-fragment",
"sha256": "d4502412107a26b8ed175ac34a7aefab6ac4019121801f148... | [] | TheoremDB P9590 · Candidate awaiting source review | null | Source-reported assessment, retrieved 8 September 2026. Frontier Science has not independently reviewed this status.
Open. No signed Lean verification is attached to this problem or its selected resolution.
This record is in the source backlog; its novelty and unresolved status have not passed source review. | null | published | incremental | hard | [
"source-candidate",
"theoremdb"
] | null | null | null | null | [
{
"kind": "website",
"citation": "TheoremDB P9590",
"url": "https://theoremdb.org/statements/P9590",
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},
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"citation": "Source formal statement (not locally verified)",
"url": "https://theoremdb.org/statements/P9590",
"not... | {
"source_record": "P9590",
"imported_at": "2026-09-08",
"status_as_of": "2026-09-08",
"evidence_label": "Source candidate · not reviewed",
"note": "Full public directory statement. Source lane: backlog; source resolution: open. Source verification claims are attributed only; no local Lean verification has be... | {"created_at":"2026-08-17T19:56:18+00:00","display_standard":{"editorial_review":"submitter_supplied","formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":9590,"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": "2026-09-08T03:33:01.059Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"9fe9072e-7a3a-4781-9074-eef508164966","format_issues":null,"number":5648} |
1.0.0 | problem:theoremdb-p9592 | problem | theoremdb-p9592 | Asymptotic sum of maximal factorial quotient remainders | {
"main": "math",
"sub": "number theory",
"slug": "number-theory",
"name": "Number theory",
"original_discipline": "mathematics"
} | [] | [
{
"code": "11",
"basis": "community-map",
"evidence": "Inferred from community field: number-theory"
}
] | For any integer $k \geq 2$ and any positive integer $n$, define $g_k(n)$ to be the maximum value of $(a_1 + \cdots + a_k) - n$, where $a_1, \ldots, a_k$ are nonnegative integers such that $a_1! \cdots a_k!$ divides $n!$. Determine whether there exists, for every $k \geq 2$, a real constant $c$ such that
$$
\sum_{n \l... | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P9592",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P9592",
"citation": null,
"doi": null,
"arxiv_id": null,
"license": "Selected packets label their content CC0-1.0; original references retain their own terms",
"notes": "Selected problem... | theorem erdos_400.parts.i :
answer(sorry) ↔ ∀ᵉ (k ≥ 2), ∃ c : ℝ,
(fun x : ℕ ↦ (∑ n ∈ Icc 1 x, (g k n : ℝ))) ~[atTop]
(fun x : ℕ ↦ c * x * Real.log x) | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem erdos_400.parts.i :\n answer(sorry) ↔ ∀ᵉ (k ≥ 2), ∃ c : ℝ,\n (fun x : ℕ ↦ (∑ n ∈ Icc 1 x, (g k n : ℝ))) ~[atTop]\n (fun x : ℕ ↦ c * x * Real.log x)",
"target": null,
"source_url": "https://theoremdb.org/statements/P9592",
"format": "declarati... | [] | TheoremDB P9592 · Candidate awaiting source review | null | Source-reported assessment, retrieved 8 September 2026. Frontier Science has not independently reviewed this status.
Open. No signed Lean verification is attached to this problem or its selected resolution.
This record is in the source backlog; its novelty and unresolved status have not passed source review. | null | published | incremental | hard | [
"source-candidate",
"theoremdb"
] | null | null | null | null | [
{
"kind": "website",
"citation": "TheoremDB P9592",
"url": "https://theoremdb.org/statements/P9592",
"note": null,
"sort_order": 0
},
{
"kind": "formalization",
"citation": "Source formal statement (not locally verified)",
"url": "https://theoremdb.org/statements/P9592",
"not... | {
"source_record": "P9592",
"imported_at": "2026-09-08",
"status_as_of": "2026-09-08",
"evidence_label": "Source candidate · not reviewed",
"note": "Full public directory statement. Source lane: backlog; source resolution: open. Source verification claims are attributed only; no local Lean verification has be... | {"created_at":"2026-08-17T19:56:22+00:00","display_standard":{"editorial_review":"submitter_supplied","formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":9592,"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": "2026-09-08T03:32:50.401Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"d2449f40-2f44-4734-8497-263dd606e2f8","format_issues":null,"number":3412} |
1.0.0 | problem:theoremdb-p9594 | problem | theoremdb-p9594 | Asymptotic characterization of the minimal collectively coprime base function | {
"main": "math",
"sub": "number theory",
"slug": "number-theory",
"name": "Number theory",
"original_discipline": "mathematics"
} | [] | [
{
"code": "11",
"basis": "community-map",
"evidence": "Inferred from community field: number-theory"
}
] | For a positive integer $n$, define $h(n)$ to be the smallest integer $m > 2$ such that the numbers $2^n - 1, 3^n - 1, \dots, m^n - 1$ are collectively coprime (that is, their greatest common divisor equals $1$). For a real number $\varepsilon > 0$, say that a property holds for eventually all positive integers $n$ if t... | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P9594",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P9594",
"citation": null,
"doi": null,
"arxiv_id": null,
"license": "Selected packets label their content CC0-1.0; original references retain their own terms",
"notes": "Selected problem... | theorem erdos_770.parts.iii : answer(sorry) ↔ ∀ ε > 0, ∀ᶠ n in atTop,
let p := sSup {m : ℕ | m.Prime ∧ m - 1 ∣ n}
p > (n : ℝ) ^ (ε : ℝ) → h n = p | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem erdos_770.parts.iii : answer(sorry) ↔ ∀ ε > 0, ∀ᶠ n in atTop,\n let p := sSup {m : ℕ | m.Prime ∧ m - 1 ∣ n}\n p > (n : ℝ) ^ (ε : ℝ) → h n = p",
"target": null,
"source_url": "https://theoremdb.org/statements/P9594",
"format": "declaration-or-fragmen... | [] | TheoremDB P9594 · Candidate awaiting source review | null | Source-reported assessment, retrieved 8 September 2026. Frontier Science has not independently reviewed this status.
Open. No signed Lean verification is attached to this problem or its selected resolution.
This record is in the source backlog; its novelty and unresolved status have not passed source review. | null | published | incremental | hard | [
"source-candidate",
"theoremdb"
] | null | null | null | null | [
{
"kind": "website",
"citation": "TheoremDB P9594",
"url": "https://theoremdb.org/statements/P9594",
"note": null,
"sort_order": 0
},
{
"kind": "formalization",
"citation": "Source formal statement (not locally verified)",
"url": "https://theoremdb.org/statements/P9594",
"not... | {
"source_record": "P9594",
"imported_at": "2026-09-08",
"status_as_of": "2026-09-08",
"evidence_label": "Source candidate · not reviewed",
"note": "Full public directory statement. Source lane: backlog; source resolution: open. Source verification claims are attributed only; no local Lean verification has be... | {"created_at":"2026-08-17T19:56:27+00:00","display_standard":{"editorial_review":"submitter_supplied","formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":9594,"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": "2026-09-08T03:32:49.943Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"23f9aa3a-a327-44b5-9fb3-68feff1d271a","format_issues":null,"number":3318} |
1.0.0 | problem:theoremdb-p9596 | problem | theoremdb-p9596 | Existence of 11 Mutually Orthogonal Latin Squares of Order 12 | {
"main": "math",
"sub": "combinatorics",
"slug": "combinatorics",
"name": "Combinatorics",
"original_discipline": "mathematics"
} | [] | [
{
"code": "05",
"basis": "community-map",
"evidence": "Inferred from community field: combinatorics"
}
] | A set of latin squares of order $n$ is called mutually orthogonal if, upon superimposing any two distinct squares in the set, each ordered pair of symbols occurs at most once. A complete set of mutually orthogonal latin squares of order $n$ consists of $n - 1$ latin squares of order $n$ that are pairwise mutually ortho... | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P9596",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P9596",
"citation": null,
"doi": null,
"arxiv_id": null,
"license": "Selected packets label their content CC0-1.0; original references retain their own terms",
"notes": "Selected problem... | theorem molsOrder12 : answer(sorry) ↔ HasCompleteMOLS 12 | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem molsOrder12 : answer(sorry) ↔ HasCompleteMOLS 12",
"target": null,
"source_url": "https://theoremdb.org/statements/P9596",
"format": "declaration-or-fragment",
"sha256": "5224d52b27484ee2abd6a9d26313c71a1e38bca8b10fdb0d06567c642a3683e1",
"license"... | [] | TheoremDB P9596 · Candidate awaiting source review | null | Source-reported assessment, retrieved 8 September 2026. Frontier Science has not independently reviewed this status.
Open. No signed Lean verification is attached to this problem or its selected resolution.
This record is in the source backlog; its novelty and unresolved status have not passed source review. | null | published | incremental | hard | [
"source-candidate",
"theoremdb"
] | null | null | null | null | [
{
"kind": "website",
"citation": "TheoremDB P9596",
"url": "https://theoremdb.org/statements/P9596",
"note": null,
"sort_order": 0
},
{
"kind": "formalization",
"citation": "Source formal statement (not locally verified)",
"url": "https://theoremdb.org/statements/P9596",
"not... | {
"source_record": "P9596",
"imported_at": "2026-09-08",
"status_as_of": "2026-09-08",
"evidence_label": "Source candidate · not reviewed",
"note": "Full public directory statement. Source lane: backlog; source resolution: open. Source verification claims are attributed only; no local Lean verification has be... | {"created_at":"2026-08-17T19:56:31+00:00","display_standard":{"editorial_review":"submitter_supplied","formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":9596,"public_status":{"evidence":{"hasFormalization":false,"hasInformalProof":false,"hasRecordedWork":false,"solutionRecord":null},"lean":{... | [
"formalization-contains-placeholder",
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] | [] | {
"created_at": "2026-09-08T03:32:15.935Z",
"updated_at": "2026-09-08T03:32:15.935Z",
"published_at": "2026-09-08T03:32:58.391Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"1ed821e2-0685-4e0f-b812-501df57d4cb9","format_issues":null,"number":5173} |
1.0.0 | problem:theoremdb-p9598 | problem | theoremdb-p9598 | Lehmer's Conjecture on the Totient Divisibility 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"
}
] | For every integer $n > 1$, determine whether the following equivalence holds: $\phi(n)$ divides $n - 1$ if and only if $n$ is prime. | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P9598",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P9598",
"citation": null,
"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_828.variants.lehmer_conjecture : answer(sorry) ↔ ∀ n > 1, φ n ∣ n - 1 ↔ Prime n | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem erdos_828.variants.lehmer_conjecture : answer(sorry) ↔ ∀ n > 1, φ n ∣ n - 1 ↔ Prime n",
"target": null,
"source_url": "https://theoremdb.org/statements/P9598",
"format": "declaration-or-fragment",
"sha256": "797fa5d040e97ea0742b7d10bb9af192b2aabf9ef7a... | [] | TheoremDB P9598 · Candidate awaiting source review | null | Source-reported assessment, retrieved 8 September 2026. Frontier Science has not independently reviewed this status.
Open. No signed Lean verification is attached to this problem or its selected resolution.
This record is in the source backlog; its novelty and unresolved status have not passed source review. | null | published | incremental | hard | [
"source-candidate",
"theoremdb"
] | null | null | null | null | [
{
"kind": "website",
"citation": "TheoremDB P9598",
"url": "https://theoremdb.org/statements/P9598",
"note": null,
"sort_order": 0
},
{
"kind": "formalization",
"citation": "Source formal statement (not locally verified)",
"url": "https://theoremdb.org/statements/P9598",
"not... | {
"source_record": "P9598",
"imported_at": "2026-09-08",
"status_as_of": "2026-09-08",
"evidence_label": "Source candidate · not reviewed",
"note": "Full public directory statement. Source lane: backlog; source resolution: open. Source verification claims are attributed only; no local Lean verification has be... | {"created_at":"2026-08-17T19:56:35+00:00","display_standard":{"editorial_review":"submitter_supplied","formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":9598,"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": "2026-09-08T03:33:03.156Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"bbca1ebe-880b-4f49-8ec9-a2f5acf597b8","format_issues":null,"number":6035} |
1.0.0 | problem:theoremdb-p9600 | problem | theoremdb-p9600 | Heath-Brown's conjecture on the sum of squared prime gaps | {
"main": "math",
"sub": "number theory",
"slug": "number-theory",
"name": "Number theory",
"original_discipline": "mathematics"
} | [] | [
{
"code": "11",
"basis": "community-map",
"evidence": "Inferred from community field: number-theory"
}
] | Let $p_n$ denote the $n$-th prime number, and define the $n$-th prime gap by $g_n = p_{n+1} - p_n$. For a positive integer $N$, let $S(N) = \sum_{n=0}^{N-1} g_n^2$. Determine whether $S(N) = O\bigl(N(\log N)^2\bigr)$ as $N \to \infty$, or equivalently, whether there exist positive constants $C$ and $N_0$ such that $S(N... | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P9600",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P9600",
"citation": null,
"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_233 :
(fun N => ((∑ n ∈ Finset.range N, (primeGap n) ^ 2) : ℝ)) =O[atTop] fun N => N * (log N)^2 | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem erdos_233 :\n (fun N => ((∑ n ∈ Finset.range N, (primeGap n) ^ 2) : ℝ)) =O[atTop] fun N => N * (log N)^2",
"target": null,
"source_url": "https://theoremdb.org/statements/P9600",
"format": "declaration-or-fragment",
"sha256": "899226f614ce0f6c82d79... | [] | TheoremDB P9600 · Candidate awaiting source review | null | Source-reported assessment, retrieved 8 September 2026. Frontier Science has not independently reviewed this status.
Open. No signed Lean verification is attached to this problem or its selected resolution.
This record is in the source backlog; its novelty and unresolved status have not passed source review. | null | published | incremental | hard | [
"source-candidate",
"theoremdb"
] | null | null | null | null | [
{
"kind": "website",
"citation": "TheoremDB P9600",
"url": "https://theoremdb.org/statements/P9600",
"note": null,
"sort_order": 0
},
{
"kind": "formalization",
"citation": "Source formal statement (not locally verified)",
"url": "https://theoremdb.org/statements/P9600",
"not... | {
"source_record": "P9600",
"imported_at": "2026-09-08",
"status_as_of": "2026-09-08",
"evidence_label": "Source candidate · not reviewed",
"note": "Full public directory statement. Source lane: backlog; source resolution: open. Source verification claims are attributed only; no local Lean verification has be... | {"created_at":"2026-08-17T19:56:40+00:00","display_standard":{"editorial_review":"submitter_supplied","formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":9600,"public_status":{"evidence":{"hasFormalization":false,"hasInformalProof":false,"hasRecordedWork":false,"solutionRecord":null},"lean":{... | [
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"current-status-unassessed",
"source-full-text-not-present"
] | [] | {
"created_at": "2026-09-08T03:32:15.935Z",
"updated_at": "2026-09-08T03:32:15.935Z",
"published_at": "2026-09-08T03:32:59.598Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"ab07f4c4-37c5-4dd4-9128-1facf01ca8f5","format_issues":null,"number":5430} |
1.0.0 | problem:theoremdb-p9602 | problem | theoremdb-p9602 | Irrationality of the Riemann zeta value at 11 | {
"main": "math",
"sub": "number theory",
"slug": "number-theory",
"name": "Number theory",
"original_discipline": "mathematics"
} | [] | [
{
"code": "11",
"basis": "community-map",
"evidence": "Inferred from community field: number-theory"
}
] | Let $\zeta(s)$ denote the Riemann zeta function, defined for $\operatorname{Re}(s) > 1$ by $\zeta(s) = \sum_{n=1}^{\infty} n^{-s}$ and extended by analytic continuation to $\mathbb{C} \setminus \{1\}$. Prove that $\zeta(11)$ is irrational. | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P9602",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P9602",
"citation": null,
"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_eleven : ∃ x, Irrational x ∧ riemannZeta 11 = x | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem irrational_eleven : ∃ x, Irrational x ∧ riemannZeta 11 = x",
"target": null,
"source_url": "https://theoremdb.org/statements/P9602",
"format": "declaration-or-fragment",
"sha256": "cadff626c6392b129af45b66b928b2e87f1cf82d7621ac53e330248ed0a80a33",
... | [] | TheoremDB P9602 · Candidate awaiting source review | null | Source-reported assessment, retrieved 8 September 2026. Frontier Science has not independently reviewed this status.
Open. No signed Lean verification is attached to this problem or its selected resolution.
This record is in the source backlog; its novelty and unresolved status have not passed source review. | null | published | incremental | hard | [
"source-candidate",
"theoremdb"
] | null | null | null | null | [
{
"kind": "website",
"citation": "TheoremDB P9602",
"url": "https://theoremdb.org/statements/P9602",
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},
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"citation": "Source formal statement (not locally verified)",
"url": "https://theoremdb.org/statements/P9602",
"not... | {
"source_record": "P9602",
"imported_at": "2026-09-08",
"status_as_of": "2026-09-08",
"evidence_label": "Source candidate · not reviewed",
"note": "Full public directory statement. Source lane: backlog; source resolution: open. Source verification claims are attributed only; no local Lean verification has be... | {"created_at":"2026-08-17T19:56:45+00:00","display_standard":{"editorial_review":"submitter_supplied","formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":9602,"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": "2026-09-08T03:33:01.811Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"e58b178b-96bd-47a0-a3ad-2964c8f25f47","format_issues":null,"number":5740} |
1.0.0 | problem:theoremdb-p9604 | problem | theoremdb-p9604 | Lower bound for the van der Waerden number W(3,20) | {
"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 · P9604",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P9604",
"citation": null,
"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_20_lower : answer(sorry) ↔ W 3 20 ≥ 389 | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem W_3_20_lower : answer(sorry) ↔ W 3 20 ≥ 389",
"target": null,
"source_url": "https://theoremdb.org/statements/P9604",
"format": "declaration-or-fragment",
"sha256": "2a84f3bfceace0c69e3a897cd65c81912d16ac4e8c859cea90137022e2231b9d",
"license": "Se... | [] | TheoremDB P9604 · Candidate awaiting source review | null | Source-reported assessment, retrieved 8 September 2026. Frontier Science has not independently reviewed this status.
Open. No signed Lean verification is attached to this problem or its selected resolution.
This record is in the source backlog; its novelty and unresolved status have not passed source review. | null | published | incremental | hard | [
"source-candidate",
"theoremdb"
] | null | null | null | null | [
{
"kind": "website",
"citation": "TheoremDB P9604",
"url": "https://theoremdb.org/statements/P9604",
"note": null,
"sort_order": 0
},
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"url": "https://theoremdb.org/statements/P9604",
"not... | {
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"note": "Full public directory statement. Source lane: backlog; source resolution: open. Source verification claims are attributed only; no local Lean verification has be... | {"created_at":"2026-08-17T19:56:49+00:00","display_standard":{"formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":9604,"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": "2026-09-08T03:33:03.537Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"9364b583-5b02-4887-87ae-1e182fd3efcf","format_issues":null,"number":6116} |
1.0.0 | problem:theoremdb-p9606 | problem | theoremdb-p9606 | Infinitely many consecutive equal values of Euler's totient function | {
"main": "math",
"sub": "number theory",
"slug": "number-theory",
"name": "Number theory",
"original_discipline": "mathematics"
} | [] | [
{
"code": "11",
"basis": "community-map",
"evidence": "Inferred from community field: number-theory"
}
] | Let $\phi$ denote Euler's totient function, which counts the number of integers between $1$ and $n$ that are coprime to $n$. For each integer $k \geq 1$, let $S_k$ denote the set of natural numbers $n$ such that $\phi(n) = \phi(n+1) = \cdots = \phi(n+k)$, or equivalently, such that $\phi(n) = \phi(n+i)$ for all integer... | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P9606",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P9606",
"citation": null,
"doi": null,
"arxiv_id": null,
"license": "Selected packets label their content CC0-1.0; original references retain their own terms",
"notes": "Selected problem... | theorem erdos_1003.variants.Icc :
answer(sorry) ↔ ∀ k ≥ 1, {n | ∀ i ∈ Set.Icc 1 k, φ n = φ (n + i)}.Infinite | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem erdos_1003.variants.Icc :\n answer(sorry) ↔ ∀ k ≥ 1, {n | ∀ i ∈ Set.Icc 1 k, φ n = φ (n + i)}.Infinite",
"target": null,
"source_url": "https://theoremdb.org/statements/P9606",
"format": "declaration-or-fragment",
"sha256": "b4f7a635d12e94513edbd68... | [] | TheoremDB P9606 · Candidate awaiting source review | null | Source-reported assessment, retrieved 8 September 2026. Frontier Science has not independently reviewed this status.
Open. No signed Lean verification is attached to this problem or its selected resolution.
This record is in the source backlog; its novelty and unresolved status have not passed source review. | null | published | incremental | hard | [
"source-candidate",
"theoremdb"
] | null | null | null | null | [
{
"kind": "website",
"citation": "TheoremDB P9606",
"url": "https://theoremdb.org/statements/P9606",
"note": null,
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},
{
"kind": "formalization",
"citation": "Source formal statement (not locally verified)",
"url": "https://theoremdb.org/statements/P9606",
"not... | {
"source_record": "P9606",
"imported_at": "2026-09-08",
"status_as_of": "2026-09-08",
"evidence_label": "Source candidate · not reviewed",
"note": "Full public directory statement. Source lane: backlog; source resolution: open. Source verification claims are attributed only; no local Lean verification has be... | {"created_at":"2026-08-17T19:56:53+00:00","display_standard":{"formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":9606,"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": "2026-09-08T03:33:00.839Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"d48de390-d70e-4482-af19-b9f75f509b77","format_issues":null,"number":5601} |
1.0.0 | problem:theoremdb-p9608 | problem | theoremdb-p9608 | Carmichael's Totient Function 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 $\varphi$ denote Euler's totient function, which counts the number of integers from $1$ to $n$ that are coprime to $n$. For a natural number $n$, we say that $n$ satisfies the Carmichael totient property if there exists a natural number $m$ with $m \neq n$ such that $\varphi(m) = \varphi(n)$. Prove that for every p... | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P9608",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P9608",
"citation": null,
"doi": null,
"arxiv_id": null,
"license": "Selected packets label their content CC0-1.0; original references retain their own terms",
"notes": "Selected problem... | theorem charmichaelTotient :
∀ ⦃n : ℕ⦄, 0 < n → CarmichaelTotientFor n | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem charmichaelTotient :\n ∀ ⦃n : ℕ⦄, 0 < n → CarmichaelTotientFor n",
"target": null,
"source_url": "https://theoremdb.org/statements/P9608",
"format": "declaration-or-fragment",
"sha256": "4ce2666dd60ebec61c7c5e64629e3b987a786182e7501859d5fb4fc84a15d... | [] | TheoremDB P9608 · Candidate awaiting source review | null | Source-reported assessment, retrieved 8 September 2026. Frontier Science has not independently reviewed this status.
Open. No signed Lean verification is attached to this problem or its selected resolution.
This record is in the source backlog; its novelty and unresolved status have not passed source review. | null | published | incremental | hard | [
"source-candidate",
"theoremdb"
] | null | null | null | null | [
{
"kind": "website",
"citation": "TheoremDB P9608",
"url": "https://theoremdb.org/statements/P9608",
"note": null,
"sort_order": 0
},
{
"kind": "formalization",
"citation": "Source formal statement (not locally verified)",
"url": "https://theoremdb.org/statements/P9608",
"not... | {
"source_record": "P9608",
"imported_at": "2026-09-08",
"status_as_of": "2026-09-08",
"evidence_label": "Source candidate · not reviewed",
"note": "Full public directory statement. Source lane: backlog; source resolution: open. Source verification claims are attributed only; no local Lean verification has be... | {"created_at":"2026-08-17T19:56:57+00:00","display_standard":{"formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":9608,"public_status":{"evidence":{"hasFormalization":false,"hasInformalProof":false,"hasRecordedWork":false,"solutionRecord":null},"lean":{"kernelAccepted":false,"label":"","reaso... | [
"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": "2026-09-08T03:32:51.005Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"d970de9d-5efa-4974-a4de-76a91616e568","format_issues":null,"number":3540} |
1.0.0 | problem:theoremdb-p9610 | problem | theoremdb-p9610 | Existence of a sparse additive subbasis with non-convergent density | {
"main": "math",
"sub": "combinatorics",
"slug": "combinatorics",
"name": "Combinatorics",
"original_discipline": "mathematics"
} | [] | [
{
"code": "05",
"basis": "community-map",
"evidence": "Inferred from community field: combinatorics"
}
] | Let $A \subseteq \mathbb{N}$ be an additive basis of order 2. An additive basis of order $h$ is a set of natural numbers such that every sufficiently large natural number can be expressed as the sum of at most $h$ elements from the set. Must there exist a strictly increasing sequence $B = \{b_1 < b_2 < \dots\} \subsete... | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P9610",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P9610",
"citation": null,
"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_326 : answer(sorry) ↔ ∀ (A : Set ℕ), A.IsAddBasisOfOrder 2 →
∃ (b : ℕ → ℕ), StrictMono b ∧ ∀ n, b n ∈ A ∧ (Set.range b).IsAddBasis ∧
∀ (x : ℝ), ¬ Tendsto (fun n ↦ (b n : ℝ) / n ^ 2) atTop (𝓝 x) | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem erdos_326 : answer(sorry) ↔ ∀ (A : Set ℕ), A.IsAddBasisOfOrder 2 →\n ∃ (b : ℕ → ℕ), StrictMono b ∧ ∀ n, b n ∈ A ∧ (Set.range b).IsAddBasis ∧\n ∀ (x : ℝ), ¬ Tendsto (fun n ↦ (b n : ℝ) / n ^ 2) atTop (𝓝 x)",
"target": null,
"source_url": "https://theor... | [] | TheoremDB P9610 · Candidate awaiting source review | null | Source-reported assessment, retrieved 8 September 2026. Frontier Science has not independently reviewed this status.
Open. No signed Lean verification is attached to this problem or its selected resolution.
This record is in the source backlog; its novelty and unresolved status have not passed source review. | null | published | incremental | hard | [
"source-candidate",
"theoremdb"
] | null | null | null | null | [
{
"kind": "website",
"citation": "TheoremDB P9610",
"url": "https://theoremdb.org/statements/P9610",
"note": null,
"sort_order": 0
},
{
"kind": "formalization",
"citation": "Source formal statement (not locally verified)",
"url": "https://theoremdb.org/statements/P9610",
"not... | {
"source_record": "P9610",
"imported_at": "2026-09-08",
"status_as_of": "2026-09-08",
"evidence_label": "Source candidate · not reviewed",
"note": "Full public directory statement. Source lane: backlog; source resolution: open. Source verification claims are attributed only; no local Lean verification has be... | {"created_at":"2026-08-17T19:57:00+00:00","display_standard":{"editorial_review":"submitter_supplied","formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":9610,"public_status":{"evidence":{"hasFormalization":false,"hasInformalProof":false,"hasRecordedWork":false,"solutionRecord":null},"lean":{... | [
"formalization-contains-placeholder",
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"created_at": "2026-09-08T03:32:15.935Z",
"updated_at": "2026-09-08T03:32:15.935Z",
"published_at": "2026-09-08T03:32:58.546Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"aede8392-3a5f-44b2-a0c9-d52e922da92e","format_issues":null,"number":5205} |
1.0.0 | problem:theoremdb-p9612 | problem | theoremdb-p9612 | Lemoine's Conjecture | {
"main": "math",
"sub": "number theory",
"slug": "number-theory",
"name": "Number theory",
"original_discipline": "mathematics"
} | [] | [
{
"code": "11",
"basis": "community-map",
"evidence": "Inferred from community field: number-theory"
}
] | Let $n$ be an odd integer greater than $6$. Determine whether there exist prime numbers $p$ and $q$ such that $n = p + 2q$. | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P9612",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P9612",
"citation": null,
"doi": null,
"arxiv_id": null,
"license": "Selected packets label their content CC0-1.0; original references retain their own terms",
"notes": "Selected problem... | theorem lemoine_conjecture (n : ℕ) (hn : 6 < n) (odd : Odd n) :
∃ (p q : ℕ), p.Prime ∧ q.Prime ∧ p + 2 * q = n | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem lemoine_conjecture (n : ℕ) (hn : 6 < n) (odd : Odd n) :\n ∃ (p q : ℕ), p.Prime ∧ q.Prime ∧ p + 2 * q = n",
"target": null,
"source_url": "https://theoremdb.org/statements/P9612",
"format": "declaration-or-fragment",
"sha256": "89a7b8abbb27745f33f8c... | [] | TheoremDB P9612 · Candidate awaiting source review | null | Source-reported assessment, retrieved 8 September 2026. Frontier Science has not independently reviewed this status.
Open. No signed Lean verification is attached to this problem or its selected resolution.
This record is in the source backlog; its novelty and unresolved status have not passed source review. | null | published | incremental | hard | [
"source-candidate",
"theoremdb"
] | null | null | null | null | [
{
"kind": "website",
"citation": "TheoremDB P9612",
"url": "https://theoremdb.org/statements/P9612",
"note": null,
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},
{
"kind": "formalization",
"citation": "Source formal statement (not locally verified)",
"url": "https://theoremdb.org/statements/P9612",
"not... | {
"source_record": "P9612",
"imported_at": "2026-09-08",
"status_as_of": "2026-09-08",
"evidence_label": "Source candidate · not reviewed",
"note": "Full public directory statement. Source lane: backlog; source resolution: open. Source verification claims are attributed only; no local Lean verification has be... | {"created_at":"2026-08-17T19:57:04+00:00","display_standard":{"editorial_review":"submitter_supplied","formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":9612,"public_status":{"evidence":{"hasFormalization":false,"hasInformalProof":false,"hasRecordedWork":false,"solutionRecord":null},"lean":{... | [
"formalization-not-compiled-during-export",
"current-status-unassessed",
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"created_at": "2026-09-08T03:32:15.935Z",
"updated_at": "2026-09-08T03:32:15.935Z",
"published_at": "2026-09-08T03:33:03.183Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"42c88e51-eacd-4dbb-a9a6-aa3fba06902e","format_issues":null,"number":6041} |
1.0.0 | problem:theoremdb-p9614 | problem | theoremdb-p9614 | Irrationality of the Riemann zeta value at 7 | {
"main": "math",
"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}$ and admits an analytic continuation to $\mathbb{C} \setminus \{1\}$. A real number is irrational if it is not equal to any ratio of two integers. Prove that $\zeta(7)$ is irrational. | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P9614",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P9614",
"citation": null,
"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_seven : ∃ x, Irrational x ∧ riemannZeta 7 = x | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem irrational_seven : ∃ x, Irrational x ∧ riemannZeta 7 = x",
"target": null,
"source_url": "https://theoremdb.org/statements/P9614",
"format": "declaration-or-fragment",
"sha256": "a3784f64f5f1542688d47735821e80af7b6a516049e325f3bcadea888f2c0b39",
"... | [] | TheoremDB P9614 · Candidate awaiting source review | null | Source-reported assessment, retrieved 8 September 2026. Frontier Science has not independently reviewed this status.
Open. No signed Lean verification is attached to this problem or its selected resolution.
This record is in the source backlog; its novelty and unresolved status have not passed source review. | null | published | incremental | hard | [
"source-candidate",
"theoremdb"
] | null | null | null | null | [
{
"kind": "website",
"citation": "TheoremDB P9614",
"url": "https://theoremdb.org/statements/P9614",
"note": null,
"sort_order": 0
},
{
"kind": "formalization",
"citation": "Source formal statement (not locally verified)",
"url": "https://theoremdb.org/statements/P9614",
"not... | {
"source_record": "P9614",
"imported_at": "2026-09-08",
"status_as_of": "2026-09-08",
"evidence_label": "Source candidate · not reviewed",
"note": "Full public directory statement. Source lane: backlog; source resolution: open. Source verification claims are attributed only; no local Lean verification has be... | {"created_at":"2026-08-17T19:57:08+00:00","display_standard":{"editorial_review":"submitter_supplied","formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":9614,"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": "2026-09-08T03:33:01.820Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"646eef68-6034-4454-a460-9a2c7ebd4632","format_issues":null,"number":5742} |
1.0.0 | problem:theoremdb-p9616 | problem | theoremdb-p9616 | Maximum guaranteed coset dimension in sumsets over the Boolean cube | {
"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$ and each real number $\alpha \in [0,1]$, define $F(n, \alpha)$ to be the infimum of $\max\{\dim(V) : V \text{ is a coset contained in } A + A\}$ over all subsets $A \subseteq \mathbb{F}_2^n$ with density at least $\alpha$, where $\mathbb{F}_2^n$ denotes the $n$-dimensional vector space ove... | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P9616",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P9616",
"citation": null,
"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_51 : answer(sorry) = guaranteedMaxCosetDim | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem green_51 : answer(sorry) = guaranteedMaxCosetDim",
"target": null,
"source_url": "https://theoremdb.org/statements/P9616",
"format": "declaration-or-fragment",
"sha256": "f9d96b1788f57f503b014a8872b20689017ce936dae9acc017564ac4b97789d8",
"license"... | [] | TheoremDB P9616 · Candidate awaiting source review | null | Source-reported assessment, retrieved 8 September 2026. Frontier Science has not independently reviewed this status.
Open. No signed Lean verification is attached to this problem or its selected resolution.
This record is in the source backlog; its novelty and unresolved status have not passed source review. | null | published | incremental | hard | [
"source-candidate",
"theoremdb"
] | null | null | null | null | [
{
"kind": "website",
"citation": "TheoremDB P9616",
"url": "https://theoremdb.org/statements/P9616",
"note": null,
"sort_order": 0
},
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"kind": "formalization",
"citation": "Source formal statement (not locally verified)",
"url": "https://theoremdb.org/statements/P9616",
"not... | {
"source_record": "P9616",
"imported_at": "2026-09-08",
"status_as_of": "2026-09-08",
"evidence_label": "Source candidate · not reviewed",
"note": "Full public directory statement. Source lane: backlog; source resolution: open. Source verification claims are attributed only; no local Lean verification has be... | {"created_at":"2026-08-17T19:57:11+00:00","display_standard":{"formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":9616,"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": "2026-09-08T03:33:03.791Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"b1a7e867-8d58-4ee5-8e09-dfe6f23bb1e2","format_issues":null,"number":6172} |
1.0.0 | problem:theoremdb-p9618 | problem | theoremdb-p9618 | Existence of a SIC-POVM in dimension 64 | {
"main": "math",
"sub": "algebra",
"slug": "algebra",
"name": "Algebra",
"original_discipline": "mathematics"
} | [] | [] | For a positive integer $d$, let $\mathbb{C}^d$ denote the $d$-dimensional complex Hilbert space with its standard inner product. A state vector in $\mathbb{C}^d$ is a vector of $L^2$ norm $1$. For two state vectors $\phi, \psi \in \mathbb{C}^d$, define their squared overlap to be $\left|\sum_{i=1}^{d} \overline{\phi_i}... | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P9618",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P9618",
"citation": null,
"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_64 : answer(sorry) ↔ HasSICPOVM 64 | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem hasSICPOVM_64 : answer(sorry) ↔ HasSICPOVM 64",
"target": null,
"source_url": "https://theoremdb.org/statements/P9618",
"format": "declaration-or-fragment",
"sha256": "c5449b66702b46f643a1a7bcd269e6397bfb0ba37cad1f9a450985067d950fb3",
"license": "... | [] | TheoremDB P9618 · Candidate awaiting source review | null | Source-reported assessment, retrieved 8 September 2026. Frontier Science has not independently reviewed this status.
Open. No signed Lean verification is attached to this problem or its selected resolution.
This record is in the source backlog; its novelty and unresolved status have not passed source review. | null | published | incremental | hard | [
"source-candidate",
"theoremdb"
] | null | null | null | null | [
{
"kind": "website",
"citation": "TheoremDB P9618",
"url": "https://theoremdb.org/statements/P9618",
"note": null,
"sort_order": 0
},
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"kind": "formalization",
"citation": "Source formal statement (not locally verified)",
"url": "https://theoremdb.org/statements/P9618",
"not... | {
"source_record": "P9618",
"imported_at": "2026-09-08",
"status_as_of": "2026-09-08",
"evidence_label": "Source candidate · not reviewed",
"note": "Full public directory statement. Source lane: backlog; source resolution: open. Source verification claims are attributed only; no local Lean verification has be... | {"created_at":"2026-08-17T19:57:15+00:00","display_standard":{"editorial_review":"submitter_supplied","formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":9618,"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": "2026-09-08T03:32:58.508Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"ad7e5c68-60c7-4bae-a24f-23738f8cde2e","format_issues":null,"number":5197} |
1.0.0 | problem:theoremdb-p9620 | problem | theoremdb-p9620 | Chowla's Cosine 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 every finite set $A$ of positive integers with $|A| = N$, define the cosine sum $S(A,\theta) = \sum_{n \in A} \cos(n\theta)$ for $\theta \in \mathbb{R}$. Does there exist an absolute constant $c > 0$ such that for all sufficiently large $N$, every finite set $A \subset \mathbb{N}$ with $0 \notin A$ and $|A| = N$ ad... | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P9620",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P9620",
"citation": null,
"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_510 :
answer(sorry) ↔ ∃ (c : ℝ) (hc : 0 < c),
∀ᶠ N in atTop, ∀ (A : Finset ℕ), 0 ∉ A → #A = N →
∃ θ, ∑ n ∈ A, cos (n * θ) < -c * sqrt N | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem erdos_510 :\n answer(sorry) ↔ ∃ (c : ℝ) (hc : 0 < c),\n ∀ᶠ N in atTop, ∀ (A : Finset ℕ), 0 ∉ A → #A = N →\n ∃ θ, ∑ n ∈ A, cos (n * θ) < -c * sqrt N",
"target": null,
"source_url": "https://theoremdb.org/statements/P9620",
"format": "declarati... | [] | TheoremDB P9620 · Candidate awaiting source review | null | Source-reported assessment, retrieved 8 September 2026. Frontier Science has not independently reviewed this status.
Open. No signed Lean verification is attached to this problem or its selected resolution.
This record is in the source backlog; its novelty and unresolved status have not passed source review. | null | published | incremental | hard | [
"source-candidate",
"theoremdb"
] | null | null | null | null | [
{
"kind": "website",
"citation": "TheoremDB P9620",
"url": "https://theoremdb.org/statements/P9620",
"note": null,
"sort_order": 0
},
{
"kind": "formalization",
"citation": "Source formal statement (not locally verified)",
"url": "https://theoremdb.org/statements/P9620",
"not... | {
"source_record": "P9620",
"imported_at": "2026-09-08",
"status_as_of": "2026-09-08",
"evidence_label": "Source candidate · not reviewed",
"note": "Full public directory statement. Source lane: backlog; source resolution: open. Source verification claims are attributed only; no local Lean verification has be... | {"created_at":"2026-08-17T19:57:18+00:00","display_standard":{"formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":9620,"public_status":{"evidence":{"hasFormalization":false,"hasInformalProof":false,"hasRecordedWork":false,"solutionRecord":null},"lean":{"kernelAccepted":false,"label":"","reaso... | [
"formalization-contains-placeholder",
"formalization-not-compiled-during-export",
"current-status-unassessed",
"source-full-text-not-present"
] | [] | {
"created_at": "2026-09-08T03:32:15.935Z",
"updated_at": "2026-09-08T03:32:15.935Z",
"published_at": "2026-09-08T03:32:51.113Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"bb353e65-e4a6-42e9-889e-d3fa43ad6a36","format_issues":null,"number":3563} |
1.0.0 | problem:theoremdb-p9622 | problem | theoremdb-p9622 | Maximum number of mutually unbiased bases in dimension six | {
"main": "math",
"sub": "combinatorics",
"slug": "combinatorics",
"name": "Combinatorics",
"original_discipline": "mathematics"
} | [] | [
{
"code": "05",
"basis": "community-map",
"evidence": "Inferred from community field: combinatorics"
}
] | Let $\mu(d)$ denote the maximum number $k$ for which there exist $k$ orthonormal bases $\mathcal{B}_1, \dots, \mathcal{B}_k$ of the complex Hilbert space $\mathbb{C}^d$ such that any two distinct bases are mutually unbiased. Two orthonormal bases $\mathcal{B}_r = \{ e_0^{(r)}, \dots, e_{d-1}^{(r)} \}$ and $\mathcal{B}_... | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P9622",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P9622",
"citation": null,
"doi": null,
"arxiv_id": null,
"license": "Selected packets label their content CC0-1.0; original references retain their own terms",
"notes": "Selected problem... | theorem mutuallyUnbiasedBases_dim6 :
IsMaxMUBCount 6 (answer(sorry)) | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem mutuallyUnbiasedBases_dim6 :\n IsMaxMUBCount 6 (answer(sorry))",
"target": null,
"source_url": "https://theoremdb.org/statements/P9622",
"format": "declaration-or-fragment",
"sha256": "1a09474135134d45de2c2983f2c594ff85f31164fae9d36f5adffd496ef554a... | [] | TheoremDB P9622 · Candidate awaiting source review | null | Source-reported assessment, retrieved 8 September 2026. Frontier Science has not independently reviewed this status.
Open. No signed Lean verification is attached to this problem or its selected resolution.
This record is in the source backlog; its novelty and unresolved status have not passed source review. | null | published | incremental | hard | [
"source-candidate",
"theoremdb"
] | null | null | null | null | [
{
"kind": "website",
"citation": "TheoremDB P9622",
"url": "https://theoremdb.org/statements/P9622",
"note": null,
"sort_order": 0
},
{
"kind": "formalization",
"citation": "Source formal statement (not locally verified)",
"url": "https://theoremdb.org/statements/P9622",
"not... | {
"source_record": "P9622",
"imported_at": "2026-09-08",
"status_as_of": "2026-09-08",
"evidence_label": "Source candidate · not reviewed",
"note": "Full public directory statement. Source lane: backlog; source resolution: open. Source verification claims are attributed only; no local Lean verification has be... | {"created_at":"2026-08-17T19:57:22+00:00","display_standard":{"formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":9622,"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": "2026-09-08T03:33:03.825Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"ca380c32-318c-4082-9ac8-f894c3992e04","format_issues":null,"number":6179} |
1.0.0 | problem:theoremdb-p9624 | problem | theoremdb-p9624 | Non-negativity of the Antihydra balance sequence | {
"main": "math",
"sub": "combinatorics",
"slug": "combinatorics",
"name": "Combinatorics",
"original_discipline": "mathematics"
} | [] | [
{
"code": "05",
"basis": "community-map",
"evidence": "Inferred from community field: combinatorics"
}
] | Define a sequence $(a_n)_{n \ge 0}$ of natural numbers by $a_0 = 8$ and $a_{n+1} = \left\lfloor \frac{3a_n}{2} \right\rfloor$ for all $n \ge 0$. Define a sequence $(b_n)_{n \ge 0}$ of integers by $b_0 = 0$ and
$$
b_{n+1} = \begin{cases} b_n + 2 & \text{if } a_n \text{ is even}, \\ b_n - 1 & \text{if } a_n \text{ is od... | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P9624",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P9624",
"citation": null,
"doi": null,
"arxiv_id": null,
"license": "Selected packets label their content CC0-1.0; original references retain their own terms",
"notes": "Selected problem... | theorem beaver_math_olympiad_problem_2_antihydra
(a : ℕ → ℕ) (b : ℕ → ℤ)
(a_ini : a 0 = 8)
(a_rec : ∀ n, a (n + 1) = (3 * a n) / 2)
(b_ini : b 0 = 0)
(b_rec : ∀ n, b (n + 1) = if a n % 2 = 0 then b n + 2 else b n - 1) :
∀ n, b n ≥ 0 | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem beaver_math_olympiad_problem_2_antihydra\n (a : ℕ → ℕ) (b : ℕ → ℤ)\n (a_ini : a 0 = 8)\n (a_rec : ∀ n, a (n + 1) = (3 * a n) / 2)\n (b_ini : b 0 = 0)\n (b_rec : ∀ n, b (n + 1) = if a n % 2 = 0 then b n + 2 else b n - 1) :\n ∀ n, b n ≥ 0",
"targe... | [] | TheoremDB P9624 · Candidate awaiting source review | null | Source-reported assessment, retrieved 8 September 2026. Frontier Science has not independently reviewed this status.
Open. No signed Lean verification is attached to this problem or its selected resolution.
This record is in the source backlog; its novelty and unresolved status have not passed source review. | null | published | incremental | hard | [
"source-candidate",
"theoremdb"
] | null | null | null | null | [
{
"kind": "website",
"citation": "TheoremDB P9624",
"url": "https://theoremdb.org/statements/P9624",
"note": null,
"sort_order": 0
},
{
"kind": "formalization",
"citation": "Source formal statement (not locally verified)",
"url": "https://theoremdb.org/statements/P9624",
"not... | {
"source_record": "P9624",
"imported_at": "2026-09-08",
"status_as_of": "2026-09-08",
"evidence_label": "Source candidate · not reviewed",
"note": "Full public directory statement. Source lane: backlog; source resolution: open. Source verification claims are attributed only; no local Lean verification has be... | {"created_at":"2026-08-17T19:57:26+00:00","display_standard":{"editorial_review":"submitter_supplied","formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":9624,"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": "2026-09-08T03:33:04.441Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"36bfcf94-f589-407c-935d-991624aaebdb","format_issues":null,"number":6310} |
1.0.0 | problem:theoremdb-p9626 | problem | theoremdb-p9626 | Upper bound for the monotone variant of Erdős distinct consecutive-sums 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 $n$, let $h(n)$ denote the maximum integer $k$ for which there exist integers $1 \le a_1 \le a_2 \le \dots \le a_k \le n$ such that all sums of the form $\sum_{u \le i \le v} a_i$ (where $1 \le u \le v \le k$) are distinct. Determine an explicit function $g : \mathbb{N} \to \mathbb{R}$ such th... | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P9626",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P9626",
"citation": null,
"doi": null,
"arxiv_id": null,
"license": "Selected packets label their content CC0-1.0; original references retain their own terms",
"notes": "Selected problem... | theorem erdos_357.variants.monotone.parts.ii.bigO_version_symm :
(fun n ↦ (h n : ℝ)) =O[atTop] (answer(sorry) : ℕ → ℝ) | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem erdos_357.variants.monotone.parts.ii.bigO_version_symm :\n (fun n ↦ (h n : ℝ)) =O[atTop] (answer(sorry) : ℕ → ℝ)",
"target": null,
"source_url": "https://theoremdb.org/statements/P9626",
"format": "declaration-or-fragment",
"sha256": "cfe91593441f4... | [] | TheoremDB P9626 · Candidate awaiting source review | null | Source-reported assessment, retrieved 8 September 2026. Frontier Science has not independently reviewed this status.
Open. No signed Lean verification is attached to this problem or its selected resolution.
This record is in the source backlog; its novelty and unresolved status have not passed source review. | null | published | incremental | hard | [
"source-candidate",
"theoremdb"
] | null | null | null | null | [
{
"kind": "website",
"citation": "TheoremDB P9626",
"url": "https://theoremdb.org/statements/P9626",
"note": null,
"sort_order": 0
},
{
"kind": "formalization",
"citation": "Source formal statement (not locally verified)",
"url": "https://theoremdb.org/statements/P9626",
"not... | {
"source_record": "P9626",
"imported_at": "2026-09-08",
"status_as_of": "2026-09-08",
"evidence_label": "Source candidate · not reviewed",
"note": "Full public directory statement. Source lane: backlog; source resolution: open. Source verification claims are attributed only; no local Lean verification has be... | {"created_at":"2026-08-17T19:57:29+00:00","display_standard":{"editorial_review":"submitter_supplied","formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":9626,"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": "2026-09-08T03:33:07.358Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"556c3157-a4c4-4252-bc77-e934bec3ecd0","format_issues":null,"number":6942} |
1.0.0 | problem:theoremdb-p9628 | problem | theoremdb-p9628 | Squarefreeness of 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 $E_n$ is defined as one plus the product of the first $n$ prime numbers, that is,
$$
E_n = 1 + p_1 p_2 \cdots p_n,
$$
where $p_i$ denotes the $i$-th prime number (with $p_1 = 2$). A positive integer is called squarefree if it is not divisible by the square of any ... | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P9628",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P9628",
"citation": null,
"doi": null,
"arxiv_id": null,
"license": "Selected packets label their content CC0-1.0; original references retain their own terms",
"notes": "Selected problem... | theorem euclid_numbers_are_square_free : answer(sorry) ↔ (∀ n, Squarefree (Euclid n)) | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem euclid_numbers_are_square_free : answer(sorry) ↔ (∀ n, Squarefree (Euclid n))",
"target": null,
"source_url": "https://theoremdb.org/statements/P9628",
"format": "declaration-or-fragment",
"sha256": "928481f044a66ceb875744989c9066a32e7e0a6929747f8b0b9... | [] | TheoremDB P9628 · Candidate awaiting source review | null | Source-reported assessment, retrieved 8 September 2026. Frontier Science has not independently reviewed this status.
Open. No signed Lean verification is attached to this problem or its selected resolution.
This record is in the source backlog; its novelty and unresolved status have not passed source review. | null | published | incremental | hard | [
"source-candidate",
"theoremdb"
] | null | null | null | null | [
{
"kind": "website",
"citation": "TheoremDB P9628",
"url": "https://theoremdb.org/statements/P9628",
"note": null,
"sort_order": 0
},
{
"kind": "formalization",
"citation": "Source formal statement (not locally verified)",
"url": "https://theoremdb.org/statements/P9628",
"not... | {
"source_record": "P9628",
"imported_at": "2026-09-08",
"status_as_of": "2026-09-08",
"evidence_label": "Source candidate · not reviewed",
"note": "Full public directory statement. Source lane: backlog; source resolution: open. Source verification claims are attributed only; no local Lean verification has be... | {"created_at":"2026-08-17T19:57:35+00:00","display_standard":{"editorial_review":"submitter_supplied","formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":9628,"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": "2026-09-08T03:33:06.047Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"02ea0caf-6a2d-455f-a32c-7f69661499df","format_issues":null,"number":6659} |
1.0.0 | problem:theoremdb-p9630 | problem | theoremdb-p9630 | Uniqueness of the primitive term in sequence A063880 | {
"main": "math",
"sub": "number theory",
"slug": "number-theory",
"name": "Number theory",
"original_discipline": "mathematics"
} | [] | [
{
"code": "11",
"basis": "community-map",
"evidence": "Inferred from community field: number-theory"
}
] | For a positive integer $n$, let $\sigma(n)$ denote the sum of all positive divisors of $n$, and let $\text{usigma}(n)$ denote the sum of the unitary divisors of $n$, where a divisor $d$ of $n$ is called unitary if $\gcd(d, n/d) = 1$. Define a number $n$ to be in sequence A063880 if $\sigma(n) = 2 \cdot \text{usigma}(n)... | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P9630",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P9630",
"citation": null,
"doi": null,
"arxiv_id": null,
"license": "Selected packets label their content CC0-1.0; original references retain their own terms",
"notes": "Selected problem... | theorem unique_primitive_108 {n : ℕ} (h : IsPrimitiveTerm n) : n = 108 | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem unique_primitive_108 {n : ℕ} (h : IsPrimitiveTerm n) : n = 108",
"target": null,
"source_url": "https://theoremdb.org/statements/P9630",
"format": "declaration-or-fragment",
"sha256": "aa0ab3a95e7a9c1546591b90332acdd4d3693ccaf2a2b5f89458cca39306e865",... | [] | TheoremDB P9630 · Candidate awaiting source review | null | Source-reported assessment, retrieved 8 September 2026. Frontier Science has not independently reviewed this status.
Open. No signed Lean verification is attached to this problem or its selected resolution.
This record is in the source backlog; its novelty and unresolved status have not passed source review. | null | published | incremental | hard | [
"source-candidate",
"theoremdb"
] | null | null | null | null | [
{
"kind": "website",
"citation": "TheoremDB P9630",
"url": "https://theoremdb.org/statements/P9630",
"note": null,
"sort_order": 0
},
{
"kind": "formalization",
"citation": "Source formal statement (not locally verified)",
"url": "https://theoremdb.org/statements/P9630",
"not... | {
"source_record": "P9630",
"imported_at": "2026-09-08",
"status_as_of": "2026-09-08",
"evidence_label": "Source candidate · not reviewed",
"note": "Full public directory statement. Source lane: backlog; source resolution: open. Source verification claims are attributed only; no local Lean verification has be... | {"created_at":"2026-08-17T19:57:39+00:00","display_standard":{"editorial_review":"submitter_supplied","formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":9630,"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": "2026-09-08T03:33:07.268Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"f4e991a8-6cf7-4597-ab1f-4f321984437d","format_issues":null,"number":6923} |
1.0.0 | problem:theoremdb-p9632 | problem | theoremdb-p9632 | Existence of the limiting ratio of distinct to repeated totient values | {
"main": "math",
"sub": "number theory",
"slug": "number-theory",
"name": "Number theory",
"original_discipline": "mathematics"
} | [] | [
{
"code": "11",
"basis": "community-map",
"evidence": "Inferred from community field: number-theory"
}
] | For each real number $x\geq 1$, let
$$
V'(x) = \#\{\phi(m) : 1\leq m\leq x\}
$$
denote the number of distinct values attained by Euler's totient function $\phi$ on the integers from $1$ to $\lfloor x\rfloor$, and let
$$
V(x) = \#\{k\leq x : k\in\phi(\mathbb{N})\}
$$
denote the number of positive integers up to $x$ ... | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P9632",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P9632",
"citation": null,
"doi": null,
"arxiv_id": null,
"license": "Selected packets label their content CC0-1.0; original references retain their own terms",
"notes": "Selected problem... | theorem erdos_417.parts.i :
answer(sorry) ↔ ∃ L : ENNReal, Tendsto (fun x ↦
({ k | k ∈ range totient ∧ (k : ℝ) ≤ x }.ncard : ENNReal) /
((totient '' { m | 1 ≤ m ∧ (m : ℝ) ≤ x }).ncard : ENNReal))
atTop (𝓝 L) | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem erdos_417.parts.i :\n answer(sorry) ↔ ∃ L : ENNReal, Tendsto (fun x ↦\n ({ k | k ∈ range totient ∧ (k : ℝ) ≤ x }.ncard : ENNReal) /\n ((totient '' { m | 1 ≤ m ∧ (m : ℝ) ≤ x }).ncard : ENNReal))\n atTop (𝓝 L)",
"target": null,
"source_url": ... | [] | TheoremDB P9632 · Candidate awaiting source review | null | Source-reported assessment, retrieved 8 September 2026. Frontier Science has not independently reviewed this status.
Open. No signed Lean verification is attached to this problem or its selected resolution.
This record is in the source backlog; its novelty and unresolved status have not passed source review. | null | published | incremental | hard | [
"source-candidate",
"theoremdb"
] | null | null | null | null | [
{
"kind": "website",
"citation": "TheoremDB P9632",
"url": "https://theoremdb.org/statements/P9632",
"note": null,
"sort_order": 0
},
{
"kind": "formalization",
"citation": "Source formal statement (not locally verified)",
"url": "https://theoremdb.org/statements/P9632",
"not... | {
"source_record": "P9632",
"imported_at": "2026-09-08",
"status_as_of": "2026-09-08",
"evidence_label": "Source candidate · not reviewed",
"note": "Full public directory statement. Source lane: backlog; source resolution: open. Source verification claims are attributed only; no local Lean verification has be... | {"created_at":"2026-08-17T19:57:44+00:00","display_standard":{"editorial_review":"submitter_supplied","formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":9632,"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": "2026-09-08T03:32:58.839Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"0524b21a-d77a-4403-a1aa-747e558c4b41","format_issues":null,"number":5266} |
1.0.0 | problem:theoremdb-p9634 | problem | theoremdb-p9634 | Erdős's Polynomial Factorization Problem for the Integers | {
"main": "math",
"sub": "algebra",
"slug": "algebra",
"name": "Algebra",
"original_discipline": "mathematics"
} | [] | [] | Does there exist a polynomial $f \in \mathbb{Z}[X]$ of degree at least $2$ and a set $A \subseteq \mathbb{Z}$ such that every integer $z$ can be written uniquely as $z = a + f(n)$ for some $a \in A$ and some positive integer $n$? Here, $\mathbb{Z}[X]$ denotes the ring of polynomials with integer coefficients, and for a... | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P9634",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P9634",
"citation": null,
"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 : answer(sorry) ↔
∃ f : ℤ[X], 2 ≤ f.degree ∧ ∃ A : Set ℤ,
∀ z, ∃! ab ∈ A ×ˢ (Set.range f.eval), z = ab.1 + ab.2 | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem erdos_477 : answer(sorry) ↔\n ∃ f : ℤ[X], 2 ≤ f.degree ∧ ∃ A : Set ℤ,\n ∀ z, ∃! ab ∈ A ×ˢ (Set.range f.eval), z = ab.1 + ab.2",
"target": null,
"source_url": "https://theoremdb.org/statements/P9634",
"format": "declaration-or-fragment",
"sha25... | [] | TheoremDB P9634 · Candidate awaiting source review | null | Source-reported assessment, retrieved 8 September 2026. Frontier Science has not independently reviewed this status.
Open. No signed Lean verification is attached to this problem or its selected resolution.
This record is in the source backlog; its novelty and unresolved status have not passed source review. | null | published | incremental | hard | [
"source-candidate",
"theoremdb"
] | null | null | null | null | [
{
"kind": "website",
"citation": "TheoremDB P9634",
"url": "https://theoremdb.org/statements/P9634",
"note": null,
"sort_order": 0
},
{
"kind": "formalization",
"citation": "Source formal statement (not locally verified)",
"url": "https://theoremdb.org/statements/P9634",
"not... | {
"source_record": "P9634",
"imported_at": "2026-09-08",
"status_as_of": "2026-09-08",
"evidence_label": "Source candidate · not reviewed",
"note": "Full public directory statement. Source lane: backlog; source resolution: open. Source verification claims are attributed only; no local Lean verification has be... | {"created_at":"2026-08-17T19:57:48+00:00","display_standard":{"editorial_review":"submitter_supplied","formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":9634,"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": "2026-09-08T03:32:56.429Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"5b4733d2-6787-45fc-a7d3-b551a8c01901","format_issues":null,"number":4738} |
1.0.0 | problem:theoremdb-p9636 | problem | theoremdb-p9636 | Density threshold for guaranteed large cosets in sumsets over the Boolean cube | {
"main": "math",
"sub": "combinatorics",
"slug": "combinatorics",
"name": "Combinatorics",
"original_discipline": "mathematics"
} | [] | [
{
"code": "05",
"basis": "community-map",
"evidence": "Inferred from community field: combinatorics"
}
] | Let $\mathbb{F}_2^n$ denote the $n$-dimensional vector space over the field with two elements. For a subset $A \subseteq \mathbb{F}_2^n$, let $A + A = \{a + a' : a, a' \in A\}$ denote its sumset, and for a subset $S \subseteq \mathbb{F}_2^n$, let the coset dimension of $S$ be the maximum dimension of a coset (i.e., a t... | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P9636",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P9636",
"citation": null,
"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_51.one_half :
answer(sorry) ↔ ∀ (k : ℝ), 0 < k →
∃ (c : ℕ), ∀ᶠ (n : ℕ) in atTop,
∀ (α : ℝ), α > (1/2 : ℝ) - k / sqrt (n : ℝ) → α ≤ 1 →
n ≤ guaranteedMaxCosetDim n α + c | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem green_51.one_half :\n answer(sorry) ↔ ∀ (k : ℝ), 0 < k →\n ∃ (c : ℕ), ∀ᶠ (n : ℕ) in atTop,\n ∀ (α : ℝ), α > (1/2 : ℝ) - k / sqrt (n : ℝ) → α ≤ 1 →\n n ≤ guaranteedMaxCosetDim n α + c",
"target": null,
"source_url": "https://theoremdb.o... | [] | TheoremDB P9636 · Candidate awaiting source review | null | Source-reported assessment, retrieved 8 September 2026. Frontier Science has not independently reviewed this status.
Open. No signed Lean verification is attached to this problem or its selected resolution.
This record is in the source backlog; its novelty and unresolved status have not passed source review. | null | published | incremental | hard | [
"source-candidate",
"theoremdb"
] | null | null | null | null | [
{
"kind": "website",
"citation": "TheoremDB P9636",
"url": "https://theoremdb.org/statements/P9636",
"note": null,
"sort_order": 0
},
{
"kind": "formalization",
"citation": "Source formal statement (not locally verified)",
"url": "https://theoremdb.org/statements/P9636",
"not... | {
"source_record": "P9636",
"imported_at": "2026-09-08",
"status_as_of": "2026-09-08",
"evidence_label": "Source candidate · not reviewed",
"note": "Full public directory statement. Source lane: backlog; source resolution: open. Source verification claims are attributed only; no local Lean verification has be... | {"created_at":"2026-08-17T20:00:06+00:00","display_standard":{"editorial_review":"submitter_supplied","formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":9636,"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": "2026-09-08T03:32:53.127Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"239065e6-a7e5-49e3-821a-9bc29c9a8c6e","format_issues":null,"number":4009} |
1.0.0 | problem:theoremdb-p9638 | problem | theoremdb-p9638 | Carbery's rectangle problem | {
"main": "math",
"sub": "analysis",
"slug": "analysis",
"name": "Analysis",
"original_discipline": "mathematics"
} | [] | [] | Let $A$ be an open subset of the unit square $[0,1]^2$ with Lebesgue measure $\alpha$. Does there exist an absolute constant $c > 0$ such that $A$ always contains four points $(x_1, y_1), (x_2, y_1), (x_2, y_2), (x_1, y_2)$ forming the vertices of an axis-parallel rectangle with area at least $c \alpha^2$? | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P9638",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P9638",
"citation": null,
"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_85 :
answer(sorry) ↔ ∃ c > 0, ∀ A : Set (ℝ × ℝ),
IsOpen A →
A ⊆ Icc 0 1 ×ˢ Icc 0 1 →
A.Nonempty →
let α := (volume A).toReal
∃ x₁ x₂ y₁ y₂,
{(x₁, y₁), (x₂, y₁), (x₂, y₂), (x₁, y₂)} ⊆ A ∧
c * α ^ 2 ≤ |x₁ - x₂| * |y₁ - y₂| | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem green_85 :\n answer(sorry) ↔ ∃ c > 0, ∀ A : Set (ℝ × ℝ),\n IsOpen A →\n A ⊆ Icc 0 1 ×ˢ Icc 0 1 →\n A.Nonempty →\n let α := (volume A).toReal\n ∃ x₁ x₂ y₁ y₂,\n {(x₁, y₁), (x₂, y₁), (x₂, y₂), (x₁, y₂)} ⊆ A ∧\n c * α ^ 2 ≤ |x₁ - x₂| * |y₁ - y₂... | [] | TheoremDB P9638 · Candidate awaiting source review | null | Source-reported assessment, retrieved 8 September 2026. Frontier Science has not independently reviewed this status.
Open. No signed Lean verification is attached to this problem or its selected resolution.
This record is in the source backlog; its novelty and unresolved status have not passed source review. | null | published | incremental | hard | [
"source-candidate",
"theoremdb"
] | null | null | null | null | [
{
"kind": "website",
"citation": "TheoremDB P9638",
"url": "https://theoremdb.org/statements/P9638",
"note": null,
"sort_order": 0
},
{
"kind": "formalization",
"citation": "Source formal statement (not locally verified)",
"url": "https://theoremdb.org/statements/P9638",
"not... | {
"source_record": "P9638",
"imported_at": "2026-09-08",
"status_as_of": "2026-09-08",
"evidence_label": "Source candidate · not reviewed",
"note": "Full public directory statement. Source lane: backlog; source resolution: open. Source verification claims are attributed only; no local Lean verification has be... | {"created_at":"2026-08-17T20:00:13+00:00","display_standard":{"formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":9638,"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": "2026-09-08T03:32:50.982Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"baa60b62-7faf-4335-b009-e52210642494","format_issues":null,"number":3535} |
1.0.0 | problem:theoremdb-p9640 | problem | theoremdb-p9640 | Surjectivity of the Hofstadter Q-like recurrence | {
"main": "math",
"sub": "number theory",
"slug": "number-theory",
"name": "Number theory",
"original_discipline": "mathematics"
} | [] | [
{
"code": "11",
"basis": "community-map",
"evidence": "Inferred from community field: number-theory"
}
] | Define a function $f : \mathbb{Z}^+ \to \mathbb{Z}^+$ by $f(1) = f(2) = 1$ and for all integers $n > 2$,
$$
f(n) = f(n - f(n-1)) + f(n - f(n-2)).
$$
(Note: It is not known whether $f(n)$ is well-defined for all $n$.) Determine whether $f$ is surjective, i.e., whether for every positive integer $m$ there exists a posi... | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P9640",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P9640",
"citation": null,
"doi": null,
"arxiv_id": null,
"license": "Selected packets label their content CC0-1.0; original references retain their own terms",
"notes": "Selected problem... | theorem erdos_422.variants.surjective : answer(sorry) ↔ f.Surjective | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem erdos_422.variants.surjective : answer(sorry) ↔ f.Surjective",
"target": null,
"source_url": "https://theoremdb.org/statements/P9640",
"format": "declaration-or-fragment",
"sha256": "d25ab44c4241bf0599deec1d5ddfb3f4ef05892a101723d408e54a15c6ce3dcb",
... | [] | TheoremDB P9640 · Candidate awaiting source review | null | Source-reported assessment, retrieved 8 September 2026. Frontier Science has not independently reviewed this status.
Open. No signed Lean verification is attached to this problem or its selected resolution.
This record is in the source backlog; its novelty and unresolved status have not passed source review. | null | published | incremental | hard | [
"source-candidate",
"theoremdb"
] | null | null | null | null | [
{
"kind": "website",
"citation": "TheoremDB P9640",
"url": "https://theoremdb.org/statements/P9640",
"note": null,
"sort_order": 0
},
{
"kind": "formalization",
"citation": "Source formal statement (not locally verified)",
"url": "https://theoremdb.org/statements/P9640",
"not... | {
"source_record": "P9640",
"imported_at": "2026-09-08",
"status_as_of": "2026-09-08",
"evidence_label": "Source candidate · not reviewed",
"note": "Full public directory statement. Source lane: backlog; source resolution: open. Source verification claims are attributed only; no local Lean verification has be... | {"created_at":"2026-08-17T20:00:17+00:00","display_standard":{"editorial_review":"submitter_supplied","formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":9640,"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": "2026-09-08T03:33:06.238Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"50011b97-86cb-44ab-aba5-f00089500ea8","format_issues":null,"number":6700} |
1.0.0 | problem:theoremdb-p9642 | problem | theoremdb-p9642 | Bijective polynomial maps from the rational plane to the rationals | {
"main": "math",
"sub": "algebra",
"slug": "algebra",
"name": "Algebra",
"original_discipline": "mathematics"
} | [] | [] | Does there exist a polynomial $f(x, y) \in \mathbb{Q}[x, y]$ such that the induced map $f: \mathbb{Q} \times \mathbb{Q} \to \mathbb{Q}$ is bijective? | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P9642",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P9642",
"citation": null,
"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_21003 :
answer(sorry) ↔ ∃ f : MvPolynomial (Fin 2) ℚ, Function.Bijective fun x ↦ f.eval x | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem mathoverflow_21003 :\n answer(sorry) ↔ ∃ f : MvPolynomial (Fin 2) ℚ, Function.Bijective fun x ↦ f.eval x",
"target": null,
"source_url": "https://theoremdb.org/statements/P9642",
"format": "declaration-or-fragment",
"sha256": "90ef1ecbb8ec89619fe6b... | [] | TheoremDB P9642 · Candidate awaiting source review | null | Source-reported assessment, retrieved 8 September 2026. Frontier Science has not independently reviewed this status.
Open. No signed Lean verification is attached to this problem or its selected resolution.
This record is in the source backlog; its novelty and unresolved status have not passed source review. | null | published | incremental | hard | [
"source-candidate",
"theoremdb"
] | null | null | null | null | [
{
"kind": "website",
"citation": "TheoremDB P9642",
"url": "https://theoremdb.org/statements/P9642",
"note": null,
"sort_order": 0
},
{
"kind": "formalization",
"citation": "Source formal statement (not locally verified)",
"url": "https://theoremdb.org/statements/P9642",
"not... | {
"source_record": "P9642",
"imported_at": "2026-09-08",
"status_as_of": "2026-09-08",
"evidence_label": "Source candidate · not reviewed",
"note": "Full public directory statement. Source lane: backlog; source resolution: open. Source verification claims are attributed only; no local Lean verification has be... | {"created_at":"2026-08-17T20:00:19+00:00","display_standard":{"editorial_review":"submitter_supplied","formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":9642,"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": "2026-09-08T03:32:50.666Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"113e2d1d-bfd0-4167-881a-bb575d21a0d6","format_issues":null,"number":3466} |
1.0.0 | problem:theoremdb-p9644 | problem | theoremdb-p9644 | Erdős's Problem 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 non-negative integer $k$, let $\mathcal{S}_k$ denote the set of natural numbers that can be written as $p + 2^{a_1} + 2^{a_2} + \cdots + 2^{a_m}$ where $p$ is a prime number, $m \leq k$, and $a_1, a_2, \ldots, a_m$ are non-negative integers (with the empty sum of powers of $2$ equal to $0$). Does there exist some... | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P9644",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P9644",
"citation": null,
"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 : answer(sorry) ↔ ∃ k, sumPrimeAndTwoPows k = Set.univ \ {0, 1} | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem erdos_10 : answer(sorry) ↔ ∃ k, sumPrimeAndTwoPows k = Set.univ \\ {0, 1}",
"target": null,
"source_url": "https://theoremdb.org/statements/P9644",
"format": "declaration-or-fragment",
"sha256": "936cd85632d5ced2e1f45772f186c735af8a7e601fbdfea4ae44444... | [] | TheoremDB P9644 · Candidate awaiting source review | null | Source-reported assessment, retrieved 8 September 2026. Frontier Science has not independently reviewed this status.
Open. No signed Lean verification is attached to this problem or its selected resolution.
This record is in the source backlog; its novelty and unresolved status have not passed source review. | null | published | incremental | hard | [
"source-candidate",
"theoremdb"
] | null | null | null | null | [
{
"kind": "website",
"citation": "TheoremDB P9644",
"url": "https://theoremdb.org/statements/P9644",
"note": null,
"sort_order": 0
},
{
"kind": "formalization",
"citation": "Source formal statement (not locally verified)",
"url": "https://theoremdb.org/statements/P9644",
"not... | {
"source_record": "P9644",
"imported_at": "2026-09-08",
"status_as_of": "2026-09-08",
"evidence_label": "Source candidate · not reviewed",
"note": "Full public directory statement. Source lane: backlog; source resolution: open. Source verification claims are attributed only; no local Lean verification has be... | {"created_at":"2026-08-17T20:00:24+00:00","display_standard":{"editorial_review":"submitter_supplied","formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":9644,"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": "2026-09-08T03:32:56.553Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"8fba0a3d-1970-40e1-81cb-12038233a789","format_issues":null,"number":4765} |
1.0.0 | problem:theoremdb-p9646 | problem | theoremdb-p9646 | Existence of large restricted-sum-avoiding subsets | {
"main": "math",
"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 finite set $A \subset \mathbb{Z}$, define $M(A)$ to be the maximum cardinality of a subset $S \subseteq A$ such that the restricted sumset $S \hat{+} S$ is disjoint from $A$, where the restricted sumset is defined as $S \hat{+} S = \{a + b : a, b \in S,\ a \neq b\}$. Does there exist, for all sufficiently large p... | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P9646",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P9646",
"citation": null,
"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_2 : answer(sorry) ↔
∀ᶠ n : ℕ in atTop, ∀ A : Finset ℤ, A.card = n →
(maxRestrictedSumAvoidingSubsetSize A : ℝ) ≥ (Real.log n) ^ 100 | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem green_2 : answer(sorry) ↔\n ∀ᶠ n : ℕ in atTop, ∀ A : Finset ℤ, A.card = n →\n (maxRestrictedSumAvoidingSubsetSize A : ℝ) ≥ (Real.log n) ^ 100",
"target": null,
"source_url": "https://theoremdb.org/statements/P9646",
"format": "declaration-or-fragm... | [] | TheoremDB P9646 · Candidate awaiting source review | null | Source-reported assessment, retrieved 8 September 2026. Frontier Science has not independently reviewed this status.
Open. No signed Lean verification is attached to this problem or its selected resolution.
This record is in the source backlog; its novelty and unresolved status have not passed source review. | null | published | incremental | hard | [
"source-candidate",
"theoremdb"
] | null | null | null | null | [
{
"kind": "website",
"citation": "TheoremDB P9646",
"url": "https://theoremdb.org/statements/P9646",
"note": null,
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"citation": "Source formal statement (not locally verified)",
"url": "https://theoremdb.org/statements/P9646",
"not... | {
"source_record": "P9646",
"imported_at": "2026-09-08",
"status_as_of": "2026-09-08",
"evidence_label": "Source candidate · not reviewed",
"note": "Full public directory statement. Source lane: backlog; source resolution: open. Source verification claims are attributed only; no local Lean verification has be... | {"created_at":"2026-08-17T20:00:25+00:00","display_standard":{"editorial_review":"submitter_supplied","formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":9646,"public_status":{"evidence":{"hasFormalization":false,"hasInformalProof":false,"hasRecordedWork":false,"solutionRecord":null},"lean":{... | [
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"source-full-text-not-present"
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"created_at": "2026-09-08T03:32:15.935Z",
"updated_at": "2026-09-08T03:32:15.935Z",
"published_at": "2026-09-08T03:32:58.769Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"4cdffee8-6a70-415c-b0d7-d0a296d91edd","format_issues":null,"number":5251} |
1.0.0 | problem:theoremdb-p9648 | problem | theoremdb-p9648 | Gilbreath's conjecture on iterated absolute differences of primes | {
"main": "math",
"sub": "number theory",
"slug": "number-theory",
"name": "Number theory",
"original_discipline": "mathematics"
} | [] | [
{
"code": "11",
"basis": "community-map",
"evidence": "Inferred from community field: number-theory"
}
] | Let $p_n$ denote the $n$th prime number, with $p_0 = 2$. Define the sequence of iterated absolute differences by setting $d^0(n) = p_n$ for all $n \geq 0$, and for each integer $k \geq 0$ and $n \geq 0$, defining
$$
d^{k+1}(n) = \bigl|d^k(n+1) - d^k(n)\bigr|.
$$
For each positive integer $k$, is $d^k(0) = 1$? Equival... | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P9648",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P9648",
"citation": null,
"doi": null,
"arxiv_id": null,
"license": "Selected packets label their content CC0-1.0; original references retain their own terms",
"notes": "Selected problem... | theorem gilbreath_conjecture (k : ℕ+) : d k 0 = 1 | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem gilbreath_conjecture (k : ℕ+) : d k 0 = 1",
"target": null,
"source_url": "https://theoremdb.org/statements/P9648",
"format": "declaration-or-fragment",
"sha256": "e9ad07866d8e9d8e60a61c7554b4270671389b9b65c503b5c23ca9c1a33ff295",
"license": "Sele... | [] | TheoremDB P9648 · Candidate awaiting source review | null | Source-reported assessment, retrieved 8 September 2026. Frontier Science has not independently reviewed this status.
Open. No signed Lean verification is attached to this problem or its selected resolution.
This record is in the source backlog; its novelty and unresolved status have not passed source review. | null | published | incremental | hard | [
"source-candidate",
"theoremdb"
] | null | null | null | null | [
{
"kind": "website",
"citation": "TheoremDB P9648",
"url": "https://theoremdb.org/statements/P9648",
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"url": "https://theoremdb.org/statements/P9648",
"not... | {
"source_record": "P9648",
"imported_at": "2026-09-08",
"status_as_of": "2026-09-08",
"evidence_label": "Source candidate · not reviewed",
"note": "Full public directory statement. Source lane: backlog; source resolution: open. Source verification claims are attributed only; no local Lean verification has be... | {"created_at":"2026-08-17T20:00:28+00:00","display_standard":{"editorial_review":"submitter_supplied","formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":9648,"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": "2026-09-08T03:32:59.322Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"8969d848-f7f7-40b7-acfe-459313170c12","format_issues":null,"number":5370} |
1.0.0 | problem:theoremdb-p9650 | problem | theoremdb-p9650 | Infinitely many binomial coefficients as products of consecutive primes for all admissible lower indices | {
"main": "math",
"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 $k \ge 2$, do there exist infinitely many integers $n$ with $k \le n - 2$ such that the binomial coefficient $\binom{n}{k}$ equals the product of consecutive prime numbers? Here, $\binom{n}{k}$ denotes the number of $k$-element subsets of an $n$-element set, and the product of consecutive primes means ... | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P9650",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P9650",
"citation": null,
"doi": null,
"arxiv_id": null,
"license": "Selected packets label their content CC0-1.0; original references retain their own terms",
"notes": "Selected problem... | theorem erdos_386.variants.forall :
answer(sorry) ↔ ∀ k ≥ 2, ∃ᶠ n in .atTop,
k ≤ n - 2 ∧ ∃ p q : ℕ, n.choose k = ∏ i ∈ .Ico p q, nth Nat.Prime i | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem erdos_386.variants.forall :\n answer(sorry) ↔ ∀ k ≥ 2, ∃ᶠ n in .atTop,\n k ≤ n - 2 ∧ ∃ p q : ℕ, n.choose k = ∏ i ∈ .Ico p q, nth Nat.Prime i",
"target": null,
"source_url": "https://theoremdb.org/statements/P9650",
"format": "declaration-or-fragme... | [] | TheoremDB P9650 · Candidate awaiting source review | null | Source-reported assessment, retrieved 8 September 2026. Frontier Science has not independently reviewed this status.
Open. No signed Lean verification is attached to this problem or its selected resolution.
This record is in the source backlog; its novelty and unresolved status have not passed source review. | null | published | incremental | hard | [
"source-candidate",
"theoremdb"
] | null | null | null | null | [
{
"kind": "website",
"citation": "TheoremDB P9650",
"url": "https://theoremdb.org/statements/P9650",
"note": null,
"sort_order": 0
},
{
"kind": "formalization",
"citation": "Source formal statement (not locally verified)",
"url": "https://theoremdb.org/statements/P9650",
"not... | {
"source_record": "P9650",
"imported_at": "2026-09-08",
"status_as_of": "2026-09-08",
"evidence_label": "Source candidate · not reviewed",
"note": "Full public directory statement. Source lane: backlog; source resolution: open. Source verification claims are attributed only; no local Lean verification has be... | {"created_at":"2026-08-17T20:00:30+00:00","display_standard":{"formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":9650,"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": "2026-09-08T03:33:00.833Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"b798050a-f251-4b3b-a0e4-2ccd1aece3f5","format_issues":null,"number":5600} |
1.0.0 | problem:theoremdb-p9652 | problem | theoremdb-p9652 | Infinitude of sums of coprime powerful numbers | {
"main": "math",
"sub": "number theory",
"slug": "number-theory",
"name": "Number theory",
"original_discipline": "mathematics"
} | [] | [
{
"code": "11",
"basis": "community-map",
"evidence": "Inferred from community field: number-theory"
}
] | For an integer $r \geq 1$, a natural number $n$ is called $r$\textit{-powerful} if every prime factor of $n$ appears with exponent at least $r$ in the prime factorization of $n$; that is, if $p$ is prime and $p \mid n$, then $p^r \mid n$. For $r \geq 4$, let $\mathcal{S}_r$ denote the collection of all finite sets $S$ ... | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P9652",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P9652",
"citation": null,
"doi": null,
"arxiv_id": null,
"license": "Selected packets label their content CC0-1.0; original references retain their own terms",
"notes": "Selected problem... | theorem erdos_939.variants.infinite : answer(sorry) ↔ ∀ r ≥ 4, (Erdos939Sums r).Infinite | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem erdos_939.variants.infinite : answer(sorry) ↔ ∀ r ≥ 4, (Erdos939Sums r).Infinite",
"target": null,
"source_url": "https://theoremdb.org/statements/P9652",
"format": "declaration-or-fragment",
"sha256": "19adf3a36f3485ba8bd91f5a12561630bd3aa44fd34ce88b... | [] | TheoremDB P9652 · Candidate awaiting source review | null | Source-reported assessment, retrieved 8 September 2026. Frontier Science has not independently reviewed this status.
Open. No signed Lean verification is attached to this problem or its selected resolution.
This record is in the source backlog; its novelty and unresolved status have not passed source review. | null | published | incremental | hard | [
"source-candidate",
"theoremdb"
] | null | null | null | null | [
{
"kind": "website",
"citation": "TheoremDB P9652",
"url": "https://theoremdb.org/statements/P9652",
"note": null,
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"kind": "formalization",
"citation": "Source formal statement (not locally verified)",
"url": "https://theoremdb.org/statements/P9652",
"not... | {
"source_record": "P9652",
"imported_at": "2026-09-08",
"status_as_of": "2026-09-08",
"evidence_label": "Source candidate · not reviewed",
"note": "Full public directory statement. Source lane: backlog; source resolution: open. Source verification claims are attributed only; no local Lean verification has be... | {"created_at":"2026-08-17T20:00:32+00:00","display_standard":{"editorial_review":"submitter_supplied","formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":9652,"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": "2026-09-08T03:33:01.266Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"8225b5c3-bc1b-4549-a074-d8f541212607","format_issues":null,"number":5693} |
1.0.0 | problem:theoremdb-p9654 | problem | theoremdb-p9654 | Non-existence of monochromatic quantum graphs with ten vertices and five colours | {
"main": "math",
"sub": "combinatorics",
"slug": "combinatorics",
"name": "Combinatorics",
"original_discipline": "mathematics"
} | [] | [
{
"code": "05",
"basis": "community-map",
"evidence": "Inferred from community field: combinatorics"
}
] | Let $N = 10$ and $D = 5$. Consider a complete graph on $N$ vertices whose edges are coloured with $D$ colours and assigned complex weights. For each vertex colouring $\iota : \{1, \ldots, N\} \to \{1, \ldots, D\}$, define the perfect-matching sum $S(\iota)$ to be the sum over all perfect matchings of the complete graph... | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P9654",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P9654",
"citation": null,
"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_d5 :
answer(sorry) ↔
¬ ∃ W : WeightsN 10 5 ℂ, EqSystemN 10 5 W | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem eqSystem10_no_solution_d5 :\n answer(sorry) ↔\n ¬ ∃ W : WeightsN 10 5 ℂ, EqSystemN 10 5 W",
"target": null,
"source_url": "https://theoremdb.org/statements/P9654",
"format": "declaration-or-fragment",
"sha256": "6c406124f69549eb564ab49b4733ac9... | [] | TheoremDB P9654 · Candidate awaiting source review | null | Source-reported assessment, retrieved 8 September 2026. Frontier Science has not independently reviewed this status.
Open. No signed Lean verification is attached to this problem or its selected resolution.
This record is in the source backlog; its novelty and unresolved status have not passed source review. | null | published | incremental | hard | [
"source-candidate",
"theoremdb"
] | null | null | null | null | [
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"kind": "website",
"citation": "TheoremDB P9654",
"url": "https://theoremdb.org/statements/P9654",
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"citation": "Source formal statement (not locally verified)",
"url": "https://theoremdb.org/statements/P9654",
"not... | {
"source_record": "P9654",
"imported_at": "2026-09-08",
"status_as_of": "2026-09-08",
"evidence_label": "Source candidate · not reviewed",
"note": "Full public directory statement. Source lane: backlog; source resolution: open. Source verification claims are attributed only; no local Lean verification has be... | {"created_at":"2026-08-17T20:00:35+00:00","display_standard":{"editorial_review":"submitter_supplied","formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":9654,"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": "2026-09-08T03:33:04.366Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"ea6545da-e52e-401a-a6b4-c4a3f88b3cac","format_issues":null,"number":6295} |
1.0.0 | problem:theoremdb-p9656 | problem | theoremdb-p9656 | Oppermann's Conjecture | {
"main": "math",
"sub": "number theory",
"slug": "number-theory",
"name": "Number theory",
"original_discipline": "mathematics"
} | [] | [
{
"code": "11",
"basis": "community-map",
"evidence": "Inferred from community field: number-theory"
}
] | For every integer $x \ge 2$, prove that there exists a prime number strictly between $x(x-1)$ and $x^2$, and there exists a prime number strictly between $x^2$ and $x(x+1)$. | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P9656",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P9656",
"citation": null,
"doi": null,
"arxiv_id": null,
"license": "Selected packets label their content CC0-1.0; original references retain their own terms",
"notes": "Selected problem... | theorem oppermann_conjecture (x : ℕ) (hx : 2 ≤ x) :
(∃ p ∈ Ioo (x * (x - 1)) (x^2), p.Prime) ∧
(∃ p ∈ Ioo (x^2) (x * (x + 1)), p.Prime) | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem oppermann_conjecture (x : ℕ) (hx : 2 ≤ x) :\n (∃ p ∈ Ioo (x * (x - 1)) (x^2), p.Prime) ∧\n (∃ p ∈ Ioo (x^2) (x * (x + 1)), p.Prime)",
"target": null,
"source_url": "https://theoremdb.org/statements/P9656",
"format": "declaration-or-fragment",
"s... | [] | TheoremDB P9656 · Candidate awaiting source review | null | Source-reported assessment, retrieved 8 September 2026. Frontier Science has not independently reviewed this status.
Open. No signed Lean verification is attached to this problem or its selected resolution.
This record is in the source backlog; its novelty and unresolved status have not passed source review. | null | published | incremental | hard | [
"source-candidate",
"theoremdb"
] | null | null | null | null | [
{
"kind": "website",
"citation": "TheoremDB P9656",
"url": "https://theoremdb.org/statements/P9656",
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"url": "https://theoremdb.org/statements/P9656",
"not... | {
"source_record": "P9656",
"imported_at": "2026-09-08",
"status_as_of": "2026-09-08",
"evidence_label": "Source candidate · not reviewed",
"note": "Full public directory statement. Source lane: backlog; source resolution: open. Source verification claims are attributed only; no local Lean verification has be... | {"created_at":"2026-08-17T20:00:39+00:00","display_standard":{"editorial_review":"submitter_supplied","formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":9656,"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": "2026-09-08T03:33:04.655Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"f30612ef-e3a1-446c-9674-cb5434e69af8","format_issues":null,"number":6357} |
1.0.0 | problem:theoremdb-p9658 | problem | theoremdb-p9658 | Antihydra sequence odd-even cumulative count inequality | {
"main": "math",
"sub": "combinatorics",
"slug": "combinatorics",
"name": "Combinatorics",
"original_discipline": "mathematics"
} | [] | [
{
"code": "05",
"basis": "community-map",
"evidence": "Inferred from community field: combinatorics"
}
] | Define the sequence $(a_n)_{n \ge 0}$ of natural numbers by $a_0 = 8$ and $a_{n+1} = \left\lfloor \frac{3a_n}{2} \right\rfloor$ for all $n \ge 0$. For any non-negative integer $n$, let $O_n$ denote the number of indices $i$ with $0 \le i < n$ such that $a_i$ is odd, and let $E_n$ denote the number of indices $i$ with $... | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P9658",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P9658",
"citation": null,
"doi": null,
"arxiv_id": null,
"license": "Selected packets label their content CC0-1.0; original references retain their own terms",
"notes": "Selected problem... | theorem beaver_math_olympiad_problem_2_antihydra.variants.set
(a : ℕ → ℕ) (a_ini : a 0 = 8)
(a_rec : ∀ n, a (n + 1) = (3 * a n) / 2) (n : ℕ) :
((Finset.Ico 0 n).filter fun x ↦ Odd (a x)).card ≤
2 * ((Finset.Ico 0 n).filter fun x ↦ Even (a x)).card | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem beaver_math_olympiad_problem_2_antihydra.variants.set\n (a : ℕ → ℕ) (a_ini : a 0 = 8)\n (a_rec : ∀ n, a (n + 1) = (3 * a n) / 2) (n : ℕ) :\n ((Finset.Ico 0 n).filter fun x ↦ Odd (a x)).card ≤\n 2 * ((Finset.Ico 0 n).filter fun x ↦ Even (a x)).card",
... | [] | TheoremDB P9658 · Candidate awaiting source review | null | Source-reported assessment, retrieved 8 September 2026. Frontier Science has not independently reviewed this status.
Open. No signed Lean verification is attached to this problem or its selected resolution.
This record is in 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 P9658",
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"citation": "Source formal statement (not locally verified)",
"url": "https://theoremdb.org/statements/P9658",
"not... | {
"source_record": "P9658",
"imported_at": "2026-09-08",
"status_as_of": "2026-09-08",
"evidence_label": "Source candidate · not reviewed",
"note": "Full public directory statement. Source lane: backlog; source resolution: open. Source verification claims are attributed only; no local Lean verification has be... | {"created_at":"2026-08-17T20:00:43+00:00","display_standard":{"editorial_review":"submitter_supplied","formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":9658,"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": "2026-09-08T03:32:49.650Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"5a547568-81e4-4552-8f6e-cb570493b7cf","format_issues":null,"number":3254} |
1.0.0 | problem:theoremdb-p9660 | problem | theoremdb-p9660 | Zero area of the Multibrot set boundary | {
"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 each natural number $n$, the multibrot set of power $n$ is defined as the set of all complex parameters $c \in \mathbb{C}$ 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 $c \in \mathbb{C}$ such that $\|(z \mapsto z^n + c)^{[k... | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P9660",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P9660",
"citation": null,
"doi": null,
"arxiv_id": null,
"license": "Selected packets label their content CC0-1.0; original references retain their own terms",
"notes": "Selected problem... | theorem volume_frontier_multibrotSet_eq_zero {n : ℕ} : volume (frontier (multibrotSet n)) = 0 | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem volume_frontier_multibrotSet_eq_zero {n : ℕ} : volume (frontier (multibrotSet n)) = 0",
"target": null,
"source_url": "https://theoremdb.org/statements/P9660",
"format": "declaration-or-fragment",
"sha256": "c94130ac891581809d7e1212e8f9526f04bcab98a62... | [] | TheoremDB P9660 · Candidate awaiting source review | null | Source-reported assessment, retrieved 8 September 2026. Frontier Science has not independently reviewed this status.
Open. No signed Lean verification is attached to this problem or its selected resolution.
This record is in the source backlog; its novelty and unresolved status have not passed source review. | null | published | incremental | hard | [
"source-candidate",
"theoremdb"
] | null | null | null | null | [
{
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"url": "https://theoremdb.org/statements/P9660",
"not... | {
"source_record": "P9660",
"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":"2026-08-17T20:00:46+00:00","display_standard":{"editorial_review":"submitter_supplied","formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":9660,"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": "2026-09-08T03:33:07.617Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"84676d1c-7775-4fe4-afdf-b1e2399fd78c","format_issues":null,"number":6998} |
1.0.0 | problem:theoremdb-p9662 | problem | theoremdb-p9662 | Polynomial upper bound for pyjama set covering numbers | {
"main": "math",
"sub": "geometry and topology",
"slug": "geometry-and-topology",
"name": "Geometry and topology",
"original_discipline": "mathematics"
} | [] | [] | For a real number $\varepsilon > 0$, the pyjama set of width $\varepsilon$ is the set of points $(x,y) \in \mathbb{R}^2$ such that the distance from $x$ to the nearest integer is at most $\varepsilon$. For each $\varepsilon > 0$, let $N(\varepsilon)$ denote the minimum number of rotated copies of the pyjama set of widt... | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P9662",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P9662",
"citation": null,
"doi": null,
"arxiv_id": null,
"license": "Selected packets label their content CC0-1.0; original references retain their own terms",
"notes": "Selected problem... | theorem green_41.variants.polynomial_bound : answer(sorry) ↔
∃ C : ℝ, ∃ ε₀ > 0, ∀ ε ∈ Ioc 0 ε₀, (minCopies ε : ℝ) ≤ ε ^ (-C) | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem green_41.variants.polynomial_bound : answer(sorry) ↔\n ∃ C : ℝ, ∃ ε₀ > 0, ∀ ε ∈ Ioc 0 ε₀, (minCopies ε : ℝ) ≤ ε ^ (-C)",
"target": null,
"source_url": "https://theoremdb.org/statements/P9662",
"format": "declaration-or-fragment",
"sha256": "802d184... | [] | TheoremDB P9662 · Candidate awaiting source review | null | Source-reported assessment, retrieved 8 September 2026. Frontier Science has not independently reviewed this status.
Open. No signed Lean verification is attached to this problem or its selected resolution.
This record is in the source backlog; its novelty and unresolved status have not passed source review. | null | published | incremental | hard | [
"source-candidate",
"theoremdb"
] | null | null | null | null | [
{
"kind": "website",
"citation": "TheoremDB P9662",
"url": "https://theoremdb.org/statements/P9662",
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"citation": "Source formal statement (not locally verified)",
"url": "https://theoremdb.org/statements/P9662",
"not... | {
"source_record": "P9662",
"imported_at": "2026-09-08",
"status_as_of": "2026-09-08",
"evidence_label": "Source candidate · not reviewed",
"note": "Full public directory statement. Source lane: backlog; source resolution: open. Source verification claims are attributed only; no local Lean verification has be... | {"created_at":"2026-08-17T20:00:49+00:00","display_standard":{"formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":9662,"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": "2026-09-08T03:33:04.911Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"65cce5b3-b74f-4443-8ea1-ce198df1b0d9","format_issues":null,"number":6413} |
1.0.0 | problem:theoremdb-p9664 | problem | theoremdb-p9664 | Exact value of the Bloch constant | {
"main": "math",
"sub": "analysis",
"slug": "analysis",
"name": "Analysis",
"original_discipline": "mathematics"
} | [] | [] | Let the Bloch constant $B$ be defined as the supremum of all real numbers $B'$ such that for every holomorphic function $f$ on the open unit disk $\{z \in \mathbb{C} : |z| < 1\}$ satisfying $f'(0) = 1$, there exists a subset $S$ of the unit disk and a point $x \in \mathbb{C}$ such that the open disk centered at $x$ wit... | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P9664",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P9664",
"citation": null,
"doi": null,
"arxiv_id": null,
"license": "Selected packets label their content CC0-1.0; original references retain their own terms",
"notes": "Selected problem... | theorem blochConstant_exact_value :
blochConstant = Real.Gamma (1 / 3) * Real.Gamma (11 / 12) /
(Real.Gamma (1 / 4) * Real.sqrt (1 + Real.sqrt 3)) | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem blochConstant_exact_value :\n blochConstant = Real.Gamma (1 / 3) * Real.Gamma (11 / 12) /\n (Real.Gamma (1 / 4) * Real.sqrt (1 + Real.sqrt 3))",
"target": null,
"source_url": "https://theoremdb.org/statements/P9664",
"format": "declaration-or-fragme... | [] | TheoremDB P9664 · Candidate awaiting source review | null | Source-reported assessment, retrieved 8 September 2026. Frontier Science has not independently reviewed this status.
Open. No signed Lean verification is attached to this problem or its selected resolution.
This record is in the source backlog; its novelty and unresolved status have not passed source review. | null | published | incremental | hard | [
"source-candidate",
"theoremdb"
] | null | null | null | null | [
{
"kind": "website",
"citation": "TheoremDB P9664",
"url": "https://theoremdb.org/statements/P9664",
"note": null,
"sort_order": 0
},
{
"kind": "formalization",
"citation": "Source formal statement (not locally verified)",
"url": "https://theoremdb.org/statements/P9664",
"not... | {
"source_record": "P9664",
"imported_at": "2026-09-08",
"status_as_of": "2026-09-08",
"evidence_label": "Source candidate · not reviewed",
"note": "Full public directory statement. Source lane: backlog; source resolution: open. Source verification claims are attributed only; no local Lean verification has be... | {"created_at":"2026-08-17T20:00:52+00:00","display_standard":{"editorial_review":"submitter_supplied","formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":9664,"public_status":{"evidence":{"hasFormalization":false,"hasInformalProof":false,"hasRecordedWork":false,"solutionRecord":null},"lean":{... | [
"formalization-not-compiled-during-export",
"current-status-unassessed",
"source-full-text-not-present"
] | [] | {
"created_at": "2026-09-08T03:32:15.935Z",
"updated_at": "2026-09-08T03:32:15.935Z",
"published_at": "2026-09-08T03:32:58.340Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"4ef19a66-1cbf-409c-81b3-6c6c509f451c","format_issues":null,"number":5163} |
1.0.0 | problem:theoremdb-p9666 | problem | theoremdb-p9666 | Erdős Problem 951 on Beurling Primes and the Prime Counting Function | {
"main": "math",
"sub": "number theory",
"slug": "number-theory",
"name": "Number theory",
"original_discipline": "mathematics"
} | [] | [
{
"code": "11",
"basis": "community-map",
"evidence": "Inferred from community field: number-theory"
}
] | For a sequence $a : \mathbb{N} \to \mathbb{R}$ with $a_0 > 1$ that is strictly increasing, we say that $a$ has the Erdős 951 property if for any two distinct finitely supported functions $k, \ell : \mathbb{N} \to \mathbb{N}$ with finite support, the corresponding Beurling integers satisfy $|B_a(k) - B_a(\ell)| \geq 1$,... | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P9666",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P9666",
"citation": null,
"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_951 : answer(sorry) ↔
∀ a : ℕ → ℝ, 1 < a 0 → StrictMono a → Erdos951Prop a →
∀ᶠ (x : ℝ) in Filter.atTop, {i : ℕ | a i ≤ x}.ncard ≤ π ⌊x⌋₊ | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem erdos_951 : answer(sorry) ↔\n ∀ a : ℕ → ℝ, 1 < a 0 → StrictMono a → Erdos951Prop a →\n ∀ᶠ (x : ℝ) in Filter.atTop, {i : ℕ | a i ≤ x}.ncard ≤ π ⌊x⌋₊",
"target": null,
"source_url": "https://theoremdb.org/statements/P9666",
"format": "declaration-or... | [] | TheoremDB P9666 · Candidate awaiting source review | null | Source-reported assessment, retrieved 8 September 2026. Frontier Science has not independently reviewed this status.
Open. No signed Lean verification is attached to this problem or its selected resolution.
This record is in the source backlog; its novelty and unresolved status have not passed source review. | null | published | incremental | hard | [
"source-candidate",
"theoremdb"
] | null | null | null | null | [
{
"kind": "website",
"citation": "TheoremDB P9666",
"url": "https://theoremdb.org/statements/P9666",
"note": null,
"sort_order": 0
},
{
"kind": "formalization",
"citation": "Source formal statement (not locally verified)",
"url": "https://theoremdb.org/statements/P9666",
"not... | {
"source_record": "P9666",
"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":"2026-08-17T20:00:56+00:00","display_standard":{"formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":9666,"public_status":{"evidence":{"hasFormalization":false,"hasInformalProof":false,"hasRecordedWork":false,"solutionRecord":null},"lean":{"kernelAccepted":false,"label":"","reaso... | [
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"created_at": "2026-09-08T03:32:15.935Z",
"updated_at": "2026-09-08T03:32:15.935Z",
"published_at": "2026-09-08T03:32:56.241Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"613bd265-0118-4ae1-bc59-c73bb653ed08","format_issues":null,"number":4698} |
1.0.0 | problem:theoremdb-p9668 | problem | theoremdb-p9668 | Tao's optimization constant 1a | {
"main": "math",
"sub": "combinatorics",
"slug": "combinatorics",
"name": "Combinatorics",
"original_discipline": "mathematics"
} | [] | [
{
"code": "05",
"basis": "community-map",
"evidence": "Inferred from community field: combinatorics"
}
] | Let $C_{1a}$ denote the supremum of all real numbers $C$ such that for every non-negative function $f : \mathbb{R} \to \mathbb{R}$, the inequality
$$
C \left(\int_{-1/4}^{1/4} f(x) \, dx\right)^2 \leq \sup_{t \in [1/2, 1]} \int_{\mathbb{R}} f(t-x) f(x) \, dx
$$
holds. Determine a real number that lies in the interval... | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P9668",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P9668",
"citation": null,
"doi": null,
"arxiv_id": null,
"license": "Selected packets label their content CC0-1.0; original references retain their own terms",
"notes": "Selected problem... | theorem mem_Ioc_c1a : answer(sorry) ∈ Set.Ioc 1.2748 C1a | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem mem_Ioc_c1a : answer(sorry) ∈ Set.Ioc 1.2748 C1a",
"target": null,
"source_url": "https://theoremdb.org/statements/P9668",
"format": "declaration-or-fragment",
"sha256": "d898ae80ae78fcbf8f25b54f3cb4dceacddc2e26e6d4ccbc79e0d79d7655a2a8",
"license"... | [] | TheoremDB P9668 · Candidate awaiting source review | null | Source-reported assessment, retrieved 8 September 2026. Frontier Science has not independently reviewed this status.
Open. No signed Lean verification is attached to this problem or its selected resolution.
This record is in the source backlog; its novelty and unresolved status have not passed source review. | null | published | incremental | hard | [
"source-candidate",
"theoremdb"
] | null | null | null | null | [
{
"kind": "website",
"citation": "TheoremDB P9668",
"url": "https://theoremdb.org/statements/P9668",
"note": null,
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},
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"citation": "Source formal statement (not locally verified)",
"url": "https://theoremdb.org/statements/P9668",
"not... | {
"source_record": "P9668",
"imported_at": "2026-09-08",
"status_as_of": "2026-09-08",
"evidence_label": "Source candidate · not reviewed",
"note": "Full public directory statement. Source lane: backlog; source resolution: open. Source verification claims are attributed only; no local Lean verification has be... | {"created_at":"2026-08-17T20:00:58+00:00","display_standard":{"formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":9668,"public_status":{"evidence":{"hasFormalization":false,"hasInformalProof":false,"hasRecordedWork":false,"solutionRecord":null},"lean":{"kernelAccepted":false,"label":"","reaso... | [
"formalization-contains-placeholder",
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"created_at": "2026-09-08T03:32:15.935Z",
"updated_at": "2026-09-08T03:32:15.935Z",
"published_at": "2026-09-08T03:33:06.269Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"4fb09604-9eb6-4c7d-94fc-aa76f1c125c7","format_issues":null,"number":6707} |
1.0.0 | problem:theoremdb-p9670 | problem | theoremdb-p9670 | Asymptotic threshold for restricted sumset coverage 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 $(k,N)$ has Property 25 if for every partition $[N] = A_1 \cup \cdots \cup A_k$ of the set $\{1,2,\dots,N\}$ into exactly $k$ nonempty parts, the union of the restricted sumsets satisfies
$$
\left|\bigcup_{i=1}^{k} (A_i \hat{+} A_i)\right| \geq \frac{1... | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P9670",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P9670",
"citation": null,
"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 : {k : ℕ → ℕ | ∀ᶠ N in atTop, Property25 (k N) N} = answer(sorry) | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem green_25 : {k : ℕ → ℕ | ∀ᶠ N in atTop, Property25 (k N) N} = answer(sorry)",
"target": null,
"source_url": "https://theoremdb.org/statements/P9670",
"format": "declaration-or-fragment",
"sha256": "4e07e60efd39df3c60758dad8374dcd30c7bb9ad0acee367a4a767... | [] | TheoremDB P9670 · Candidate awaiting source review | null | Source-reported assessment, retrieved 8 September 2026. Frontier Science has not independently reviewed this status.
Open. No signed Lean verification is attached to this problem or its selected resolution.
This record is in the source backlog; its novelty and unresolved status have not passed source review. | null | published | incremental | hard | [
"source-candidate",
"theoremdb"
] | null | null | null | null | [
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"kind": "website",
"citation": "TheoremDB P9670",
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"url": "https://theoremdb.org/statements/P9670",
"not... | {
"source_record": "P9670",
"imported_at": "2026-09-08",
"status_as_of": "2026-09-08",
"evidence_label": "Source candidate · not reviewed",
"note": "Full public directory statement. Source lane: backlog; source resolution: open. Source verification claims are attributed only; no local Lean verification has be... | {"created_at":"2026-08-17T20:01:01+00:00","display_standard":{"editorial_review":"submitter_supplied","formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":9670,"public_status":{"evidence":{"hasFormalization":false,"hasInformalProof":false,"hasRecordedWork":false,"solutionRecord":null},"lean":{... | [
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] | [] | {
"created_at": "2026-09-08T03:32:15.935Z",
"updated_at": "2026-09-08T03:32:15.935Z",
"published_at": "2026-09-08T03:32:50.409Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"c5b38240-52f9-4466-ba7d-e0a08622ddf1","format_issues":null,"number":3414} |
1.0.0 | problem:theoremdb-p9672 | problem | theoremdb-p9672 | Non-existence of monochromatic quantum graphs with ten vertices and eight colours | {
"main": "math",
"sub": "combinatorics",
"slug": "combinatorics",
"name": "Combinatorics",
"original_discipline": "mathematics"
} | [] | [
{
"code": "05",
"basis": "community-map",
"evidence": "Inferred from community field: combinatorics"
}
] | Let $N = 10$ and $D = 8$. Consider a complete graph on $N$ vertices whose edges are coloured with $D$ colours and assigned complex weights. For each vertex colouring $\iota$ (an assignment of one of the $D$ colours to each of the $N$ vertices), define the perfect-matching sum as the sum over all perfect matchings of th... | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P9672",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P9672",
"citation": null,
"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_d8 :
answer(sorry) ↔
¬ ∃ W : WeightsN 10 8 ℂ, EqSystemN 10 8 W | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem eqSystem10_no_solution_d8 :\n answer(sorry) ↔\n ¬ ∃ W : WeightsN 10 8 ℂ, EqSystemN 10 8 W",
"target": null,
"source_url": "https://theoremdb.org/statements/P9672",
"format": "declaration-or-fragment",
"sha256": "b8541e30b2071aacc6571ad8f7176fb... | [] | TheoremDB P9672 · Candidate awaiting source review | null | Source-reported assessment, retrieved 8 September 2026. Frontier Science has not independently reviewed this status.
Open. No signed Lean verification is attached to this problem or its selected resolution.
This record is in 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 P9672",
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"citation": "Source formal statement (not locally verified)",
"url": "https://theoremdb.org/statements/P9672",
"not... | {
"source_record": "P9672",
"imported_at": "2026-09-08",
"status_as_of": "2026-09-08",
"evidence_label": "Source candidate · not reviewed",
"note": "Full public directory statement. Source lane: backlog; source resolution: open. Source verification claims are attributed only; no local Lean verification has be... | {"created_at":"2026-08-17T20:01:03+00:00","display_standard":{"editorial_review":"submitter_supplied","formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":9672,"public_status":{"evidence":{"hasFormalization":false,"hasInformalProof":false,"hasRecordedWork":false,"solutionRecord":null},"lean":{... | [
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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": "2026-09-08T03:33:04.361Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"764ea0e5-9bf6-4b88-9823-5ddbb63ff918","format_issues":null,"number":6294} |
1.0.0 | problem:theoremdb-p9674 | problem | theoremdb-p9674 | Asymptotic growth 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}{\gcd(a,b)} : a, b \in A \right\}
$$
has cardinality at least $m$. Determine an explicit function $f : \mathbb{N} \to \mathbb{R}$ such that $... | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P9674",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P9674",
"citation": null,
"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 :
(fun n ↦ (cofactorThreshold n : ℝ)) =Θ[atTop] (answer(sorry) : ℕ → ℝ) | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem erdos_539 :\n (fun n ↦ (cofactorThreshold n : ℝ)) =Θ[atTop] (answer(sorry) : ℕ → ℝ)",
"target": null,
"source_url": "https://theoremdb.org/statements/P9674",
"format": "declaration-or-fragment",
"sha256": "9d036850c5f89c76cfb76f569c7fdf5dc23887b43e... | [] | TheoremDB P9674 · Candidate awaiting source review | null | Source-reported assessment, retrieved 8 September 2026. Frontier Science has not independently reviewed this status.
Open. No signed Lean verification is attached to this problem or its selected resolution.
This record is in the source backlog; its novelty and unresolved status have not passed source review. | null | published | incremental | hard | [
"source-candidate",
"theoremdb"
] | null | null | null | null | [
{
"kind": "website",
"citation": "TheoremDB P9674",
"url": "https://theoremdb.org/statements/P9674",
"note": null,
"sort_order": 0
},
{
"kind": "formalization",
"citation": "Source formal statement (not locally verified)",
"url": "https://theoremdb.org/statements/P9674",
"not... | {
"source_record": "P9674",
"imported_at": "2026-09-08",
"status_as_of": "2026-09-08",
"evidence_label": "Source candidate · not reviewed",
"note": "Full public directory statement. Source lane: backlog; source resolution: open. Source verification claims are attributed only; no local Lean verification has be... | {"created_at":"2026-08-17T20:01:06+00:00","display_standard":{"editorial_review":"submitter_supplied","formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":9674,"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": "2026-09-08T03:32:50.158Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"e17e5e39-c9a9-4c30-baee-94aea4adf7af","format_issues":null,"number":3363} |
1.0.0 | problem:theoremdb-p9676 | problem | theoremdb-p9676 | Selfridge's conjecture on the monotonicity of Fermat number prime factors | {
"main": "math",
"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$, let $F_n = 2^{2^n} + 1$ denote the $n$-th Fermat number, and let $f(n)$ denote the number of distinct prime factors of $F_n$. Determine whether the sequence $(f(n))_{n \geq 0}$ is monotone non-decreasing, or prove that it is not. | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P9676",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P9676",
"citation": null,
"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_seq_conjecture : ¬ Monotone fermatFactors | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem selfridge_seq_conjecture : ¬ Monotone fermatFactors",
"target": null,
"source_url": "https://theoremdb.org/statements/P9676",
"format": "declaration-or-fragment",
"sha256": "04b34f51e7e1bea392617f96a57c0de106b027d6d53edb7d779c8cb3fd177f4f",
"licen... | [] | TheoremDB P9676 · Candidate awaiting source review | null | Source-reported assessment, retrieved 8 September 2026. Frontier Science has not independently reviewed this status.
Open. No signed Lean verification is attached to this problem or its selected resolution.
This record is in the source backlog; its novelty and unresolved status have not passed source review. | null | published | incremental | hard | [
"source-candidate",
"theoremdb"
] | null | null | null | null | [
{
"kind": "website",
"citation": "TheoremDB P9676",
"url": "https://theoremdb.org/statements/P9676",
"note": null,
"sort_order": 0
},
{
"kind": "formalization",
"citation": "Source formal statement (not locally verified)",
"url": "https://theoremdb.org/statements/P9676",
"not... | {
"source_record": "P9676",
"imported_at": "2026-09-08",
"status_as_of": "2026-09-08",
"evidence_label": "Source candidate · not reviewed",
"note": "Full public directory statement. Source lane: backlog; source resolution: open. Source verification claims are attributed only; no local Lean verification has be... | {"created_at":"2026-08-17T20:01:10+00:00","display_standard":{"editorial_review":"submitter_supplied","formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":9676,"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": "2026-09-08T03:33:05.760Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"14cba151-8eb3-4b84-94f4-7b645b16ddb2","format_issues":null,"number":6597} |
1.0.0 | problem:theoremdb-p9678 | problem | theoremdb-p9678 | Bounds for the monochromatic sum-product Ramsey threshold | {
"main": "math",
"sub": "combinatorics",
"slug": "combinatorics",
"name": "Combinatorics",
"original_discipline": "mathematics"
} | [] | [
{
"code": "05",
"basis": "community-map",
"evidence": "Inferred from community field: combinatorics"
}
] | For each positive integer $r$, let $N_0(r)$ denote the smallest positive integer $N$ such that every $r$-colouring of $\{1, 2, \ldots, N\}$ contains integers $x, y \geq 3$ for which $x + y$ and $xy$ both belong to $\{1, 2, \ldots, N\}$ and receive the same colour. The existence of $N_0(r)$ for all $r$ follows from a th... | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P9678",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P9678",
"citation": null,
"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_22 :
let ans := (answer(sorry) : ℕ → ℝ)
∀ᶠ r in atTop, N₀ r ≤ ans r ∧
ans =o[atTop] GreenSawhneyBound | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem green_22 :\n let ans := (answer(sorry) : ℕ → ℝ)\n ∀ᶠ r in atTop, N₀ r ≤ ans r ∧\n ans =o[atTop] GreenSawhneyBound",
"target": null,
"source_url": "https://theoremdb.org/statements/P9678",
"format": "declaration-or-fragment",
"sha256": "1e2832... | [] | TheoremDB P9678 · Candidate awaiting source review | null | Source-reported assessment, retrieved 8 September 2026. Frontier Science has not independently reviewed this status.
Open. No signed Lean verification is attached to this problem or its selected resolution.
This record is in the source backlog; its novelty and unresolved status have not passed source review. | null | published | incremental | hard | [
"source-candidate",
"theoremdb"
] | null | null | null | null | [
{
"kind": "website",
"citation": "TheoremDB P9678",
"url": "https://theoremdb.org/statements/P9678",
"note": null,
"sort_order": 0
},
{
"kind": "formalization",
"citation": "Source formal statement (not locally verified)",
"url": "https://theoremdb.org/statements/P9678",
"not... | {
"source_record": "P9678",
"imported_at": "2026-09-08",
"status_as_of": "2026-09-08",
"evidence_label": "Source candidate · not reviewed",
"note": "Full public directory statement. Source lane: backlog; source resolution: open. Source verification claims are attributed only; no local Lean verification has be... | {"created_at":"2026-08-17T20:01:13+00:00","display_standard":{"editorial_review":"submitter_supplied","formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":9678,"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": "2026-09-08T03:32:50.753Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"766c5c7b-48db-44b7-8c8c-f86578d0ce99","format_issues":null,"number":3485} |
1.0.0 | problem:theoremdb-p9680 | problem | theoremdb-p9680 | Non-existence of a ternary-integer monochromatic quantum graph for six vertices and five 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. The complete graph $K_N$ on vertices $\{1,\dots,N\}$ has $\binom{N}{2}$ edges. A function $W$ assigning to each edge a value in a coefficient domain $\alpha$ is called a *weight function*. For each vertex-colouring $\iota:\{1,\dots,N\}\to\{1,\dots,D\... | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P9680",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P9680",
"citation": null,
"doi": null,
"arxiv_id": null,
"license": "Selected packets label their content CC0-1.0; original references retain their own terms",
"notes": "Selected problem... | theorem eqSystem6_no_solution_d5_trinary_int :
answer(sorry) ↔
¬ ∃ W : WeightsN 6 5 ℤ,
(∀ e, W e = (-1 : ℤ) ∨ W e = 0 ∨ W e = 1) ∧
EqSystemN 6 5 W | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem eqSystem6_no_solution_d5_trinary_int :\n answer(sorry) ↔\n ¬ ∃ W : WeightsN 6 5 ℤ,\n (∀ e, W e = (-1 : ℤ) ∨ W e = 0 ∨ W e = 1) ∧\n EqSystemN 6 5 W",
"target": null,
"source_url": "https://theoremdb.org/statements/P9680",
"forma... | [] | TheoremDB P9680 · Candidate awaiting source review | null | Source-reported assessment, retrieved 8 September 2026. Frontier Science has not independently reviewed this status.
Open. No signed Lean verification is attached to this problem or its selected resolution.
This record is in the source backlog; its novelty and unresolved status have not passed source review. | null | published | incremental | hard | [
"source-candidate",
"theoremdb"
] | null | null | null | null | [
{
"kind": "website",
"citation": "TheoremDB P9680",
"url": "https://theoremdb.org/statements/P9680",
"note": null,
"sort_order": 0
},
{
"kind": "formalization",
"citation": "Source formal statement (not locally verified)",
"url": "https://theoremdb.org/statements/P9680",
"not... | {
"source_record": "P9680",
"imported_at": "2026-09-08",
"status_as_of": "2026-09-08",
"evidence_label": "Source candidate · not reviewed",
"note": "Full public directory statement. Source lane: backlog; source resolution: open. Source verification claims are attributed only; no local Lean verification has be... | {"created_at":"2026-08-17T20:01:17+00:00","display_standard":{"editorial_review":"submitter_supplied","formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":9680,"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": "2026-09-08T03:33:04.265Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"3c1e8db4-4ae8-4f90-93f9-4e42e4021eed","format_issues":null,"number":6275} |
1.0.0 | problem:theoremdb-p9682 | problem | theoremdb-p9682 | Oppermann's Conjecture (Part I) | {
"main": "math",
"sub": "number theory",
"slug": "number-theory",
"name": "Number theory",
"original_discipline": "mathematics"
} | [] | [
{
"code": "11",
"basis": "community-map",
"evidence": "Inferred from community field: number-theory"
}
] | For every integer $x \geq 2$, prove that there exists a prime number $p$ satisfying $x(x-1) < p < x^2$. | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P9682",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P9682",
"citation": null,
"doi": null,
"arxiv_id": null,
"license": "Selected packets label their content CC0-1.0; original references retain their own terms",
"notes": "Selected problem... | theorem oppermann_conjecture.parts.i (x : ℕ) (hx : 2 ≤ x) :
∃ p ∈ Ioo (x * (x - 1)) (x^2), p.Prime | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem oppermann_conjecture.parts.i (x : ℕ) (hx : 2 ≤ x) :\n ∃ p ∈ Ioo (x * (x - 1)) (x^2), p.Prime",
"target": null,
"source_url": "https://theoremdb.org/statements/P9682",
"format": "declaration-or-fragment",
"sha256": "6cc023ab79a3cfa3b6867fc0b1410a763... | [] | TheoremDB P9682 · Candidate awaiting source review | null | Source-reported assessment, retrieved 8 September 2026. Frontier Science has not independently reviewed this status.
Open. No signed Lean verification is attached to this problem or its selected resolution.
This record is in the source backlog; its novelty and unresolved status have not passed source review. | null | published | incremental | hard | [
"source-candidate",
"theoremdb"
] | null | null | null | null | [
{
"kind": "website",
"citation": "TheoremDB P9682",
"url": "https://theoremdb.org/statements/P9682",
"note": null,
"sort_order": 0
},
{
"kind": "formalization",
"citation": "Source formal statement (not locally verified)",
"url": "https://theoremdb.org/statements/P9682",
"not... | {
"source_record": "P9682",
"imported_at": "2026-09-08",
"status_as_of": "2026-09-08",
"evidence_label": "Source candidate · not reviewed",
"note": "Full public directory statement. Source lane: backlog; source resolution: open. Source verification claims are attributed only; no local Lean verification has be... | {"created_at":"2026-08-17T20:01:21+00:00","display_standard":{"editorial_review":"submitter_supplied","formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":9682,"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": "2026-09-08T03:33:04.659Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"a150fc48-e1a8-41a8-aeea-bed14d4853db","format_issues":null,"number":6358} |
1.0.0 | problem:theoremdb-p9684 | problem | theoremdb-p9684 | Exact rank of the Elkies-Klagsbrun 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 = x^3 - 27006183241630922218434652145297453784768054621836357954737385\,x + 55258058551342376475736699591118191821521067032535079608372404779149413277716173425636721497.
$$
Assuming the generalized Riemann hypothesis, pr... | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P9684",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P9684",
"citation": null,
"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_elkiesKlagsbrun29 : finrank ℤ elkiesKlagsbrun29⟮ℚ⟯ = 29 | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem rank_elkiesKlagsbrun29 : finrank ℤ elkiesKlagsbrun29⟮ℚ⟯ = 29",
"target": null,
"source_url": "https://theoremdb.org/statements/P9684",
"format": "declaration-or-fragment",
"sha256": "f04a9b5dcaf13b6ea9e06434202a74db494de43a6eb3c866a05d584583394570",
... | [] | TheoremDB P9684 · Candidate awaiting source review | null | Source-reported assessment, retrieved 8 September 2026. Frontier Science has not independently reviewed this status.
Open. No signed Lean verification is attached to this problem or its selected resolution.
This record is in the source backlog; its novelty and unresolved status have not passed source review. | null | published | incremental | hard | [
"source-candidate",
"theoremdb"
] | null | null | null | null | [
{
"kind": "website",
"citation": "TheoremDB P9684",
"url": "https://theoremdb.org/statements/P9684",
"note": null,
"sort_order": 0
},
{
"kind": "formalization",
"citation": "Source formal statement (not locally verified)",
"url": "https://theoremdb.org/statements/P9684",
"not... | {
"source_record": "P9684",
"imported_at": "2026-09-08",
"status_as_of": "2026-09-08",
"evidence_label": "Source candidate · not reviewed",
"note": "Full public directory statement. Source lane: backlog; source resolution: open. Source verification claims are attributed only; no local Lean verification has be... | {"created_at":"2026-08-17T20:01:27+00:00","display_standard":{"formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":9684,"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": "2026-09-08T03:32:57.319Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"d7161d66-daa3-4048-a92a-51e27328e645","format_issues":null,"number":4936} |
1.0.0 | problem:theoremdb-p9686 | problem | theoremdb-p9686 | Transcendence of $e^e$ | {
"main": "math",
"sub": "number theory",
"slug": "number-theory",
"name": "Number theory",
"original_discipline": "mathematics"
} | [] | [
{
"code": "11",
"basis": "community-map",
"evidence": "Inferred from community field: number-theory"
}
] | Let $e$ denote the base of the natural logarithm, so that $e = \exp(1)$. A real number $\alpha$ is said to be transcendental over the rational numbers $\mathbb{Q}$ if there is no non-zero polynomial $P(x)$ with rational coefficients such that $P(\alpha) = 0$. Prove that the number $e^e = \exp(\exp(1))$ is transcendenta... | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P9686",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P9686",
"citation": null,
"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_exp_transcendental : Transcendental ℚ (exp (exp 1)) | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem exp_exp_transcendental : Transcendental ℚ (exp (exp 1))",
"target": null,
"source_url": "https://theoremdb.org/statements/P9686",
"format": "declaration-or-fragment",
"sha256": "68e8dabcb1dfcbb70e5e2606a9bfbd70fe2e523d12baf390709bf114be7c6096",
"l... | [] | TheoremDB P9686 · Candidate awaiting source review | null | Source-reported assessment, retrieved 8 September 2026. Frontier Science has not independently reviewed this status.
Open. No signed Lean verification is attached to this problem or its selected resolution.
This record is in the source backlog; its novelty and unresolved status have not passed source review. | null | published | incremental | hard | [
"source-candidate",
"theoremdb"
] | null | null | null | null | [
{
"kind": "website",
"citation": "TheoremDB P9686",
"url": "https://theoremdb.org/statements/P9686",
"note": null,
"sort_order": 0
},
{
"kind": "formalization",
"citation": "Source formal statement (not locally verified)",
"url": "https://theoremdb.org/statements/P9686",
"not... | {
"source_record": "P9686",
"imported_at": "2026-09-08",
"status_as_of": "2026-09-08",
"evidence_label": "Source candidate · not reviewed",
"note": "Full public directory statement. Source lane: backlog; source resolution: open. Source verification claims are attributed only; no local Lean verification has be... | {"created_at":"2026-08-17T20:01:33+00:00","display_standard":{"editorial_review":"submitter_supplied","formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":9686,"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": "2026-09-08T03:33:06.973Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"191246d6-cc11-4dfb-9f02-15d29bfb55be","format_issues":null,"number":6858} |
1.0.0 | problem:theoremdb-p9688 | problem | theoremdb-p9688 | Existence of SIC-POVMs in Every Finite Dimension | {
"main": "math",
"sub": "algebra",
"slug": "algebra",
"name": "Algebra",
"original_discipline": "mathematics"
} | [] | [] | For each positive integer $d$, let $\mathbb{C}^d$ denote the $d$-dimensional complex Hilbert space with the standard inner product. A state vector in $\mathbb{C}^d$ is a unit vector, that is, a vector $\psi \in \mathbb{C}^d$ with $\|\psi\| = 1$. The squared overlap between two state vectors $\varphi, \psi \in \mathbb{C... | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P9688",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P9688",
"citation": null,
"doi": null,
"arxiv_id": null,
"license": "Selected packets label their content CC0-1.0; original references retain their own terms",
"notes": "Selected problem... | theorem sicPOVMs :
answer(sorry) ↔ ∀ d : ℕ, 1 ≤ d → HasSICPOVM d | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem sicPOVMs :\n answer(sorry) ↔ ∀ d : ℕ, 1 ≤ d → HasSICPOVM d",
"target": null,
"source_url": "https://theoremdb.org/statements/P9688",
"format": "declaration-or-fragment",
"sha256": "116bdac3566f1b870d1c526713998ab8d2095b6bf5e448b144a671f9d85c4741",
... | [] | TheoremDB P9688 · Candidate awaiting source review | null | Source-reported assessment, retrieved 8 September 2026. Frontier Science has not independently reviewed this status.
Open. No signed Lean verification is attached to this problem or its selected resolution.
This record is in the source backlog; its novelty and unresolved status have not passed source review. | null | published | incremental | hard | [
"source-candidate",
"theoremdb"
] | null | null | null | null | [
{
"kind": "website",
"citation": "TheoremDB P9688",
"url": "https://theoremdb.org/statements/P9688",
"note": null,
"sort_order": 0
},
{
"kind": "formalization",
"citation": "Source formal statement (not locally verified)",
"url": "https://theoremdb.org/statements/P9688",
"not... | {
"source_record": "P9688",
"imported_at": "2026-09-08",
"status_as_of": "2026-09-08",
"evidence_label": "Source candidate · not reviewed",
"note": "Full public directory statement. Source lane: backlog; source resolution: open. Source verification claims are attributed only; no local Lean verification has be... | {"created_at":"2026-08-17T20:01:38+00:00","display_standard":{"editorial_review":"submitter_supplied","formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":9688,"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": "2026-09-08T03:32:58.811Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"6f8acef9-0264-4617-9d49-05cb4032bd6d","format_issues":null,"number":5260} |
1.0.0 | problem:theoremdb-p9690 | problem | theoremdb-p9690 | Lower bounds for the multiplicity of totient values | {
"main": "math",
"sub": "number theory",
"slug": "number-theory",
"name": "Number theory",
"original_discipline": "mathematics"
} | [] | [
{
"code": "11",
"basis": "community-map",
"evidence": "Inferred from community field: number-theory"
}
] | For each positive integer $n$, let $g(n)$ denote the number of positive integers $m$ such that Euler's totient function $\varphi(m)$ equals $n$. Determine whether the following statement holds: for every real number $\varepsilon > 0$, the set $\{ n \in \mathbb{N} : g(n) > n^{1-\varepsilon} \}$ is infinite. | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P9690",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P9690",
"citation": null,
"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_821 :
answer(sorry) ↔ ∀ ε > (0 : ℝ), { n : ℕ | (g n : ℝ) > (n : ℝ) ^ (1 - ε) }.Infinite | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem erdos_821 :\n answer(sorry) ↔ ∀ ε > (0 : ℝ), { n : ℕ | (g n : ℝ) > (n : ℝ) ^ (1 - ε) }.Infinite",
"target": null,
"source_url": "https://theoremdb.org/statements/P9690",
"format": "declaration-or-fragment",
"sha256": "cd3da19efea5d2172bfbe4d23034bd... | [] | TheoremDB P9690 · Candidate awaiting source review | null | Source-reported assessment, retrieved 8 September 2026. Frontier Science has not independently reviewed this status.
Open. No signed Lean verification is attached to this problem or its selected resolution.
This record is in the source backlog; its novelty and unresolved status have not passed source review. | null | published | incremental | hard | [
"source-candidate",
"theoremdb"
] | null | null | null | null | [
{
"kind": "website",
"citation": "TheoremDB P9690",
"url": "https://theoremdb.org/statements/P9690",
"note": null,
"sort_order": 0
},
{
"kind": "formalization",
"citation": "Source formal statement (not locally verified)",
"url": "https://theoremdb.org/statements/P9690",
"not... | {
"source_record": "P9690",
"imported_at": "2026-09-08",
"status_as_of": "2026-09-08",
"evidence_label": "Source candidate · not reviewed",
"note": "Full public directory statement. Source lane: backlog; source resolution: open. Source verification claims are attributed only; no local Lean verification has be... | {"created_at":"2026-08-17T20:01:48+00:00","display_standard":{"formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":9690,"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": "2026-09-08T03:33:03.587Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"edf3d133-b4c7-4eb4-8040-50f1cbcdc8b7","format_issues":null,"number":6126} |
1.0.0 | problem:theoremdb-p9692 | problem | theoremdb-p9692 | Infinitude of Wall–Sun–Sun Primes | {
"main": "math",
"sub": "number theory",
"slug": "number-theory",
"name": "Number theory",
"original_discipline": "mathematics"
} | [] | [
{
"code": "11",
"basis": "community-map",
"evidence": "Inferred from community field: number-theory"
}
] | Let the Lucas numbers be defined by $L_0 = 2$, $L_1 = 1$, and $L_n = L_{n-1} + L_{n-2}$ for $n \geq 2$. A prime $p$ is called a Wall–Sun–Sun prime if $L_p \equiv 1 \pmod{p^2}$. Prove that the set of Wall–Sun–Sun primes is infinite. | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P9692",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P9692",
"citation": null,
"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_isWallSunSunPrime : {p : ℕ | IsWallSunSunPrime p}.Infinite | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem infinite_isWallSunSunPrime : {p : ℕ | IsWallSunSunPrime p}.Infinite",
"target": null,
"source_url": "https://theoremdb.org/statements/P9692",
"format": "declaration-or-fragment",
"sha256": "cd8ef432de2697b5762d69242cd0dff0368ae1fad806840fd819641282a9a... | [] | TheoremDB P9692 · Candidate awaiting source review | null | Source-reported assessment, retrieved 8 September 2026. Frontier Science has not independently reviewed this status.
Open. No signed Lean verification is attached to this problem or its selected resolution.
This record is in the source backlog; its novelty and unresolved status have not passed source review. | null | published | incremental | hard | [
"source-candidate",
"theoremdb"
] | null | null | null | null | [
{
"kind": "website",
"citation": "TheoremDB P9692",
"url": "https://theoremdb.org/statements/P9692",
"note": null,
"sort_order": 0
},
{
"kind": "formalization",
"citation": "Source formal statement (not locally verified)",
"url": "https://theoremdb.org/statements/P9692",
"not... | {
"source_record": "P9692",
"imported_at": "2026-09-08",
"status_as_of": "2026-09-08",
"evidence_label": "Source candidate · not reviewed",
"note": "Full public directory statement. Source lane: backlog; source resolution: open. Source verification claims are attributed only; no local Lean verification has be... | {"created_at":"2026-08-17T20:01:50+00:00","display_standard":{"editorial_review":"submitter_supplied","formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":9692,"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": "2026-09-08T03:33:01.276Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"7cd8db2a-a5e0-479c-b8e5-5ff9d66296d1","format_issues":null,"number":5695} |
1.0.0 | problem:theoremdb-p9694 | problem | theoremdb-p9694 | Irrationality of the Euler Totient Generating Series at 1/2 | {
"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 $\phi : \mathbb{N} \to \mathbb{N}$ denote Euler's totient function, which counts the number of integers between $1$ and $n$ that are coprime to $n$. Consider the infinite series
$$
S = \sum_{n=0}^{\infty} \frac{\phi(n)}{2^n}.
$$
Is $S$ irrational? | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P9694",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P9694",
"citation": null,
"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_249 : answer(sorry) ↔ Irrational (∑' n : ℕ, (φ n) / (2 ^ n)) | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem erdos_249 : answer(sorry) ↔ Irrational (∑' n : ℕ, (φ n) / (2 ^ n))",
"target": null,
"source_url": "https://theoremdb.org/statements/P9694",
"format": "declaration-or-fragment",
"sha256": "e02d20d7dfcf8108942753ccb1421f7d7703d2569936c94a68a25314c5813a... | [] | TheoremDB P9694 · Candidate awaiting source review | null | Source-reported assessment, retrieved 8 September 2026. Frontier Science has not independently reviewed this status.
Open. No signed Lean verification is attached to this problem or its selected resolution.
This record is in the source backlog; its novelty and unresolved status have not passed source review. | null | published | incremental | hard | [
"source-candidate",
"theoremdb"
] | null | null | null | null | [
{
"kind": "website",
"citation": "TheoremDB P9694",
"url": "https://theoremdb.org/statements/P9694",
"note": null,
"sort_order": 0
},
{
"kind": "formalization",
"citation": "Source formal statement (not locally verified)",
"url": "https://theoremdb.org/statements/P9694",
"not... | {
"source_record": "P9694",
"imported_at": "2026-09-08",
"status_as_of": "2026-09-08",
"evidence_label": "Source candidate · not reviewed",
"note": "Full public directory statement. Source lane: backlog; source resolution: open. Source verification claims are attributed only; no local Lean verification has be... | {"created_at":"2026-08-17T20:01:51+00:00","display_standard":{"editorial_review":"submitter_supplied","formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":9694,"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": "2026-09-08T03:33:01.761Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"c2f704ab-4827-4d71-ab7c-889847cf1d38","format_issues":null,"number":5729} |
1.0.0 | problem:theoremdb-p9696 | problem | theoremdb-p9696 | Infinitude of safe primes | {
"main": "math",
"sub": "number theory",
"slug": "number-theory",
"name": "Number theory",
"original_discipline": "mathematics"
} | [] | [
{
"code": "11",
"basis": "community-map",
"evidence": "Inferred from community field: number-theory"
}
] | A prime number $p$ is called a safe prime if $2p+1$ is also prime. Prove that there are infinitely many safe primes. | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P9696",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P9696",
"citation": null,
"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_safe_primes :
Infinite {p : ℕ | Prime p ∧ Prime (2 * p + 1)} | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem infinite_safe_primes :\n Infinite {p : ℕ | Prime p ∧ Prime (2 * p + 1)}",
"target": null,
"source_url": "https://theoremdb.org/statements/P9696",
"format": "declaration-or-fragment",
"sha256": "1f04ef37ee11beabdaa657ae27b8286f172b0910f2696b4990700c... | [] | TheoremDB P9696 · Candidate awaiting source review | null | Source-reported assessment, retrieved 8 September 2026. Frontier Science has not independently reviewed this status.
Open. No signed Lean verification is attached to this problem or its selected resolution.
This record is in the source backlog; its novelty and unresolved status have not passed source review. | null | published | incremental | hard | [
"source-candidate",
"theoremdb"
] | null | null | null | null | [
{
"kind": "website",
"citation": "TheoremDB P9696",
"url": "https://theoremdb.org/statements/P9696",
"note": null,
"sort_order": 0
},
{
"kind": "formalization",
"citation": "Source formal statement (not locally verified)",
"url": "https://theoremdb.org/statements/P9696",
"not... | {
"source_record": "P9696",
"imported_at": "2026-09-08",
"status_as_of": "2026-09-08",
"evidence_label": "Source candidate · not reviewed",
"note": "Full public directory statement. Source lane: backlog; source resolution: open. Source verification claims are attributed only; no local Lean verification has be... | {"created_at":"2026-08-17T20:01:52+00:00","display_standard":{"editorial_review":"submitter_supplied","formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":9696,"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": "2026-09-08T03:33:01.257Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"2be9c1cd-7c74-403b-9416-747a7bf676ad","format_issues":null,"number":5691} |
1.0.0 | problem:theoremdb-p9698 | problem | theoremdb-p9698 | Infinitely many solutions to a two-power congruence modulo the exponent | {
"main": "math",
"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 $k > 1$, let $S_k$ denote the set of positive integers $n$ such that $2^n \equiv k \pmod{n}$. Determine whether $S_k$ is infinite for every $k > 1$. | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P9698",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P9698",
"citation": null,
"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_479 : answer(sorry) ↔ ∀ᵉ (k > 1), { n | 2 ^ n ≡ k [MOD n]}.Infinite | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem erdos_479 : answer(sorry) ↔ ∀ᵉ (k > 1), { n | 2 ^ n ≡ k [MOD n]}.Infinite",
"target": null,
"source_url": "https://theoremdb.org/statements/P9698",
"format": "declaration-or-fragment",
"sha256": "89d2744f84bbab232ef5035c5de7945ce2c663c2f601f6a04550b39... | [] | TheoremDB P9698 · Candidate awaiting source review | null | Source-reported assessment, retrieved 8 September 2026. Frontier Science has not independently reviewed this status.
Open. No signed Lean verification is attached to this problem or its selected resolution.
This record is in the source backlog; its novelty and unresolved status have not passed source review. | null | published | incremental | hard | [
"source-candidate",
"theoremdb"
] | null | null | null | null | [
{
"kind": "website",
"citation": "TheoremDB P9698",
"url": "https://theoremdb.org/statements/P9698",
"note": null,
"sort_order": 0
},
{
"kind": "formalization",
"citation": "Source formal statement (not locally verified)",
"url": "https://theoremdb.org/statements/P9698",
"not... | {
"source_record": "P9698",
"imported_at": "2026-09-08",
"status_as_of": "2026-09-08",
"evidence_label": "Source candidate · not reviewed",
"note": "Full public directory statement. Source lane: backlog; source resolution: open. Source verification claims are attributed only; no local Lean verification has be... | {"created_at":"2026-08-17T20:01:55+00:00","display_standard":{"editorial_review":"submitter_supplied","formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":9698,"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": "2026-09-08T03:33:01.050Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"c3ef10b9-8510-431c-9987-3f80f3b00419","format_issues":null,"number":5646} |
1.0.0 | problem:theoremdb-p9700 | problem | theoremdb-p9700 | Independent Domination Number Bound for Graphs with Even Maximum Degree | {
"main": "math",
"sub": "combinatorics",
"slug": "combinatorics",
"name": "Combinatorics",
"original_discipline": "mathematics"
} | [] | [
{
"code": "05",
"basis": "community-map",
"evidence": "Inferred from community field: combinatorics"
}
] | Let $G$ be a finite simple graph with vertex set $V$. Assume that $G$ has no isolated vertices, so the minimum degree of $G$ is positive. Let $D$ denote the maximum degree of $G$, and suppose that $D$ is even. Let $n = |V|$ be the number of vertices, and let $i(G)$ denote the independent domination number of $G$. Prove... | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P9700",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P9700",
"citation": null,
"doi": null,
"arxiv_id": null,
"license": "Selected packets label their content CC0-1.0; original references retain their own terms",
"notes": "Selected problem... | theorem independentDominationEven (hIso : 0 < G.minDegree) (hEven : Even G.maxDegree) :
let D := G.maxDegree
let i := G.indepDominationNumber
let n := Fintype.card V
(D + 2)^2 * i ≤ (D^2 + 4) * n | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem independentDominationEven (hIso : 0 < G.minDegree) (hEven : Even G.maxDegree) :\n let D := G.maxDegree\n let i := G.indepDominationNumber\n let n := Fintype.card V\n (D + 2)^2 * i ≤ (D^2 + 4) * n",
"target": null,
"source_url": "https://theoremdb.... | [] | TheoremDB P9700 · Candidate awaiting source review | null | Source-reported assessment, retrieved 8 September 2026. Frontier Science has not independently reviewed this status.
Open. No signed Lean verification is attached to this problem or its selected resolution.
This record is in the source backlog; its novelty and unresolved status have not passed source review. | null | published | incremental | hard | [
"source-candidate",
"theoremdb"
] | null | null | null | null | [
{
"kind": "website",
"citation": "TheoremDB P9700",
"url": "https://theoremdb.org/statements/P9700",
"note": null,
"sort_order": 0
},
{
"kind": "formalization",
"citation": "Source formal statement (not locally verified)",
"url": "https://theoremdb.org/statements/P9700",
"not... | {
"source_record": "P9700",
"imported_at": "2026-09-08",
"status_as_of": "2026-09-08",
"evidence_label": "Source candidate · not reviewed",
"note": "Full public directory statement. Source lane: backlog; source resolution: open. Source verification claims are attributed only; no local Lean verification has be... | {"created_at":"2026-08-17T20:01:56+00:00","display_standard":{"formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":9700,"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"
] | [] | {
"created_at": "2026-09-08T03:32:15.935Z",
"updated_at": "2026-09-08T03:32:15.935Z",
"published_at": "2026-09-08T03:33:00.789Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"4200a278-2978-4cbb-af27-91cb71ae80c8","format_issues":null,"number":5590} |
1.0.0 | problem:theoremdb-p9702 | problem | theoremdb-p9702 | Representability of all large integers as sums of at most $r$ $r$-powerful numbers | {
"main": "math",
"sub": "number theory",
"slug": "number-theory",
"name": "Number theory",
"original_discipline": "mathematics"
} | [] | [
{
"code": "11",
"basis": "community-map",
"evidence": "Inferred from community field: number-theory"
}
] | For an integer $r \ge 2$, a positive integer $n$ is called $r$-powerful (or $r$-full) if for every prime $p$ dividing $n$, the highest power of $p$ dividing $n$ is at least $p^r$; equivalently, $n$ is divisible by the $r$th power of each of its prime divisors. Determine whether it is true that for every integer $r \ge ... | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P9702",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P9702",
"citation": null,
"doi": null,
"arxiv_id": null,
"license": "Selected packets label their content CC0-1.0; original references retain their own terms",
"notes": "Selected problem... | theorem erdos_940.variants.large_integers :
answer(sorry) ↔
∀ r ≥ 3, (∀ᶠ x in atTop, ∃ (S : Multiset ℕ), S.card ≤ r ∧ (∀ s ∈ S, r.Full s) ∧ x = S.sum) | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem erdos_940.variants.large_integers :\n answer(sorry) ↔\n ∀ r ≥ 3, (∀ᶠ x in atTop, ∃ (S : Multiset ℕ), S.card ≤ r ∧ (∀ s ∈ S, r.Full s) ∧ x = S.sum)",
"target": null,
"source_url": "https://theoremdb.org/statements/P9702",
"format": "declaration-or-fr... | [] | TheoremDB P9702 · Candidate awaiting source review | null | Source-reported assessment, retrieved 8 September 2026. Frontier Science has not independently reviewed this status.
Open. No signed Lean verification is attached to this problem or its selected resolution.
This record is in the source backlog; its novelty and unresolved status have not passed source review. | null | published | incremental | hard | [
"source-candidate",
"theoremdb"
] | null | null | null | null | [
{
"kind": "website",
"citation": "TheoremDB P9702",
"url": "https://theoremdb.org/statements/P9702",
"note": null,
"sort_order": 0
},
{
"kind": "formalization",
"citation": "Source formal statement (not locally verified)",
"url": "https://theoremdb.org/statements/P9702",
"not... | {
"source_record": "P9702",
"imported_at": "2026-09-08",
"status_as_of": "2026-09-08",
"evidence_label": "Source candidate · not reviewed",
"note": "Full public directory statement. Source lane: backlog; source resolution: open. Source verification claims are attributed only; no local Lean verification has be... | {"created_at":"2026-08-17T20:01:57+00:00","display_standard":{"editorial_review":"submitter_supplied","formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":9702,"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": "2026-09-08T03:33:05.617Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"a53eebd7-1a31-434b-a62c-e5989da6ca59","format_issues":null,"number":6566} |
1.0.0 | problem:theoremdb-p9704 | problem | theoremdb-p9704 | Erdős Problem 41 on Distinct Triple 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"
}
] | A set $A \subseteq \mathbb{N}$ is said to satisfy the triple distinct sums condition if for any two finite subsets $I, J \subseteq A$ with $|I| = |J| = 3$, the equality $\sum_{i \in I} i = \sum_{j \in J} j$ implies $I = J$. In other words, all sums of three elements from $A$ are distinct aside from trivial coincidences... | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P9704",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P9704",
"citation": null,
"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_41 (A : Set ℕ) (h_triple : NtupleCondition A 3) (h_infinite : A.Infinite) :
Filter.atTop.liminf (fun N => (A ∩ Icc 1 N).ncard / (N : ℝ)^(1/3 : ℝ)) = 0 | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem erdos_41 (A : Set ℕ) (h_triple : NtupleCondition A 3) (h_infinite : A.Infinite) :\n Filter.atTop.liminf (fun N => (A ∩ Icc 1 N).ncard / (N : ℝ)^(1/3 : ℝ)) = 0",
"target": null,
"source_url": "https://theoremdb.org/statements/P9704",
"format": "declarat... | [] | TheoremDB P9704 · Candidate awaiting source review | null | Source-reported assessment, retrieved 8 September 2026. Frontier Science has not independently reviewed this status.
Open. No signed Lean verification is attached to this problem or its selected resolution.
This record is in the source backlog; its novelty and unresolved status have not passed source review. | null | published | incremental | hard | [
"source-candidate",
"theoremdb"
] | null | null | null | null | [
{
"kind": "website",
"citation": "TheoremDB P9704",
"url": "https://theoremdb.org/statements/P9704",
"note": null,
"sort_order": 0
},
{
"kind": "formalization",
"citation": "Source formal statement (not locally verified)",
"url": "https://theoremdb.org/statements/P9704",
"not... | {
"source_record": "P9704",
"imported_at": "2026-09-08",
"status_as_of": "2026-09-08",
"evidence_label": "Source candidate · not reviewed",
"note": "Full public directory statement. Source lane: backlog; source resolution: open. Source verification claims are attributed only; no local Lean verification has be... | {"created_at":"2026-08-17T20:01:57+00:00","display_standard":{"formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":9704,"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": "2026-09-08T03:32:55.443Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"95d8a35d-2d79-4609-9592-382795d8868a","format_issues":null,"number":4520} |
1.0.0 | problem:theoremdb-p9706 | problem | theoremdb-p9706 | Non-existence of monochromatic quantum graphs with six vertices and four colours over the complex numbers | {
"main": "math",
"sub": "combinatorics",
"slug": "combinatorics",
"name": "Combinatorics",
"original_discipline": "mathematics"
} | [] | [
{
"code": "05",
"basis": "community-map",
"evidence": "Inferred from community field: combinatorics"
}
] | Let $N = 6$ and $D = 4$. Consider a complete graph on the vertex set $\{0, 1, \dots, N-1\}$. A weighting is a function $W$ that assigns to each unordered pair of distinct vertices $\{i,j\}$ and each ordered pair of colours $(c, c') \in \{0, 1, \dots, D-1\}^2$ a complex number $W(\{i,j\}, c, c')$. For each vertex colour... | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P9706",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P9706",
"citation": null,
"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_d4 :
answer(sorry) ↔
¬ ∃ W : WeightsN 6 4 ℂ, EqSystemN 6 4 W | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem eqSystem6_no_solution_d4 :\n answer(sorry) ↔\n ¬ ∃ W : WeightsN 6 4 ℂ, EqSystemN 6 4 W",
"target": null,
"source_url": "https://theoremdb.org/statements/P9706",
"format": "declaration-or-fragment",
"sha256": "41e60528da138423df7b6d70d0fdb8d94d... | [] | TheoremDB P9706 · Candidate awaiting source review | null | Source-reported assessment, retrieved 8 September 2026. Frontier Science has not independently reviewed this status.
Open. No signed Lean verification is attached to this problem or its selected resolution.
This record is in the source backlog; its novelty and unresolved status have not passed source review. | null | published | incremental | hard | [
"source-candidate",
"theoremdb"
] | null | null | null | null | [
{
"kind": "website",
"citation": "TheoremDB P9706",
"url": "https://theoremdb.org/statements/P9706",
"note": null,
"sort_order": 0
},
{
"kind": "formalization",
"citation": "Source formal statement (not locally verified)",
"url": "https://theoremdb.org/statements/P9706",
"not... | {
"source_record": "P9706",
"imported_at": "2026-09-08",
"status_as_of": "2026-09-08",
"evidence_label": "Source candidate · not reviewed",
"note": "Full public directory statement. Source lane: backlog; source resolution: open. Source verification claims are attributed only; no local Lean verification has be... | {"created_at":"2026-08-17T20:01:58+00:00","display_standard":{"editorial_review":"submitter_supplied","formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":9706,"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": "2026-09-08T03:33:04.337Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"86757a6f-70a6-4bbd-b514-1728c936d4f1","format_issues":null,"number":6289} |
1.0.0 | problem:theoremdb-p9708 | problem | theoremdb-p9708 | Smallest containing circle for fifteen 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 open disk $\{p \in \mathbb{R}^2 : p_0^2 + p_1^2 < 1\}$ in the Euclidean plane. For a positive integer $n$ and sets $s, S \subseteq \mathbb{R}^2$, a packing of $n$ copies of $s$ inside $S$ consists of $n$ isometric embeddings $\phi_1, \ldots, \phi_n : \mathbb{R}^2 \to \mathbb{R}^2$ such that the... | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P9708",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P9708",
"citation": null,
"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_fifteen_circle_packing_in_circle :
IsLeast {r : ℝ≥0 | Nonempty (Packing 15 UnitCircle (Circle r))} answer(sorry) | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem least_fifteen_circle_packing_in_circle :\n IsLeast {r : ℝ≥0 | Nonempty (Packing 15 UnitCircle (Circle r))} answer(sorry)",
"target": null,
"source_url": "https://theoremdb.org/statements/P9708",
"format": "declaration-or-fragment",
"sha256": "eb15f... | [] | TheoremDB P9708 · Candidate awaiting source review | null | Source-reported assessment, retrieved 8 September 2026. Frontier Science has not independently reviewed this status.
Open. No signed Lean verification is attached to this problem or its selected resolution.
This record is in the source backlog; its novelty and unresolved status have not passed source review. | null | published | incremental | hard | [
"source-candidate",
"theoremdb"
] | null | null | null | null | [
{
"kind": "website",
"citation": "TheoremDB P9708",
"url": "https://theoremdb.org/statements/P9708",
"note": null,
"sort_order": 0
},
{
"kind": "formalization",
"citation": "Source formal statement (not locally verified)",
"url": "https://theoremdb.org/statements/P9708",
"not... | {
"source_record": "P9708",
"imported_at": "2026-09-08",
"status_as_of": "2026-09-08",
"evidence_label": "Source candidate · not reviewed",
"note": "Full public directory statement. Source lane: backlog; source resolution: open. Source verification claims are attributed only; no local Lean verification has be... | {"created_at":"2026-08-17T20:01:59+00:00","display_standard":{"editorial_review":"submitter_supplied","formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":9708,"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": "2026-09-08T03:33:05.892Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"851baa81-3c13-41a3-b458-adb83c46a982","format_issues":null,"number":6625} |
1.0.0 | problem:theoremdb-p9710 | problem | theoremdb-p9710 | A weaker variant of Erdős's prime-counting inequality | {
"main": "math",
"sub": "number theory",
"slug": "number-theory",
"name": "Number theory",
"original_discipline": "mathematics"
} | [] | [
{
"code": "11",
"basis": "community-map",
"evidence": "Inferred from community field: number-theory"
}
] | For a positive real number $\varepsilon$, does there exist, for all sufficiently large real numbers $x$, a real number $y < x$ such that
$$
\pi(x) < \pi(y) + \varepsilon \cdot \pi(x - y)?
$$
Here $\pi(t)$ denotes the prime-counting function, which gives the number of primes not exceeding $t$. | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P9710",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P9710",
"citation": null,
"doi": null,
"arxiv_id": null,
"license": "Selected packets label their content CC0-1.0; original references retain their own terms",
"notes": "Selected problem... | theorem erdos_950.variants.weaker_pi :
answer(sorry) ↔ ∀ ε > 0, ∀ᶠ x in atTop, ∃ y < x,
(π x : ℝ) < (π y : ℝ) + ε * (π (x - y) : ℝ) | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem erdos_950.variants.weaker_pi :\n answer(sorry) ↔ ∀ ε > 0, ∀ᶠ x in atTop, ∃ y < x,\n (π x : ℝ) < (π y : ℝ) + ε * (π (x - y) : ℝ)",
"target": null,
"source_url": "https://theoremdb.org/statements/P9710",
"format": "declaration-or-fragment",
"sha... | [] | TheoremDB P9710 · Candidate awaiting source review | null | Source-reported assessment, retrieved 8 September 2026. Frontier Science has not independently reviewed this status.
Open. No signed Lean verification is attached to this problem or its selected resolution.
This record is in the source backlog; its novelty and unresolved status have not passed source review. | null | published | incremental | hard | [
"source-candidate",
"theoremdb"
] | null | null | null | null | [
{
"kind": "website",
"citation": "TheoremDB P9710",
"url": "https://theoremdb.org/statements/P9710",
"note": null,
"sort_order": 0
},
{
"kind": "formalization",
"citation": "Source formal statement (not locally verified)",
"url": "https://theoremdb.org/statements/P9710",
"not... | {
"source_record": "P9710",
"imported_at": "2026-09-08",
"status_as_of": "2026-09-08",
"evidence_label": "Source candidate · not reviewed",
"note": "Full public directory statement. Source lane: backlog; source resolution: open. Source verification claims are attributed only; no local Lean verification has be... | {"created_at":"2026-08-17T20:02:00+00:00","display_standard":{"editorial_review":"submitter_supplied","formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":9710,"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": "2026-09-08T03:32:49.444Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"c9b6f51f-e3be-4cdb-b3ef-119cb0827aff","format_issues":null,"number":3209} |
1.0.0 | problem:theoremdb-p9712 | problem | theoremdb-p9712 | Dense Planar Set with Rational Pairwise Distances | {
"main": "math",
"sub": "geometry and topology",
"slug": "geometry-and-topology",
"name": "Geometry and topology",
"original_discipline": "mathematics"
} | [] | [] | A subset of the complex plane is identified with a subset of the Euclidean plane $\mathbb{R}^2$ via the standard identification of $\mathbb{C}$ with $\mathbb{R}^2$. The distance between two points is the usual Euclidean distance. A set is dense if its closure equals the whole space. Determine whether there exists a den... | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P9712",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P9712",
"citation": null,
"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_212 : answer(sorry) ↔
∃ u : Set ℂ, Dense u ∧ u.Pairwise fun c₁ c₂ => dist c₁ c₂ ∈ Set.range Rat.cast | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem erdos_212 : answer(sorry) ↔\n ∃ u : Set ℂ, Dense u ∧ u.Pairwise fun c₁ c₂ => dist c₁ c₂ ∈ Set.range Rat.cast",
"target": null,
"source_url": "https://theoremdb.org/statements/P9712",
"format": "declaration-or-fragment",
"sha256": "1ccef9c7cb78baa50... | [] | TheoremDB P9712 · Candidate awaiting source review | null | Source-reported assessment, retrieved 8 September 2026. Frontier Science has not independently reviewed this status.
Open. No signed Lean verification is attached to this problem or its selected resolution.
This record is in the source backlog; its novelty and unresolved status have not passed source review. | null | published | incremental | hard | [
"source-candidate",
"theoremdb"
] | null | null | null | null | [
{
"kind": "website",
"citation": "TheoremDB P9712",
"url": "https://theoremdb.org/statements/P9712",
"note": null,
"sort_order": 0
},
{
"kind": "formalization",
"citation": "Source formal statement (not locally verified)",
"url": "https://theoremdb.org/statements/P9712",
"not... | {
"source_record": "P9712",
"imported_at": "2026-09-08",
"status_as_of": "2026-09-08",
"evidence_label": "Source candidate · not reviewed",
"note": "Full public directory statement. Source lane: backlog; source resolution: open. Source verification claims are attributed only; no local Lean verification has be... | {"created_at":"2026-08-17T20:02:01+00:00","display_standard":{"editorial_review":"submitter_supplied","formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":9712,"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": "2026-09-08T03:32:53.049Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"8c026b50-931d-414a-a14e-dece760ce173","format_issues":null,"number":3993} |
1.0.0 | problem:theoremdb-p9714 | problem | theoremdb-p9714 | Asymptotic vanishing of the maximal covering exponent in prime residue systems | {
"main": "math",
"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$ and real number $\varepsilon$, let $P(n,\varepsilon)$ denote the property that there exists a choice of residue class $a_p$ for every prime $p$ with $n^{\varepsilon} < 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}$. De... | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P9714",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P9714",
"citation": null,
"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.ii : answer(sorry) ↔
epsilonFunction =o[atTop] (fun (n : ℕ) ↦ (1 : ℝ)) | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem erdos_688.parts.ii : answer(sorry) ↔\n epsilonFunction =o[atTop] (fun (n : ℕ) ↦ (1 : ℝ))",
"target": null,
"source_url": "https://theoremdb.org/statements/P9714",
"format": "declaration-or-fragment",
"sha256": "4c79cce2e4cefd7bd2d2211aca63691a2c526... | [] | TheoremDB P9714 · Candidate awaiting source review | null | Source-reported assessment, retrieved 8 September 2026. Frontier Science has not independently reviewed this status.
Open. No signed Lean verification is attached to this problem or its selected resolution.
This record is in the source backlog; its novelty and unresolved status have not passed source review. | null | published | incremental | hard | [
"source-candidate",
"theoremdb"
] | null | null | null | null | [
{
"kind": "website",
"citation": "TheoremDB P9714",
"url": "https://theoremdb.org/statements/P9714",
"note": null,
"sort_order": 0
},
{
"kind": "formalization",
"citation": "Source formal statement (not locally verified)",
"url": "https://theoremdb.org/statements/P9714",
"not... | {
"source_record": "P9714",
"imported_at": "2026-09-08",
"status_as_of": "2026-09-08",
"evidence_label": "Source candidate · not reviewed",
"note": "Full public directory statement. Source lane: backlog; source resolution: open. Source verification claims are attributed only; no local Lean verification has be... | {"created_at":"2026-08-17T20:02:02+00:00","display_standard":{"editorial_review":"submitter_supplied","formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":9714,"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": "2026-09-08T03:32:50.514Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"e3eb84a3-c405-4d02-9926-e7e309350a36","format_issues":null,"number":3432} |
1.0.0 | problem:theoremdb-p9716 | problem | theoremdb-p9716 | Existence of an absolutely maximally entangled state for eight parties of local dimension six | {
"main": "math",
"sub": "combinatorics",
"slug": "combinatorics",
"name": "Combinatorics",
"original_discipline": "mathematics"
} | [] | [
{
"code": "05",
"basis": "community-map",
"evidence": "Inferred from community field: combinatorics"
}
] | Let $n$ and $d$ be positive integers. An $n$-partite pure state $\psi$ with local dimension $d$ is called absolutely maximally entangled (AME) if, for every subset of at most $\lfloor n/2 \rfloor$ parties, the reduced density matrix obtained by tracing out the complementary parties is maximally mixed. Does there exist ... | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P9716",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P9716",
"citation": null,
"doi": null,
"arxiv_id": null,
"license": "Selected packets label their content CC0-1.0; original references retain their own terms",
"notes": "Selected problem... | theorem ame_8_6_open :
answer(sorry) ↔ ExistsAME 8 6 | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem ame_8_6_open :\n answer(sorry) ↔ ExistsAME 8 6",
"target": null,
"source_url": "https://theoremdb.org/statements/P9716",
"format": "declaration-or-fragment",
"sha256": "2f2119dd01af67710e2ab1a43c69a4885f423272486f5063488f692d0dd16721",
"license... | [] | TheoremDB P9716 · Candidate awaiting source review | null | Source-reported assessment, retrieved 8 September 2026. Frontier Science has not independently reviewed this status.
Open. No signed Lean verification is attached to this problem or its selected resolution.
This record is in the source backlog; its novelty and unresolved status have not passed source review. | null | published | incremental | hard | [
"source-candidate",
"theoremdb"
] | null | null | null | null | [
{
"kind": "website",
"citation": "TheoremDB P9716",
"url": "https://theoremdb.org/statements/P9716",
"note": null,
"sort_order": 0
},
{
"kind": "formalization",
"citation": "Source formal statement (not locally verified)",
"url": "https://theoremdb.org/statements/P9716",
"not... | {
"source_record": "P9716",
"imported_at": "2026-09-08",
"status_as_of": "2026-09-08",
"evidence_label": "Source candidate · not reviewed",
"note": "Full public directory statement. Source lane: backlog; source resolution: open. Source verification claims are attributed only; no local Lean verification has be... | {"created_at":"2026-08-17T20:02:02+00:00","display_standard":{"editorial_review":"submitter_supplied","formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":9716,"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": "2026-09-08T03:32:58.627Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"f391fd76-357f-4620-8459-bce9ea04db94","format_issues":null,"number":5222} |
1.0.0 | problem:theoremdb-p9718 | problem | theoremdb-p9718 | The Kakeya set conjecture | {
"main": "math",
"sub": "analysis",
"slug": "analysis",
"name": "Analysis",
"original_discipline": "mathematics"
} | [] | [] | Let $n$ be a positive integer. A subset $S$ of $\mathbb{R}^n$ is called a Kakeya set if for every unit vector $v \in \mathbb{R}^n$ (that is, every $v$ with $\|v\| = 1$), there exists a point $a \in \mathbb{R}^n$ such that the line segment from $a$ to $a + v$ is contained in $S$. Prove that every Kakeya set in $\mathbb{... | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P9718",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P9718",
"citation": null,
"doi": null,
"arxiv_id": null,
"license": "Selected packets label their content CC0-1.0; original references retain their own terms",
"notes": "Selected problem... | theorem kakeya_set_conjecture (n : ℕ) (hn : n > 0) :
KakeyaSetConjectureDim n | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem kakeya_set_conjecture (n : ℕ) (hn : n > 0) :\n KakeyaSetConjectureDim n",
"target": null,
"source_url": "https://theoremdb.org/statements/P9718",
"format": "declaration-or-fragment",
"sha256": "321fa1108cbf169ced2e9a90a40f29dcdd08a200c534bdca24266a... | [] | TheoremDB P9718 · Candidate awaiting source review | null | Source-reported assessment, retrieved 8 September 2026. Frontier Science has not independently reviewed this status.
Open. No signed Lean verification is attached to this problem or its selected resolution.
This record is in the source backlog; its novelty and unresolved status have not passed source review. | null | published | incremental | hard | [
"source-candidate",
"theoremdb"
] | null | null | null | null | [
{
"kind": "website",
"citation": "TheoremDB P9718",
"url": "https://theoremdb.org/statements/P9718",
"note": null,
"sort_order": 0
},
{
"kind": "formalization",
"citation": "Source formal statement (not locally verified)",
"url": "https://theoremdb.org/statements/P9718",
"not... | {
"source_record": "P9718",
"imported_at": "2026-09-08",
"status_as_of": "2026-09-08",
"evidence_label": "Source candidate · not reviewed",
"note": "Full public directory statement. Source lane: backlog; source resolution: open. Source verification claims are attributed only; no local Lean verification has be... | {"created_at":"2026-08-17T20:02:03+00:00","display_standard":{"formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":9718,"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": "2026-09-08T03:33:06.575Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"845cb42a-2505-464b-851f-e96aed46c2b6","format_issues":null,"number":6772} |
1.0.0 | problem:theoremdb-p9720 | problem | theoremdb-p9720 | Polignac's Conjecture | {
"main": "math",
"sub": "number theory",
"slug": "number-theory",
"name": "Number theory",
"original_discipline": "mathematics"
} | [] | [
{
"code": "11",
"basis": "community-map",
"evidence": "Inferred from community field: number-theory"
}
] | For every natural number $k$, does there exist infinitely many prime numbers $p$ such that $p + 2k$ is also prime? | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P9720",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P9720",
"citation": null,
"doi": null,
"arxiv_id": null,
"license": "Selected packets label their content CC0-1.0; original references retain their own terms",
"notes": "Selected problem... | theorem polignac_conjecture (k : ℕ) :
Infinite {p : ℕ | p.Prime ∧ (p + 2 * k).Prime} | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem polignac_conjecture (k : ℕ) :\n Infinite {p : ℕ | p.Prime ∧ (p + 2 * k).Prime}",
"target": null,
"source_url": "https://theoremdb.org/statements/P9720",
"format": "declaration-or-fragment",
"sha256": "a8ef81a9769c511ec51bb9805d17085a0e0431f6bcc80f8... | [] | TheoremDB P9720 · Candidate awaiting source review | null | Source-reported assessment, retrieved 8 September 2026. Frontier Science has not independently reviewed this status.
Open. No signed Lean verification is attached to this problem or its selected resolution.
This record is in the source backlog; its novelty and unresolved status have not passed source review. | null | published | incremental | hard | [
"source-candidate",
"theoremdb"
] | null | null | null | null | [
{
"kind": "website",
"citation": "TheoremDB P9720",
"url": "https://theoremdb.org/statements/P9720",
"note": null,
"sort_order": 0
},
{
"kind": "formalization",
"citation": "Source formal statement (not locally verified)",
"url": "https://theoremdb.org/statements/P9720",
"not... | {
"source_record": "P9720",
"imported_at": "2026-09-08",
"status_as_of": "2026-09-08",
"evidence_label": "Source candidate · not reviewed",
"note": "Full public directory statement. Source lane: backlog; source resolution: open. Source verification claims are attributed only; no local Lean verification has be... | {"created_at":"2026-08-17T20:02:04+00:00","display_standard":{"formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":9720,"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": "2026-09-08T03:33:04.860Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"6fbd6bc1-2299-41f1-94ce-0b6ab9d773e0","format_issues":null,"number":6402} |
1.0.0 | problem:theoremdb-p9722 | problem | theoremdb-p9722 | Maximum Size of a Zero-Sum Signed Unit Fraction Set with No Proper Zero-Sum Subset | {
"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 $A$ be a subset of $\{1, 2, \ldots, N\}$. A function $\delta: A \to \{-1, 1\}$ is called a signing of $A$. The signed sum of $A$ with respect to $\delta$ is $\sum_{n \in A} \frac{\delta(n)}{n}$. A subset $A' \subseteq A$ is called a zero-sum subset if $\sum_{n \in A'} \frac{\delta(n)}{n}... | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P9722",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P9722",
"citation": null,
"doi": null,
"arxiv_id": null,
"license": "Selected packets label their content CC0-1.0; original references retain their own terms",
"notes": "Selected problem... | theorem erdos_319 (N : ℕ) : IsGreatest
{ #A | (A) (_ : A ⊆ Finset.Icc 1 N)
(_ : ∃ δ : ℕ → ℤˣ, ∑ n ∈ A, (δ n : ℚ) / n = 0 ∧
∀ A' ⊂ A, A'.Nonempty → ∑ n ∈ A', (δ n : ℚ) / n ≠ 0) }
answer(sorry) | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem erdos_319 (N : ℕ) : IsGreatest\n { #A | (A) (_ : A ⊆ Finset.Icc 1 N)\n (_ : ∃ δ : ℕ → ℤˣ, ∑ n ∈ A, (δ n : ℚ) / n = 0 ∧\n ∀ A' ⊂ A, A'.Nonempty → ∑ n ∈ A', (δ n : ℚ) / n ≠ 0) }\n answer(sorry)",
"target": null,
"source_url": "https://theoremd... | [] | TheoremDB P9722 · Candidate awaiting source review | null | Source-reported assessment, retrieved 8 September 2026. Frontier Science has not independently reviewed this status.
Open. No signed Lean verification is attached to this problem or its selected resolution.
This record is in the source backlog; its novelty and unresolved status have not passed source review. | null | published | incremental | hard | [
"source-candidate",
"theoremdb"
] | null | null | null | null | [
{
"kind": "website",
"citation": "TheoremDB P9722",
"url": "https://theoremdb.org/statements/P9722",
"note": null,
"sort_order": 0
},
{
"kind": "formalization",
"citation": "Source formal statement (not locally verified)",
"url": "https://theoremdb.org/statements/P9722",
"not... | {
"source_record": "P9722",
"imported_at": "2026-09-08",
"status_as_of": "2026-09-08",
"evidence_label": "Source candidate · not reviewed",
"note": "Full public directory statement. Source lane: backlog; source resolution: open. Source verification claims are attributed only; no local Lean verification has be... | {"created_at":"2026-08-17T20:02:04+00:00","display_standard":{"formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":9722,"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": "2026-09-08T03:33:03.849Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"abe763df-f552-4988-9c62-7f77e3ece649","format_issues":null,"number":6184} |
1.0.0 | problem:theoremdb-p9724 | problem | theoremdb-p9724 | Asymptotic Growth of the Practical Number Function for 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"
}
] | A positive integer $n$ is called practical if every integer $m$ with $1 \le m \le n$ can be written as a sum of distinct positive divisors of $n$. For a practical number $n$, define $h(n)$ to be the maximum, over all integers $m$ with $1 \le m \le n$, of the minimum number of divisors of $n$ needed to represent $m$ as ... | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P9724",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P9724",
"citation": null,
"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_18b : answer(sorry) ↔
∀ ε : ℝ, 0 < ε → ∀ᶠ n : ℕ in atTop, (practicalH n.factorial : ℝ) < (n : ℝ) ^ ε | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem erdos_18b : answer(sorry) ↔\n ∀ ε : ℝ, 0 < ε → ∀ᶠ n : ℕ in atTop, (practicalH n.factorial : ℝ) < (n : ℝ) ^ ε",
"target": null,
"source_url": "https://theoremdb.org/statements/P9724",
"format": "declaration-or-fragment",
"sha256": "b553c17320ff32b5b... | [] | TheoremDB P9724 · Candidate awaiting source review | null | Source-reported assessment, retrieved 8 September 2026. Frontier Science has not independently reviewed this status.
Open. No signed Lean verification is attached to this problem or its selected resolution.
This record is in the source backlog; its novelty and unresolved status have not passed source review. | null | published | incremental | hard | [
"source-candidate",
"theoremdb"
] | null | null | null | null | [
{
"kind": "website",
"citation": "TheoremDB P9724",
"url": "https://theoremdb.org/statements/P9724",
"note": null,
"sort_order": 0
},
{
"kind": "formalization",
"citation": "Source formal statement (not locally verified)",
"url": "https://theoremdb.org/statements/P9724",
"not... | {
"source_record": "P9724",
"imported_at": "2026-09-08",
"status_as_of": "2026-09-08",
"evidence_label": "Source candidate · not reviewed",
"note": "Full public directory statement. Source lane: backlog; source resolution: open. Source verification claims are attributed only; no local Lean verification has be... | {"created_at":"2026-08-17T20:02:05+00:00","display_standard":{"editorial_review":"submitter_supplied","formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":9724,"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": "2026-09-08T03:32:50.249Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"eb1480f8-5228-4f0f-8279-ae759d1fb1b6","format_issues":null,"number":3381} |
1.0.0 | problem:theoremdb-p9726 | problem | theoremdb-p9726 | Dickson's conjecture on simultaneous prime values of linear polynomials | {
"main": "math",
"sub": "number theory",
"slug": "number-theory",
"name": "Number theory",
"original_discipline": "mathematics"
} | [] | [
{
"code": "11",
"basis": "community-map",
"evidence": "Inferred from community field: number-theory"
}
] | Let $f_1, f_2, \ldots, f_k$ be a finite collection of linear polynomials with integer coefficients, where each $f_i(n) = a_i n + b_i$ with $a_i \neq 0$. For each $i$, suppose the Bunyakovsky condition holds: the coefficients $a_i$ and $b_i$ are coprime, and the polynomial $f_i$ is irreducible over the integers (which f... | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P9726",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P9726",
"citation": null,
"doi": null,
"arxiv_id": null,
"license": "Selected packets label their content CC0-1.0; original references retain their own terms",
"notes": "Selected problem... | theorem dickson_conjecture (fs : Finset ℤ[X]) (hfs : ∀ f ∈ fs, f.degree = 1 ∧ BunyakovskyCondition f)
(hfs' : SchinzelCondition fs) : Infinite {n : ℕ | ∀ f ∈ fs, (f.eval (n : ℤ)).natAbs.Prime} | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem dickson_conjecture (fs : Finset ℤ[X]) (hfs : ∀ f ∈ fs, f.degree = 1 ∧ BunyakovskyCondition f)\n (hfs' : SchinzelCondition fs) : Infinite {n : ℕ | ∀ f ∈ fs, (f.eval (n : ℤ)).natAbs.Prime}",
"target": null,
"source_url": "https://theoremdb.org/statements/P97... | [] | TheoremDB P9726 · Candidate awaiting source review | null | Source-reported assessment, retrieved 8 September 2026. Frontier Science has not independently reviewed this status.
Open. No signed Lean verification is attached to this problem or its selected resolution.
This record is in the source backlog; its novelty and unresolved status have not passed source review. | null | published | incremental | hard | [
"source-candidate",
"theoremdb"
] | null | null | null | null | [
{
"kind": "website",
"citation": "TheoremDB P9726",
"url": "https://theoremdb.org/statements/P9726",
"note": null,
"sort_order": 0
},
{
"kind": "formalization",
"citation": "Source formal statement (not locally verified)",
"url": "https://theoremdb.org/statements/P9726",
"not... | {
"source_record": "P9726",
"imported_at": "2026-09-08",
"status_as_of": "2026-09-08",
"evidence_label": "Source candidate · not reviewed",
"note": "Full public directory statement. Source lane: backlog; source resolution: open. Source verification claims are attributed only; no local Lean verification has be... | {"created_at":"2026-08-17T20:02:05+00:00","display_standard":{"editorial_review":"submitter_supplied","formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":9726,"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": "2026-09-08T03:32:53.150Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"4270381d-b2eb-453a-9de0-7f82b1e9df14","format_issues":null,"number":4014} |
1.0.0 | problem:theoremdb-p9728 | problem | theoremdb-p9728 | Szabó's strong bound for arithmetic intersection sets | {
"main": "math",
"sub": "combinatorics",
"slug": "combinatorics",
"name": "Combinatorics",
"original_discipline": "mathematics"
} | [] | [
{
"code": "05",
"basis": "community-map",
"evidence": "Inferred from community field: combinatorics"
}
] | For a positive integer $N$, an arithmetic intersection set is a collection $\{A_1, \ldots, A_t\}$ of subsets of $\{1, \ldots, N\}$ such that for every $i \neq j$, the intersection $A_i \cap A_j$ is a non-empty arithmetic progression. Let $t(N)$ denote the maximum possible size of an arithmetic intersection set for a gi... | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P9728",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P9728",
"citation": null,
"doi": null,
"arxiv_id": null,
"license": "Selected packets label their content CC0-1.0; original references retain their own terms",
"notes": "Selected problem... | theorem erdos_272.variants.szabo_strong :
(fun N ↦ (maxArithInterCard N - N ^ 2 / 2 : ℝ)) =O[atTop] fun N : ℕ ↦ (N : ℝ) | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem erdos_272.variants.szabo_strong :\n (fun N ↦ (maxArithInterCard N - N ^ 2 / 2 : ℝ)) =O[atTop] fun N : ℕ ↦ (N : ℝ)",
"target": null,
"source_url": "https://theoremdb.org/statements/P9728",
"format": "declaration-or-fragment",
"sha256": "9d485fd4d617... | [] | TheoremDB P9728 · Candidate awaiting source review | null | Source-reported assessment, retrieved 8 September 2026. Frontier Science has not independently reviewed this status.
Open. No signed Lean verification is attached to this problem or its selected resolution.
This record is in the source backlog; its novelty and unresolved status have not passed source review. | null | published | incremental | hard | [
"source-candidate",
"theoremdb"
] | null | null | null | null | [
{
"kind": "website",
"citation": "TheoremDB P9728",
"url": "https://theoremdb.org/statements/P9728",
"note": null,
"sort_order": 0
},
{
"kind": "formalization",
"citation": "Source formal statement (not locally verified)",
"url": "https://theoremdb.org/statements/P9728",
"not... | {
"source_record": "P9728",
"imported_at": "2026-09-08",
"status_as_of": "2026-09-08",
"evidence_label": "Source candidate · not reviewed",
"note": "Full public directory statement. Source lane: backlog; source resolution: open. Source verification claims are attributed only; no local Lean verification has be... | {"created_at":"2026-08-17T20:02:07+00:00","display_standard":{"formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":9728,"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": "2026-09-08T03:33:06.260Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"baafa353-cfd9-4d70-845a-e7992dd43634","format_issues":null,"number":6705} |
1.0.0 | problem:theoremdb-p9730 | problem | theoremdb-p9730 | Infinitude of balanced primes of order 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 each positive integer $k$, let $p_k$ denote the $k$-th prime number. Determine whether the set of natural numbers $n$ for which $\dfrac{p_n + p_{n+2}}{2}$ is prime is infinite. | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P9730",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P9730",
"citation": null,
"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 :
answer(sorry) ↔ {n : ℕ | Prime ((Nat.nth Prime n + Nat.nth Prime (n + 2)) / 2)}.Infinite | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem balanced_primes :\n answer(sorry) ↔ {n : ℕ | Prime ((Nat.nth Prime n + Nat.nth Prime (n + 2)) / 2)}.Infinite",
"target": null,
"source_url": "https://theoremdb.org/statements/P9730",
"format": "declaration-or-fragment",
"sha256": "c9d350a44fedcd091... | [] | TheoremDB P9730 · Candidate awaiting source review | null | Source-reported assessment, retrieved 8 September 2026. Frontier Science has not independently reviewed this status.
Open. No signed Lean verification is attached to this problem or its selected resolution.
This record is in the source backlog; its novelty and unresolved status have not passed source review. | null | published | incremental | hard | [
"source-candidate",
"theoremdb"
] | null | null | null | null | [
{
"kind": "website",
"citation": "TheoremDB P9730",
"url": "https://theoremdb.org/statements/P9730",
"note": null,
"sort_order": 0
},
{
"kind": "formalization",
"citation": "Source formal statement (not locally verified)",
"url": "https://theoremdb.org/statements/P9730",
"not... | {
"source_record": "P9730",
"imported_at": "2026-09-08",
"status_as_of": "2026-09-08",
"evidence_label": "Source candidate · not reviewed",
"note": "Full public directory statement. Source lane: backlog; source resolution: open. Source verification claims are attributed only; no local Lean verification has be... | {"created_at":"2026-08-17T20:02:08+00:00","display_standard":{"editorial_review":"submitter_supplied","formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":9730,"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": "2026-09-08T03:33:01.090Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"60ec05b6-3f68-4c94-b796-53b0e3c6dc82","format_issues":null,"number":5655} |
1.0.0 | problem:theoremdb-p9732 | problem | theoremdb-p9732 | Erdős's Divisor-Count Maximization Problem | {
"main": "math",
"sub": "number theory",
"slug": "number-theory",
"name": "Number theory",
"original_discipline": "mathematics"
} | [] | [
{
"code": "11",
"basis": "community-map",
"evidence": "Inferred from community field: number-theory"
}
] | For a positive integer $m$, let $\tau(m)$ denote the number of positive divisors of $m$. Determine whether there exists an integer $n > 24$ such that
$$
\max_{0 \leq m < n} \bigl(m + \tau(m)\bigr) \leq n + 2.
$$
More precisely, does the set $\{n \in \mathbb{N} : n > 24 \text{ and } \max_{0 \leq m < n}(m + \tau(m)) ... | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P9732",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P9732",
"citation": null,
"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 : answer(sorry) ↔ ∃ n > 24, ⨆ m : Fin n, m + σ 0 m ≤ n + 2 | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem erdos_647 : answer(sorry) ↔ ∃ n > 24, ⨆ m : Fin n, m + σ 0 m ≤ n + 2",
"target": null,
"source_url": "https://theoremdb.org/statements/P9732",
"format": "declaration-or-fragment",
"sha256": "d77bb73e549ff8242646e51d0ae589b1d11338cdfc6bdbec3eff95efcf79... | [] | TheoremDB P9732 · Candidate awaiting source review | null | Source-reported assessment, retrieved 8 September 2026. Frontier Science has not independently reviewed this status.
Open. No signed Lean verification is attached to this problem or its selected resolution.
This record is in the source backlog; its novelty and unresolved status have not passed source review. | null | published | incremental | hard | [
"source-candidate",
"theoremdb"
] | null | null | null | null | [
{
"kind": "website",
"citation": "TheoremDB P9732",
"url": "https://theoremdb.org/statements/P9732",
"note": null,
"sort_order": 0
},
{
"kind": "formalization",
"citation": "Source formal statement (not locally verified)",
"url": "https://theoremdb.org/statements/P9732",
"not... | {
"source_record": "P9732",
"imported_at": "2026-09-08",
"status_as_of": "2026-09-08",
"evidence_label": "Source candidate · not reviewed",
"note": "Full public directory statement. Source lane: backlog; source resolution: open. Source verification claims are attributed only; no local Lean verification has be... | {"created_at":"2026-08-17T20:02:08+00:00","display_standard":{"editorial_review":"submitter_supplied","formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":9732,"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": "2026-09-08T03:32:56.374Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"12e476b3-febe-4fcf-999b-66582d2abe79","format_issues":null,"number":4727} |
1.0.0 | problem:theoremdb-p9734 | problem | theoremdb-p9734 | Converse of Tunnell's theorem for odd squarefree 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 an odd squarefree natural number. Define $A_n = \{(x, y, z) \in \mathbb{Z}^3 : n = 2x^2 + y^2 + 32z^2\}$ and $B_n = \{(x, y, z) \in \mathbb{Z}^3 : n = 2x^2 + y^2 + 8z^2\}$. A natural number $n$ is called a congruent number if there exists a right triangle with rational sides $a$, $b$, and hypotenuse $c$ such... | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P9734",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P9734",
"citation": null,
"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_odd_converse (n : ℕ) (hsqf : Squarefree n) (hodd : Odd n) :
2 * (A n).ncard = (B n).ncard → congruentNumber n | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem Tunnell_odd_converse (n : ℕ) (hsqf : Squarefree n) (hodd : Odd n) :\n 2 * (A n).ncard = (B n).ncard → congruentNumber n",
"target": null,
"source_url": "https://theoremdb.org/statements/P9734",
"format": "declaration-or-fragment",
"sha256": "7b35a3... | [] | TheoremDB P9734 · Candidate awaiting source review | null | Source-reported assessment, retrieved 8 September 2026. Frontier Science has not independently reviewed this status.
Open. No signed Lean verification is attached to this problem or its selected resolution.
This record is in the source backlog; its novelty and unresolved status have not passed source review. | null | published | incremental | hard | [
"source-candidate",
"theoremdb"
] | null | null | null | null | [
{
"kind": "website",
"citation": "TheoremDB P9734",
"url": "https://theoremdb.org/statements/P9734",
"note": null,
"sort_order": 0
},
{
"kind": "formalization",
"citation": "Source formal statement (not locally verified)",
"url": "https://theoremdb.org/statements/P9734",
"not... | {
"source_record": "P9734",
"imported_at": "2026-09-08",
"status_as_of": "2026-09-08",
"evidence_label": "Source candidate · not reviewed",
"note": "Full public directory statement. Source lane: backlog; source resolution: open. Source verification claims are attributed only; no local Lean verification has be... | {"created_at":"2026-08-17T20:02:09+00:00","display_standard":{"editorial_review":"submitter_supplied","formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":9734,"public_status":{"evidence":{"hasFormalization":false,"hasInformalProof":false,"hasRecordedWork":false,"solutionRecord":null},"lean":{... | [
"formalization-not-compiled-during-export",
"current-status-unassessed",
"source-full-text-not-present"
] | [] | {
"created_at": "2026-09-08T03:32:15.935Z",
"updated_at": "2026-09-08T03:32:15.935Z",
"published_at": "2026-09-08T03:32:52.702Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"62d449ea-e247-4663-bc0b-1a608967e80b","format_issues":null,"number":3917} |
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