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1.0.0 | problem:theoremdb-p8938 | problem | theoremdb-p8938 | The New Mersenne Conjecture | {
"main": "math",
"sub": "number theory",
"slug": "number-theory",
"name": "Number theory",
"original_discipline": "mathematics"
} | [] | [
{
"code": "11",
"basis": "community-map",
"evidence": "Inferred from community field: number-theory"
}
] | Let $p$ be an odd natural number. The Mersenne number with exponent $p$ is $M_p = 2^p - 1$. We say that $p$ gives a Wagstaff prime if $(2^p + 1)/3$ is prime. We say that $p$ 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 ... | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P8938",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P8938",
"citation": null,
"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 (p : ℕ) (hp : Odd p) :
NewMersenneConjectureStatement p | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem new_mersenne_conjecture (p : ℕ) (hp : Odd p) :\n NewMersenneConjectureStatement p",
"target": null,
"source_url": "https://theoremdb.org/statements/P8938",
"format": "declaration-or-fragment",
"sha256": "f5f29cbbeedcd2d657abcb76891c774b9e9a49b7327c... | [] | TheoremDB P8938 · Candidate awaiting source review | null | Source-reported assessment, retrieved 8 September 2026. Frontier Science has not independently reviewed this status.
Open. No signed Lean verification is attached to this problem or its selected resolution.
This record is in the source backlog; its novelty and unresolved status have not passed source review. | null | published | incremental | hard | [
"source-candidate",
"theoremdb"
] | null | null | null | null | [
{
"kind": "website",
"citation": "TheoremDB P8938",
"url": "https://theoremdb.org/statements/P8938",
"note": null,
"sort_order": 0
},
{
"kind": "formalization",
"citation": "Source formal statement (not locally verified)",
"url": "https://theoremdb.org/statements/P8938",
"not... | {
"source_record": "P8938",
"imported_at": "2026-09-08",
"status_as_of": "2026-09-08",
"evidence_label": "Source candidate · not reviewed",
"note": "Full public directory statement. Source lane: backlog; source resolution: open. Source verification claims are attributed only; no local Lean verification has be... | {"created_at":"2026-08-17T19:40:58+00:00","display_standard":{"formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":8938,"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": "2026-09-08T03:33:06.662Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"094f7a04-56cc-45a9-95ae-b98cf038e6a7","format_issues":null,"number":6791} |
1.0.0 | problem:theoremdb-p8940 | problem | theoremdb-p8940 | Effective abc conjecture with explicit multiplicative constant | {
"main": "math",
"sub": "number theory",
"slug": "number-theory",
"name": "Number theory",
"original_discipline": "mathematics"
} | [] | [
{
"code": "11",
"basis": "community-map",
"evidence": "Inferred from community field: number-theory"
}
] | Let $\varepsilon$ be a positive real number. The radical of a positive integer $n$, denoted $\operatorname{rad}(n)$, is the product of the distinct prime factors of $n$. Determine whether there exists a real constant $K$ such that for all positive integers $a$, $b$, and $c$ satisfying $a + b = c$ and $\gcd(a,b) = \gcd(... | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P8940",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P8940",
"citation": null,
"doi": null,
"arxiv_id": null,
"license": "Selected packets label their content CC0-1.0; original references retain their own terms",
"notes": "Selected problem... | theorem abc.variants.lt_constant_mul (ε : ℝ) (hε : 0 < ε) : ∃ K,
∀ (a b c : ℕ), 0 < a → 0 < b → 0 < c → ({a, b, c} : Set ℕ).Pairwise Nat.Coprime → a + b = c →
c < K * (radical <| a * b * c : ℝ)^(1 + ε) | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem abc.variants.lt_constant_mul (ε : ℝ) (hε : 0 < ε) : ∃ K,\n ∀ (a b c : ℕ), 0 < a → 0 < b → 0 < c → ({a, b, c} : Set ℕ).Pairwise Nat.Coprime → a + b = c →\n c < K * (radical <| a * b * c : ℝ)^(1 + ε)",
"target": null,
"source_url": "https://theoremdb.org/... | [] | TheoremDB P8940 · Candidate awaiting source review | null | Source-reported assessment, retrieved 8 September 2026. Frontier Science has not independently reviewed this status.
Open. No signed Lean verification is attached to this problem or its selected resolution.
This record is in the source backlog; its novelty and unresolved status have not passed source review. | null | published | incremental | hard | [
"source-candidate",
"theoremdb"
] | null | null | null | null | [
{
"kind": "website",
"citation": "TheoremDB P8940",
"url": "https://theoremdb.org/statements/P8940",
"note": null,
"sort_order": 0
},
{
"kind": "formalization",
"citation": "Source formal statement (not locally verified)",
"url": "https://theoremdb.org/statements/P8940",
"not... | {
"source_record": "P8940",
"imported_at": "2026-09-08",
"status_as_of": "2026-09-08",
"evidence_label": "Source candidate · not reviewed",
"note": "Full public directory statement. Source lane: backlog; source resolution: open. Source verification claims are attributed only; no local Lean verification has be... | {"created_at":"2026-08-17T19:41:01+00:00","display_standard":{"formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":8940,"public_status":{"evidence":{"hasFormalization":false,"hasInformalProof":false,"hasRecordedWork":false,"solutionRecord":null},"lean":{"kernelAccepted":false,"label":"","reaso... | [
"formalization-not-compiled-during-export",
"current-status-unassessed",
"source-full-text-not-present"
] | [] | {
"created_at": "2026-09-08T03:32:15.935Z",
"updated_at": "2026-09-08T03:32:15.935Z",
"published_at": "2026-09-08T03:32:54.714Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"184b3301-c9df-4e4f-bb6b-a78df98a5571","format_issues":null,"number":4362} |
1.0.0 | problem:theoremdb-p8942 | problem | theoremdb-p8942 | The gap conjecture for group growth | {
"main": "math",
"sub": "algebra",
"slug": "algebra",
"name": "Algebra",
"original_discipline": "mathematics"
} | [] | [] | Let $G$ be a group and let $S \subseteq G$ be a finite set whose closure generates $G$, i.e., the subgroup generated by $S$ equals $G$. Suppose $G$ has superpolynomial growth. Then there exists a positive integer $C$ such that for all sufficiently large natural numbers $n$,
$$
e^{\sqrt{n}} \leq \gamma_S(Cn),
$$
whe... | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P8942",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P8942",
"citation": null,
"doi": null,
"arxiv_id": null,
"license": "Selected packets label their content CC0-1.0; original references retain their own terms",
"notes": "Selected problem... | theorem gap_conjecture :
∀ (G : Type) [Group G] (S : Set G), S.Finite → Subgroup.closure S = ⊤ →
HasSuperPolynomialGrowth G →
∃ C : ℕ, 0 < C ∧
∀ᶠ n : ℕ in atTop, Real.exp (Real.sqrt (n : ℝ)) ≤
(GrowthFunction S (C * n) : ℝ) | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem gap_conjecture :\n ∀ (G : Type) [Group G] (S : Set G), S.Finite → Subgroup.closure S = ⊤ →\n HasSuperPolynomialGrowth G →\n ∃ C : ℕ, 0 < C ∧\n ∀ᶠ n : ℕ in atTop, Real.exp (Real.sqrt (n : ℝ)) ≤\n (GrowthFunction S (C * n) : ℝ)",
"targe... | [] | TheoremDB P8942 · Candidate awaiting source review | null | Source-reported assessment, retrieved 8 September 2026. Frontier Science has not independently reviewed this status.
Open. No signed Lean verification is attached to this problem or its selected resolution.
This record is in the source backlog; its novelty and unresolved status have not passed source review. | null | published | incremental | hard | [
"source-candidate",
"theoremdb"
] | null | null | null | null | [
{
"kind": "website",
"citation": "TheoremDB P8942",
"url": "https://theoremdb.org/statements/P8942",
"note": null,
"sort_order": 0
},
{
"kind": "formalization",
"citation": "Source formal statement (not locally verified)",
"url": "https://theoremdb.org/statements/P8942",
"not... | {
"source_record": "P8942",
"imported_at": "2026-09-08",
"status_as_of": "2026-09-08",
"evidence_label": "Source candidate · not reviewed",
"note": "Full public directory statement. Source lane: backlog; source resolution: open. Source verification claims are attributed only; no local Lean verification has be... | {"created_at":"2026-08-17T19:41:04+00:00","display_standard":{"formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":8942,"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": "2026-09-08T03:33:06.507Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"5c495776-94da-47bb-9e92-4c7538762014","format_issues":null,"number":6757} |
1.0.0 | problem:theoremdb-p8944 | problem | theoremdb-p8944 | DeLaViña's Conjecture 133 on the Path Number of a Connected Graph | {
"main": "math",
"sub": "combinatorics",
"slug": "combinatorics",
"name": "Combinatorics",
"original_discipline": "mathematics"
} | [] | [
{
"code": "05",
"basis": "community-map",
"evidence": "Inferred from community field: combinatorics"
}
] | Let $G$ be a finite simple connected graph with at least two vertices. The radius of $G$, denoted $\operatorname{rad}(G)$, is the minimum eccentricity among all vertices of $G$, expressed as a natural number. The path number of $G$, denoted $\operatorname{path}(G)$, is the number of vertices in a largest induced path i... | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P8944",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P8944",
"citation": null,
"doi": null,
"arxiv_id": null,
"license": "Selected packets label their content CC0-1.0; original references retain their own terms",
"notes": "Selected problem... | theorem conjecture133 (G : SimpleGraph α) [DecidableRel G.Adj] (h : G.Connected) :
let rad := G.radius.toNat
let hasC4 := ∃ a b c d : α, a ≠ b ∧ a ≠ c ∧ a ≠ d ∧ b ≠ c ∧ b ≠ d ∧ c ≠ d ∧
G.Adj a b ∧ G.Adj b c ∧ G.Adj c d ∧ G.Adj d a
let cC4 : ℕ := if hasC4 then 0 else 1
(rad : ℝ) + (⌊l G⌋ : ℝ) ^ cC4... | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem conjecture133 (G : SimpleGraph α) [DecidableRel G.Adj] (h : G.Connected) :\n let rad := G.radius.toNat\n let hasC4 := ∃ a b c d : α, a ≠ b ∧ a ≠ c ∧ a ≠ d ∧ b ≠ c ∧ b ≠ d ∧ c ≠ d ∧\n G.Adj a b ∧ G.Adj b c ∧ G.Adj c d ∧ G.Adj d a\n let cC4 : ℕ := if hasC... | [] | TheoremDB P8944 · Candidate awaiting source review | null | Source-reported assessment, retrieved 8 September 2026. Frontier Science has not independently reviewed this status.
Open. No signed Lean verification is attached to this problem or its selected resolution.
This record is in the source backlog; its novelty and unresolved status have not passed source review. | null | published | incremental | hard | [
"source-candidate",
"theoremdb"
] | null | null | null | null | [
{
"kind": "website",
"citation": "TheoremDB P8944",
"url": "https://theoremdb.org/statements/P8944",
"note": null,
"sort_order": 0
},
{
"kind": "formalization",
"citation": "Source formal statement (not locally verified)",
"url": "https://theoremdb.org/statements/P8944",
"not... | {
"source_record": "P8944",
"imported_at": "2026-09-08",
"status_as_of": "2026-09-08",
"evidence_label": "Source candidate · not reviewed",
"note": "Full public directory statement. Source lane: backlog; source resolution: open. Source verification claims are attributed only; no local Lean verification has be... | {"created_at":"2026-08-17T19:41:07+00:00","display_standard":{"editorial_review":"submitter_supplied","formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":8944,"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": "2026-09-08T03:32:53.040Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"7cfdef84-5ab6-48be-9bc2-8eb0b34e958c","format_issues":null,"number":3991} |
1.0.0 | problem:theoremdb-p8946 | problem | theoremdb-p8946 | Bounded Gaps Between Gaussian 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 Gaussian integer is a complex number of the form $a + bi$ where $a$ and $b$ are integers. The norm of a Gaussian integer $z = a + bi$ is defined as $N(z) = a^2 + b^2$. A Gaussian prime is a Gaussian integer that cannot be factored into a product of two non-unit Gaussian integers. Does there exist an infinite sequence... | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P8946",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P8946",
"citation": null,
"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_952 : answer(sorry) ↔
∃ (x : ℕ → GaussianInt) (C : ℤ),
Function.Injective x ∧
∀ n, Prime (x n) ∧ (x (n + 1) - x n).norm < C | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem erdos_952 : answer(sorry) ↔\n ∃ (x : ℕ → GaussianInt) (C : ℤ),\n Function.Injective x ∧\n ∀ n, Prime (x n) ∧ (x (n + 1) - x n).norm < C",
"target": null,
"source_url": "https://theoremdb.org/statements/P8946",
"format": "declaration-or-fragment",... | [] | TheoremDB P8946 · Candidate awaiting source review | null | Source-reported assessment, retrieved 8 September 2026. Frontier Science has not independently reviewed this status.
Open. No signed Lean verification is attached to this problem or its selected resolution.
This record is in the source backlog; its novelty and unresolved status have not passed source review. | null | published | incremental | hard | [
"source-candidate",
"theoremdb"
] | null | null | null | null | [
{
"kind": "website",
"citation": "TheoremDB P8946",
"url": "https://theoremdb.org/statements/P8946",
"note": null,
"sort_order": 0
},
{
"kind": "formalization",
"citation": "Source formal statement (not locally verified)",
"url": "https://theoremdb.org/statements/P8946",
"not... | {
"source_record": "P8946",
"imported_at": "2026-09-08",
"status_as_of": "2026-09-08",
"evidence_label": "Source candidate · not reviewed",
"note": "Full public directory statement. Source lane: backlog; source resolution: open. Source verification claims are attributed only; no local Lean verification has be... | {"created_at":"2026-08-17T19:41:10+00:00","display_standard":{"editorial_review":"submitter_supplied","formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":8946,"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": "2026-09-08T03:32:50.716Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"5cddea8b-9314-4427-ae5c-13ba10f00cf3","format_issues":null,"number":3477} |
1.0.0 | problem:theoremdb-p8948 | problem | theoremdb-p8948 | Asymptotic formula for admissible prime constellations | {
"main": "math",
"sub": "number theory",
"slug": "number-theory",
"name": "Number theory",
"original_discipline": "mathematics"
} | [] | [
{
"code": "11",
"basis": "community-map",
"evidence": "Inferred from community field: number-theory"
}
] | Let $k$ be a non-negative integer and let $m = (m_0, m_1, \dots, m_k)$ be a tuple of natural numbers with $m_0 = 0$ and $m_i > 0$ for all $i \neq 0$. A prime constellation for $m$ is a tuple $(p, p + 2m_1, \dots, p + 2m_k)$ such that every entry is a prime number. Such a prime constellation is called admissible if, for... | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P8948",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P8948",
"citation": null,
"doi": null,
"arxiv_id": null,
"license": "Selected packets label their content CC0-1.0; original references retain their own terms",
"notes": "Selected problem... | theorem first_hardy_littlewood_conjecture {k : ℕ} (m : Fin k.succ → ℕ) :
FirstHardyLittlewoodConjectureFor m | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem first_hardy_littlewood_conjecture {k : ℕ} (m : Fin k.succ → ℕ) :\n FirstHardyLittlewoodConjectureFor m",
"target": null,
"source_url": "https://theoremdb.org/statements/P8948",
"format": "declaration-or-fragment",
"sha256": "f9256c6cdb2883b1e78debf... | [] | TheoremDB P8948 · Candidate awaiting source review | null | Source-reported assessment, retrieved 8 September 2026. Frontier Science has not independently reviewed this status.
Open. No signed Lean verification is attached to this problem or its selected resolution.
This record is in the source backlog; its novelty and unresolved status have not passed source review. | null | published | incremental | hard | [
"source-candidate",
"theoremdb"
] | null | null | null | null | [
{
"kind": "website",
"citation": "TheoremDB P8948",
"url": "https://theoremdb.org/statements/P8948",
"note": null,
"sort_order": 0
},
{
"kind": "formalization",
"citation": "Source formal statement (not locally verified)",
"url": "https://theoremdb.org/statements/P8948",
"not... | {
"source_record": "P8948",
"imported_at": "2026-09-08",
"status_as_of": "2026-09-08",
"evidence_label": "Source candidate · not reviewed",
"note": "Full public directory statement. Source lane: backlog; source resolution: open. Source verification claims are attributed only; no local Lean verification has be... | {"created_at":"2026-08-17T19:41:13+00:00","display_standard":{"editorial_review":"submitter_supplied","formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":8948,"public_status":{"evidence":{"hasFormalization":false,"hasInformalProof":false,"hasRecordedWork":false,"solutionRecord":null},"lean":{... | [
"formalization-not-compiled-during-export",
"current-status-unassessed",
"source-full-text-not-present"
] | [] | {
"created_at": "2026-09-08T03:32:15.935Z",
"updated_at": "2026-09-08T03:32:15.935Z",
"published_at": "2026-09-08T03:32:50.034Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"e686e722-d1fd-4af8-92db-4b0fb89d4d0b","format_issues":null,"number":3337} |
1.0.0 | problem:theoremdb-p8950 | problem | theoremdb-p8950 | Explicit large Steiner system with $t > 5$ | {
"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 $t$, $k$, $n$ with $n > k > t > 5$, $t < 10$, and $n < 200$, a Steiner system $S(t, k, n)$ is a collection of $k$-element subsets (called blocks) of an $n$-element set such that every $t$-element subset is contained in exactly one block. Determine whether there exists a Steiner system $S(t, k, n)$... | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P8950",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P8950",
"citation": null,
"doi": null,
"arxiv_id": null,
"license": "Selected packets label their content CC0-1.0; original references retain their own terms",
"notes": "Selected problem... | def large_steiner_systems : LargeSteinerSystemWitness | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "def large_steiner_systems : LargeSteinerSystemWitness",
"target": null,
"source_url": "https://theoremdb.org/statements/P8950",
"format": "declaration-or-fragment",
"sha256": "85460abb62472218845770cba12b31663e0ccc7eed019849c8ba1bd3a8cc43fe",
"license": "... | [] | TheoremDB P8950 · Candidate awaiting source review | null | Source-reported assessment, retrieved 8 September 2026. Frontier Science has not independently reviewed this status.
Open. No signed Lean verification is attached to this problem or its selected resolution.
This record is in the source backlog; its novelty and unresolved status have not passed source review. | null | published | incremental | hard | [
"source-candidate",
"theoremdb"
] | null | null | null | null | [
{
"kind": "website",
"citation": "TheoremDB P8950",
"url": "https://theoremdb.org/statements/P8950",
"note": null,
"sort_order": 0
},
{
"kind": "formalization",
"citation": "Source formal statement (not locally verified)",
"url": "https://theoremdb.org/statements/P8950",
"not... | {
"source_record": "P8950",
"imported_at": "2026-09-08",
"status_as_of": "2026-09-08",
"evidence_label": "Source candidate · not reviewed",
"note": "Full public directory statement. Source lane: backlog; source resolution: open. Source verification claims are attributed only; no local Lean verification has be... | {"created_at":"2026-08-17T19:41:15+00:00","display_standard":{"editorial_review":"submitter_supplied","formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":8950,"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": "2026-09-08T03:32:58.889Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"b2321bd5-7348-4cef-8aea-13fb08da322c","format_issues":null,"number":5277} |
1.0.0 | problem:theoremdb-p8952 | problem | theoremdb-p8952 | Inscribed rectangle problem | {
"main": "math",
"sub": "geometry and topology",
"slug": "geometry-and-topology",
"name": "Geometry and topology",
"original_discipline": "mathematics"
} | [] | [] | Let a Jordan curve be a simple closed curve in the plane, represented as an embedding $\gamma : S^1 \to \mathbb{R}^2$. Four points $a,b,c,d \in \mathbb{R}^2$ are said to form a rectangle with aspect ratio $r > 0$ (with $a$ opposite to $c$) if the line segments from $a$ to $c$ and from $b$ to $d$ have the same midpoint ... | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P8952",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P8952",
"citation": null,
"doi": null,
"arxiv_id": null,
"license": "Selected packets label their content CC0-1.0; original references retain their own terms",
"notes": "Selected problem... | theorem inscribed_rectangle_problem :
answer(sorry) ↔ ∀ (γ : Circle → ℝ²) (hγ : IsEmbedding γ) (r : ℝ) (hr : r > 0),
∃ t₁ t₂ t₃ t₄, IsRectangle (γ t₁) (γ t₂) (γ t₃) (γ t₄) r | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem inscribed_rectangle_problem :\n answer(sorry) ↔ ∀ (γ : Circle → ℝ²) (hγ : IsEmbedding γ) (r : ℝ) (hr : r > 0),\n ∃ t₁ t₂ t₃ t₄, IsRectangle (γ t₁) (γ t₂) (γ t₃) (γ t₄) r",
"target": null,
"source_url": "https://theoremdb.org/statements/P8952",
"fo... | [] | TheoremDB P8952 · Candidate awaiting source review | null | Source-reported assessment, retrieved 8 September 2026. Frontier Science has not independently reviewed this status.
Open. No signed Lean verification is attached to this problem or its selected resolution.
This record is in the source backlog; its novelty and unresolved status have not passed source review. | null | published | incremental | hard | [
"source-candidate",
"theoremdb"
] | null | null | null | null | [
{
"kind": "website",
"citation": "TheoremDB P8952",
"url": "https://theoremdb.org/statements/P8952",
"note": null,
"sort_order": 0
},
{
"kind": "formalization",
"citation": "Source formal statement (not locally verified)",
"url": "https://theoremdb.org/statements/P8952",
"not... | {
"source_record": "P8952",
"imported_at": "2026-09-08",
"status_as_of": "2026-09-08",
"evidence_label": "Source candidate · not reviewed",
"note": "Full public directory statement. Source lane: backlog; source resolution: open. Source verification claims are attributed only; no local Lean verification has be... | {"created_at":"2026-08-17T19:41:15+00:00","display_standard":{"editorial_review":"submitter_supplied","formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":8952,"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": "2026-09-08T03:33:01.290Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"0fc09f51-6c49-47b9-9c2d-3b5c65d4e27a","format_issues":null,"number":5698} |
1.0.0 | problem:theoremdb-p8954 | problem | theoremdb-p8954 | Obligatory 3-uniform hypergraphs and 2-colorability | {
"main": "math",
"sub": "combinatorics",
"slug": "combinatorics",
"name": "Combinatorics",
"original_discipline": "mathematics"
} | [] | [
{
"code": "05",
"basis": "community-map",
"evidence": "Inferred from community field: combinatorics"
}
] | Let a 3-uniform hypergraph on a vertex set $V$ be a collection of 3-element subsets of $V$, called hyperedges. A hypergraph is 2-colorable if its vertices can be colored with two colors such that no hyperedge is monochromatic. The chromatic cardinal of a hypergraph is the smallest cardinal $\kappa$ such that there exis... | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P8954",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P8954",
"citation": null,
"doi": null,
"arxiv_id": null,
"license": "Selected packets label their content CC0-1.0; original references retain their own terms",
"notes": "Selected problem... | theorem erdos_593.variants.obligatory_implies_two_colorable : answer(sorry) ↔
∀ (W : Type) [Fintype W] (F : ThreeUniformHypergraph W),
IsObligatory F → F.IsTwoColorable | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem erdos_593.variants.obligatory_implies_two_colorable : answer(sorry) ↔\n ∀ (W : Type) [Fintype W] (F : ThreeUniformHypergraph W),\n IsObligatory F → F.IsTwoColorable",
"target": null,
"source_url": "https://theoremdb.org/statements/P8954",
"format"... | [] | TheoremDB P8954 · Candidate awaiting source review | null | Source-reported assessment, retrieved 8 September 2026. Frontier Science has not independently reviewed this status.
Open. No signed Lean verification is attached to this problem or its selected resolution.
This record is in the source backlog; its novelty and unresolved status have not passed source review. | null | published | incremental | hard | [
"source-candidate",
"theoremdb"
] | null | null | null | null | [
{
"kind": "website",
"citation": "TheoremDB P8954",
"url": "https://theoremdb.org/statements/P8954",
"note": null,
"sort_order": 0
},
{
"kind": "formalization",
"citation": "Source formal statement (not locally verified)",
"url": "https://theoremdb.org/statements/P8954",
"not... | {
"source_record": "P8954",
"imported_at": "2026-09-08",
"status_as_of": "2026-09-08",
"evidence_label": "Source candidate · not reviewed",
"note": "Full public directory statement. Source lane: backlog; source resolution: open. Source verification claims are attributed only; no local Lean verification has be... | {"created_at":"2026-08-17T19:41:17+00:00","display_standard":{"editorial_review":"submitter_supplied","formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":8954,"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": "2026-09-08T03:33:04.604Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"a34052ee-87e6-4e87-9afb-21a75f14274a","format_issues":null,"number":6346} |
1.0.0 | problem:theoremdb-p8956 | problem | theoremdb-p8956 | A weaker variant of Erdős's problem on sums of three $k$th powers | {
"main": "math",
"sub": "number theory",
"slug": "number-theory",
"name": "Number theory",
"original_discipline": "mathematics"
} | [] | [
{
"code": "11",
"basis": "community-map",
"evidence": "Inferred from community field: number-theory"
}
] | For positive integers $k$ and $n$, say that $n$ is a sum of three $k$th powers if there exist nonnegative integers $a$, $b$, $c$ such that $a^k + b^k + c^k = n$. For each $k \geq 3$ and each real $x > 0$, let $f_{k,3}(x)$ denote the number of integers $n$ with $1 \leq n \leq x$ that are sums of three $k$th powers. Dete... | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P8956",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P8956",
"citation": null,
"doi": null,
"arxiv_id": null,
"license": "Selected packets label their content CC0-1.0; original references retain their own terms",
"notes": "Selected problem... | theorem erdos_325.variants.weaker :
answer(sorry) ↔ ∀ ε > 0, ∀ k : ℕ, 3 ≤ k → (fun x : ℕ => (x : ℝ) ^ ((3 / k : ℝ) - ε)) =O[atTop]
(fun x => (cardIsSumThreePowerBelow k x : ℝ)) | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem erdos_325.variants.weaker :\n answer(sorry) ↔ ∀ ε > 0, ∀ k : ℕ, 3 ≤ k → (fun x : ℕ => (x : ℝ) ^ ((3 / k : ℝ) - ε)) =O[atTop]\n (fun x => (cardIsSumThreePowerBelow k x : ℝ))",
"target": null,
"source_url": "https://theoremdb.org/statements/P8956",
... | [] | TheoremDB P8956 · Candidate awaiting source review | null | Source-reported assessment, retrieved 8 September 2026. Frontier Science has not independently reviewed this status.
Open. No signed Lean verification is attached to this problem or its selected resolution.
This record is in the source backlog; its novelty and unresolved status have not passed source review. | null | published | incremental | hard | [
"source-candidate",
"theoremdb"
] | null | null | null | null | [
{
"kind": "website",
"citation": "TheoremDB P8956",
"url": "https://theoremdb.org/statements/P8956",
"note": null,
"sort_order": 0
},
{
"kind": "formalization",
"citation": "Source formal statement (not locally verified)",
"url": "https://theoremdb.org/statements/P8956",
"not... | {
"source_record": "P8956",
"imported_at": "2026-09-08",
"status_as_of": "2026-09-08",
"evidence_label": "Source candidate · not reviewed",
"note": "Full public directory statement. Source lane: backlog; source resolution: open. Source verification claims are attributed only; no local Lean verification has be... | {"created_at":"2026-08-17T19:41:18+00:00","display_standard":{"editorial_review":"submitter_supplied","formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":8956,"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": "2026-09-08T03:32:49.449Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"4dfba2b8-3f06-4b24-b8b9-8dcbcff0fa23","format_issues":null,"number":3210} |
1.0.0 | problem:theoremdb-p8958 | problem | theoremdb-p8958 | Erdős Problem 445 on Short Intervals with Multiplicative Inverses Modulo Primes | {
"main": "math",
"sub": "number theory",
"slug": "number-theory",
"name": "Number theory",
"original_discipline": "mathematics"
} | [] | [
{
"code": "11",
"basis": "community-map",
"evidence": "Inferred from community field: number-theory"
}
] | For a real number $c > 0$, a prime $p$, and a natural number $n$, we say that the Erdős 445 property holds if there exist natural numbers $a$ and $b$ such that $n < a < n + p^c$, $n < b < n + p^c$, and $ab \equiv 1 \pmod{p}$. Determine whether the following statement is true: for every real number $c > 1/2$, for all su... | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P8958",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P8958",
"citation": null,
"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_445 :
answer(sorry) ↔ ∀ c : ℝ, c > 1 / 2 →
∀ᶠ p : ℕ in atTop, p.Prime → ∀ n : ℕ, Erdos445Prop c p n | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem erdos_445 :\n answer(sorry) ↔ ∀ c : ℝ, c > 1 / 2 →\n ∀ᶠ p : ℕ in atTop, p.Prime → ∀ n : ℕ, Erdos445Prop c p n",
"target": null,
"source_url": "https://theoremdb.org/statements/P8958",
"format": "declaration-or-fragment",
"sha256": "8200b0abd85... | [] | TheoremDB P8958 · Candidate awaiting source review | null | Source-reported assessment, retrieved 8 September 2026. Frontier Science has not independently reviewed this status.
Open. No signed Lean verification is attached to this problem or its selected resolution.
This record is in the source backlog; its novelty and unresolved status have not passed source review. | null | published | incremental | hard | [
"source-candidate",
"theoremdb"
] | null | null | null | null | [
{
"kind": "website",
"citation": "TheoremDB P8958",
"url": "https://theoremdb.org/statements/P8958",
"note": null,
"sort_order": 0
},
{
"kind": "formalization",
"citation": "Source formal statement (not locally verified)",
"url": "https://theoremdb.org/statements/P8958",
"not... | {
"source_record": "P8958",
"imported_at": "2026-09-08",
"status_as_of": "2026-09-08",
"evidence_label": "Source candidate · not reviewed",
"note": "Full public directory statement. Source lane: backlog; source resolution: open. Source verification claims are attributed only; no local Lean verification has be... | {"created_at":"2026-08-17T19:41:20+00:00","display_standard":{"formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":8958,"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": "2026-09-08T03:32:55.497Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"36ed40b5-0ca1-4eb4-8cff-429eac1375de","format_issues":null,"number":4532} |
1.0.0 | problem:theoremdb-p8960 | problem | theoremdb-p8960 | Erdős's problem on multiplicative cardinality sets of high density | {
"main": "math",
"sub": "number theory",
"slug": "number-theory",
"name": "Number theory",
"original_discipline": "mathematics"
} | [] | [
{
"code": "11",
"basis": "community-map",
"evidence": "Inferred from community field: number-theory"
}
] | A set $A \subseteq \mathbb{N}$ of natural numbers is called a multiplicative cardinality set if whenever $a_1 \cdots a_r = b_1 \cdots b_s$ with $a_i, b_j \in A$ (allowing repetitions), it necessarily follows that $r = s$. Equivalently, for any two finite multisets $a$ and $b$ drawn from $A$, equality of their products ... | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P8960",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P8960",
"citation": null,
"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_786.parts.i : answer(sorry) ↔ ∀ ε > 0, ε ≤ 1 →
∃ (A : Set ℕ) (δ : ℝ), 0 ∉ A ∧ 1 - ε < δ ∧ A.HasDensity δ ∧ A.IsMulCardSet | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem erdos_786.parts.i : answer(sorry) ↔ ∀ ε > 0, ε ≤ 1 →\n ∃ (A : Set ℕ) (δ : ℝ), 0 ∉ A ∧ 1 - ε < δ ∧ A.HasDensity δ ∧ A.IsMulCardSet",
"target": null,
"source_url": "https://theoremdb.org/statements/P8960",
"format": "declaration-or-fragment",
"sha256... | [] | TheoremDB P8960 · Candidate awaiting source review | null | Source-reported assessment, retrieved 8 September 2026. Frontier Science has not independently reviewed this status.
Open. No signed Lean verification is attached to this problem or its selected resolution.
This record is in the source backlog; its novelty and unresolved status have not passed source review. | null | published | incremental | hard | [
"source-candidate",
"theoremdb"
] | null | null | null | null | [
{
"kind": "website",
"citation": "TheoremDB P8960",
"url": "https://theoremdb.org/statements/P8960",
"note": null,
"sort_order": 0
},
{
"kind": "formalization",
"citation": "Source formal statement (not locally verified)",
"url": "https://theoremdb.org/statements/P8960",
"not... | {
"source_record": "P8960",
"imported_at": "2026-09-08",
"status_as_of": "2026-09-08",
"evidence_label": "Source candidate · not reviewed",
"note": "Full public directory statement. Source lane: backlog; source resolution: open. Source verification claims are attributed only; no local Lean verification has be... | {"created_at":"2026-08-17T19:43:17+00:00","display_standard":{"editorial_review":"submitter_supplied","formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":8960,"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": "2026-09-08T03:32:56.509Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"74cb358e-641f-4827-a3cd-0a0721d10b6c","format_issues":null,"number":4756} |
1.0.0 | problem:theoremdb-p8962 | problem | theoremdb-p8962 | Erdős–Gallai cycle decomposition conjecture | {
"main": "math",
"sub": "combinatorics",
"slug": "combinatorics",
"name": "Combinatorics",
"original_discipline": "mathematics"
} | [] | [
{
"code": "05",
"basis": "community-map",
"evidence": "Inferred from community field: combinatorics"
}
] | Let $G$ be a finite simple graph on $n$ vertices. A subgraph $H$ of $G$ is called a cycle or an edge if either $H$ is connected and every vertex of $H$ has degree exactly $2$, or $H$ has exactly one edge. A decomposition of $G$ into subgraphs is a collection $D$ of subgraphs of $G$ whose edge sets are pairwise disjoint... | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P8962",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P8962",
"citation": null,
"doi": null,
"arxiv_id": null,
"license": "Selected packets label their content CC0-1.0; original references retain their own terms",
"notes": "Selected problem... | theorem erdos_184 :
∃ f : ℕ → ℝ,
(f =O[atTop] fun n : ℕ ↦ (n : ℝ)) ∧
∀ {V : Type*} [Fintype V] [DecidableEq V] (G : SimpleGraph V),
∃ (D : Finset G.Subgraph),
(∀ H ∈ D, IsCycleOrEdge H.coe) ∧
IsDecomposition G D ∧
(D.card : ℝ) ≤ f (Fintype.card V) | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem erdos_184 :\n ∃ f : ℕ → ℝ,\n (f =O[atTop] fun n : ℕ ↦ (n : ℝ)) ∧\n ∀ {V : Type*} [Fintype V] [DecidableEq V] (G : SimpleGraph V),\n ∃ (D : Finset G.Subgraph),\n (∀ H ∈ D, IsCycleOrEdge H.coe) ∧\n IsDecomposition G D ∧\n (D.card ... | [] | TheoremDB P8962 · Candidate awaiting source review | null | Source-reported assessment, retrieved 8 September 2026. Frontier Science has not independently reviewed this status.
Open. No signed Lean verification is attached to this problem or its selected resolution.
This record is in the source backlog; its novelty and unresolved status have not passed source review. | null | published | incremental | hard | [
"source-candidate",
"theoremdb"
] | null | null | null | null | [
{
"kind": "website",
"citation": "TheoremDB P8962",
"url": "https://theoremdb.org/statements/P8962",
"note": null,
"sort_order": 0
},
{
"kind": "formalization",
"citation": "Source formal statement (not locally verified)",
"url": "https://theoremdb.org/statements/P8962",
"not... | {
"source_record": "P8962",
"imported_at": "2026-09-08",
"status_as_of": "2026-09-08",
"evidence_label": "Source candidate · not reviewed",
"note": "Full public directory statement. Source lane: backlog; source resolution: open. Source verification claims are attributed only; no local Lean verification has be... | {"created_at":"2026-08-17T19:43:46+00:00","display_standard":{"editorial_review":"submitter_supplied","formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":8962,"public_status":{"evidence":{"hasFormalization":false,"hasInformalProof":false,"hasRecordedWork":false,"solutionRecord":null},"lean":{... | [
"formalization-not-compiled-during-export",
"current-status-unassessed",
"source-full-text-not-present"
] | [] | {
"created_at": "2026-09-08T03:32:15.935Z",
"updated_at": "2026-09-08T03:32:15.935Z",
"published_at": "2026-09-08T03:32:56.664Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"315df91c-175b-4c28-8381-631b50144ef1","format_issues":null,"number":4788} |
1.0.0 | problem:theoremdb-p8964 | problem | theoremdb-p8964 | Characterization of Sequence A56777 by Prime Quadruples | {
"main": "math",
"sub": "number theory",
"slug": "number-theory",
"name": "Number theory",
"original_discipline": "mathematics"
} | [] | [
{
"code": "11",
"basis": "community-map",
"evidence": "Inferred from community field: number-theory"
}
] | Let $n$ be a natural number. We say that $n$ is in the sequence A56777 if $n$ is composite, $n > 1$, and $n$ satisfies both $\varphi(n+12) = \varphi(n) + 12$ and $\sigma(n+12) = \sigma(n) + 12$, where $\varphi$ denotes Euler's totient function and $\sigma$ denotes the sum-of-divisors function. We say that $n$ comes fro... | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P8964",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P8964",
"citation": null,
"doi": null,
"arxiv_id": null,
"license": "Selected packets label their content CC0-1.0; original references retain their own terms",
"notes": "Selected problem... | theorem comesFromPrimeQuadruple_of_a {n : ℕ} (h : A n) : ComesFromPrimeQuadruple n | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem comesFromPrimeQuadruple_of_a {n : ℕ} (h : A n) : ComesFromPrimeQuadruple n",
"target": null,
"source_url": "https://theoremdb.org/statements/P8964",
"format": "declaration-or-fragment",
"sha256": "90b9d00dad441382a6e5818fd015655d881159a2fe15acd7b63571... | [] | TheoremDB P8964 · Candidate awaiting source review | null | Source-reported assessment, retrieved 8 September 2026. Frontier Science has not independently reviewed this status.
Open. No signed Lean verification is attached to this problem or its selected resolution.
This record is in the source backlog; its novelty and unresolved status have not passed source review. | null | published | incremental | hard | [
"source-candidate",
"theoremdb"
] | null | null | null | null | [
{
"kind": "website",
"citation": "TheoremDB P8964",
"url": "https://theoremdb.org/statements/P8964",
"note": null,
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},
{
"kind": "formalization",
"citation": "Source formal statement (not locally verified)",
"url": "https://theoremdb.org/statements/P8964",
"not... | {
"source_record": "P8964",
"imported_at": "2026-09-08",
"status_as_of": "2026-09-08",
"evidence_label": "Source candidate · not reviewed",
"note": "Full public directory statement. Source lane: backlog; source resolution: open. Source verification claims are attributed only; no local Lean verification has be... | {"created_at":"2026-08-17T19:43:52+00:00","display_standard":{"formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":8964,"public_status":{"evidence":{"hasFormalization":false,"hasInformalProof":false,"hasRecordedWork":false,"solutionRecord":null},"lean":{"kernelAccepted":false,"label":"","reaso... | [
"formalization-not-compiled-during-export",
"current-status-unassessed",
"source-full-text-not-present"
] | [] | {
"created_at": "2026-09-08T03:32:15.935Z",
"updated_at": "2026-09-08T03:32:15.935Z",
"published_at": "2026-09-08T03:32:51.094Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"12183d49-c3a2-4ad0-ab57-69cc0d19fde1","format_issues":null,"number":3559} |
1.0.0 | problem:theoremdb-p8966 | problem | theoremdb-p8966 | Existence of a finite magma satisfying Equation 677 but not Equation 255 | {
"main": "math",
"sub": "algebra",
"slug": "algebra",
"name": "Algebra",
"original_discipline": "mathematics"
} | [] | [] | A magma is a set equipped with a single binary operation, denoted here by $\diamond$. For a magma $(G, \diamond)$, consider the following two identities:
$\textbf{Equation 677}$: For all $x, y \in G$,
$$
x = y \diamond \bigl(x \diamond ((y \diamond x) \diamond y)\bigr).
$$
$\textbf{Equation 255}$: For all $x \in G$... | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P8966",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P8966",
"citation": null,
"doi": null,
"arxiv_id": null,
"license": "Selected packets label their content CC0-1.0; original references retain their own terms",
"notes": "Selected problem... | theorem Finite.Equation677_not_implies_Equation255 :
∃ (G : Type) (_ : Magma G), Finite G ∧ Equation677 G ∧ ¬ Equation255 G | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem Finite.Equation677_not_implies_Equation255 :\n ∃ (G : Type) (_ : Magma G), Finite G ∧ Equation677 G ∧ ¬ Equation255 G",
"target": null,
"source_url": "https://theoremdb.org/statements/P8966",
"format": "declaration-or-fragment",
"sha256": "7849f8ca... | [] | TheoremDB P8966 · Candidate awaiting source review | null | Source-reported assessment, retrieved 8 September 2026. Frontier Science has not independently reviewed this status.
Open. No signed Lean verification is attached to this problem or its selected resolution.
This record is in the source backlog; its novelty and unresolved status have not passed source review. | null | published | incremental | hard | [
"source-candidate",
"theoremdb"
] | null | null | null | null | [
{
"kind": "website",
"citation": "TheoremDB P8966",
"url": "https://theoremdb.org/statements/P8966",
"note": null,
"sort_order": 0
},
{
"kind": "formalization",
"citation": "Source formal statement (not locally verified)",
"url": "https://theoremdb.org/statements/P8966",
"not... | {
"source_record": "P8966",
"imported_at": "2026-09-08",
"status_as_of": "2026-09-08",
"evidence_label": "Source candidate · not reviewed",
"note": "Full public directory statement. Source lane: backlog; source resolution: open. Source verification claims are attributed only; no local Lean verification has be... | {"created_at":"2026-08-17T19:43:55+00:00","display_standard":{"editorial_review":"submitter_supplied","formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":8966,"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": "2026-09-08T03:32:58.436Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"8e20ff1b-2ccc-4d3d-a573-b2f4dc7518a6","format_issues":null,"number":5182} |
1.0.0 | problem:theoremdb-p8968 | problem | theoremdb-p8968 | Distinct Pairwise Sums of Integer Polynomial 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"
}
] | Does there exist a polynomial $f \in \mathbb{Z}[x]$ such that the sums $f(a) + f(b)$ are distinct for all pairs of nonnegative integers $(a,b)$ with $a < b$? Equivalently, does there exist a polynomial $f \in \mathbb{Z}[x]$ for which the map $(a,b) \mapsto f(a) + f(b)$ is injective on the set $\{(a,b) \in \mathbb{N} \t... | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P8968",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P8968",
"citation": null,
"doi": null,
"arxiv_id": null,
"license": "Selected packets label their content CC0-1.0; original references retain their own terms",
"notes": "Selected problem... | theorem erdos_324 : answer(sorry) ↔
∃ f : ℤ[X], {(a, b) : ℕ × ℕ | a < b}.InjOn fun (a, b) => f.eval (a : ℤ) + f.eval (b : ℤ) | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem erdos_324 : answer(sorry) ↔\n ∃ f : ℤ[X], {(a, b) : ℕ × ℕ | a < b}.InjOn fun (a, b) => f.eval (a : ℤ) + f.eval (b : ℤ)",
"target": null,
"source_url": "https://theoremdb.org/statements/P8968",
"format": "declaration-or-fragment",
"sha256": "748053b... | [] | TheoremDB P8968 · Candidate awaiting source review | null | Source-reported assessment, retrieved 8 September 2026. Frontier Science has not independently reviewed this status.
Open. No signed Lean verification is attached to this problem or its selected resolution.
This record is in the source backlog; its novelty and unresolved status have not passed source review. | null | published | incremental | hard | [
"source-candidate",
"theoremdb"
] | null | null | null | null | [
{
"kind": "website",
"citation": "TheoremDB P8968",
"url": "https://theoremdb.org/statements/P8968",
"note": null,
"sort_order": 0
},
{
"kind": "formalization",
"citation": "Source formal statement (not locally verified)",
"url": "https://theoremdb.org/statements/P8968",
"not... | {
"source_record": "P8968",
"imported_at": "2026-09-08",
"status_as_of": "2026-09-08",
"evidence_label": "Source candidate · not reviewed",
"note": "Full public directory statement. Source lane: backlog; source resolution: open. Source verification claims are attributed only; no local Lean verification has be... | {"created_at":"2026-08-17T19:43:56+00:00","display_standard":{"editorial_review":"submitter_supplied","formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":8968,"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": "2026-09-08T03:32:53.236Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"2b97a697-0857-46db-b431-c8ef860db42b","format_issues":null,"number":4032} |
1.0.0 | problem:theoremdb-p8970 | problem | theoremdb-p8970 | Moser's worm problem | {
"main": "math",
"sub": "geometry and topology",
"slug": "geometry-and-topology",
"name": "Geometry and topology",
"original_discipline": "mathematics"
} | [] | [] | A worm is defined as the range of a 1-Lipschitz function from the interval $[0,1]$ to $\mathbb{R}^2$; equivalently, it is a curve of length at most 1. A worm cover is a measurable subset $X \subseteq \mathbb{R}^2$ with the property that for every worm $w$, there exists an orientation-preserving rigid motion (a proper E... | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P8970",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P8970",
"citation": null,
"doi": null,
"arxiv_id": null,
"license": "Selected packets label their content CC0-1.0; original references retain their own terms",
"notes": "Selected problem... | theorem mosers_worm_problem :
IsGLB {v | ∃ X ∈ WormCovers, volume X = v} answer(sorry) | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem mosers_worm_problem :\n IsGLB {v | ∃ X ∈ WormCovers, volume X = v} answer(sorry)",
"target": null,
"source_url": "https://theoremdb.org/statements/P8970",
"format": "declaration-or-fragment",
"sha256": "fe9e539df18e741feaa67ca3bf619100beecf352ba042... | [] | TheoremDB P8970 · Candidate awaiting source review | null | Source-reported assessment, retrieved 8 September 2026. Frontier Science has not independently reviewed this status.
Open. No signed Lean verification is attached to this problem or its selected resolution.
This record is in the source backlog; its novelty and unresolved status have not passed source review. | null | published | incremental | hard | [
"source-candidate",
"theoremdb"
] | null | null | null | null | [
{
"kind": "website",
"citation": "TheoremDB P8970",
"url": "https://theoremdb.org/statements/P8970",
"note": null,
"sort_order": 0
},
{
"kind": "formalization",
"citation": "Source formal statement (not locally verified)",
"url": "https://theoremdb.org/statements/P8970",
"not... | {
"source_record": "P8970",
"imported_at": "2026-09-08",
"status_as_of": "2026-09-08",
"evidence_label": "Source candidate · not reviewed",
"note": "Full public directory statement. Source lane: backlog; source resolution: open. Source verification claims are attributed only; no local Lean verification has be... | {"created_at":"2026-08-17T19:44:00+00:00","display_standard":{"editorial_review":"submitter_supplied","formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":8970,"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": "2026-09-08T03:33:04.119Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"241dbd12-c331-4cc2-a7ff-6c5d7604a052","format_issues":null,"number":6242} |
1.0.0 | problem:theoremdb-p8972 | problem | theoremdb-p8972 | Growth rate of the Hofstadter-Conway $10,000 sequence | {
"main": "math",
"sub": "number theory",
"slug": "number-theory",
"name": "Number theory",
"original_discipline": "mathematics"
} | [] | [
{
"code": "11",
"basis": "community-map",
"evidence": "Inferred from community field: number-theory"
}
] | Define a sequence $f$ on the positive integers by $f(1)=f(2)=1$ and, for $n>2$,
$$
f(n)=f(n-f(n-1))+f(n-f(n-2)).
$$
(It is not known whether $f(n)$ is well-defined for all $n$.) Determine a function $g:\mathbb{Z}^+\to\mathbb{R}$ such that $f(n)=O(g(n))$ as $n\to\infty$, or determine the optimal asymptotic upper bound... | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P8972",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P8972",
"citation": null,
"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.growth_rate :
(fun n ↦ (f n : ℝ)) =O[atTop] (answer(sorry) : ℕ+ → ℝ) | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem erdos_422.variants.growth_rate :\n (fun n ↦ (f n : ℝ)) =O[atTop] (answer(sorry) : ℕ+ → ℝ)",
"target": null,
"source_url": "https://theoremdb.org/statements/P8972",
"format": "declaration-or-fragment",
"sha256": "4b1816ec76441d6a5473ae1b329f8d1d0ed1... | [] | TheoremDB P8972 · Candidate awaiting source review | null | Source-reported assessment, retrieved 8 September 2026. Frontier Science has not independently reviewed this status.
Open. No signed Lean verification is attached to this problem or its selected resolution.
This record is in the source backlog; its novelty and unresolved status have not passed source review. | null | published | incremental | hard | [
"source-candidate",
"theoremdb"
] | null | null | null | null | [
{
"kind": "website",
"citation": "TheoremDB P8972",
"url": "https://theoremdb.org/statements/P8972",
"note": null,
"sort_order": 0
},
{
"kind": "formalization",
"citation": "Source formal statement (not locally verified)",
"url": "https://theoremdb.org/statements/P8972",
"not... | {
"source_record": "P8972",
"imported_at": "2026-09-08",
"status_as_of": "2026-09-08",
"evidence_label": "Source candidate · not reviewed",
"note": "Full public directory statement. Source lane: backlog; source resolution: open. Source verification claims are attributed only; no local Lean verification has be... | {"created_at":"2026-08-17T19:44:03+00:00","display_standard":{"editorial_review":"submitter_supplied","formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":8972,"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": "2026-09-08T03:32:59.520Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"c9cadd07-35a7-4787-b42c-4ac44d01267a","format_issues":null,"number":5413} |
1.0.0 | problem:theoremdb-p8974 | problem | theoremdb-p8974 | Non-existence of monochromatic quantum graphs with eight 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 $N$ be a positive even integer. A weight function on the complete graph $K_N$ with $D$ colours and coefficients in a commutative ring $\alpha$ is a function $W$ that assigns to each unordered pair of distinct vertices $\{i,j\}$ and each colour $c \in \{0, 1, \dots, D-1\}$ a weight $W(\{i,j\}, c) \in \alpha$. For ea... | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P8974",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P8974",
"citation": null,
"doi": null,
"arxiv_id": null,
"license": "Selected packets label their content CC0-1.0; original references retain their own terms",
"notes": "Selected problem... | theorem eqSystem8_no_solution_d3 :
answer(sorry) ↔
¬ ∃ W : WeightsN 8 3 ℂ, EqSystemN 8 3 W | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem eqSystem8_no_solution_d3 :\n answer(sorry) ↔\n ¬ ∃ W : WeightsN 8 3 ℂ, EqSystemN 8 3 W",
"target": null,
"source_url": "https://theoremdb.org/statements/P8974",
"format": "declaration-or-fragment",
"sha256": "dd18cb5540effe5d4d9ad9126fb1c9e527... | [] | TheoremDB P8974 · Candidate awaiting source review | null | Source-reported assessment, retrieved 8 September 2026. Frontier Science has not independently reviewed this status.
Open. No signed Lean verification is attached to this problem or its selected resolution.
This record is in the source backlog; its novelty and unresolved status have not passed source review. | null | published | incremental | hard | [
"source-candidate",
"theoremdb"
] | null | null | null | null | [
{
"kind": "website",
"citation": "TheoremDB P8974",
"url": "https://theoremdb.org/statements/P8974",
"note": null,
"sort_order": 0
},
{
"kind": "formalization",
"citation": "Source formal statement (not locally verified)",
"url": "https://theoremdb.org/statements/P8974",
"not... | {
"source_record": "P8974",
"imported_at": "2026-09-08",
"status_as_of": "2026-09-08",
"evidence_label": "Source candidate · not reviewed",
"note": "Full public directory statement. Source lane: backlog; source resolution: open. Source verification claims are attributed only; no local Lean verification has be... | {"created_at":"2026-08-17T19:44:06+00:00","display_standard":{"editorial_review":"submitter_supplied","formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":8974,"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": "2026-09-08T03:33:04.305Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"53eba773-100d-43ec-8a3f-86e7e45a8b79","format_issues":null,"number":6283} |
1.0.0 | problem:theoremdb-p8976 | problem | theoremdb-p8976 | Convergence of the Flint Hills series | {
"main": "math",
"sub": "analysis",
"slug": "analysis",
"name": "Analysis",
"original_discipline": "mathematics"
} | [] | [] | Consider the series whose terms are indexed by the natural numbers starting from $n = 0$, with the $n$th term given by
$$
\frac{1}{(n+1)^3 \sin^2(n+1)}.
$$
Does this series converge? That is, does the infinite sum
$$
\sum_{n=0}^{\infty} \frac{1}{(n+1)^3 \sin^2(n+1)}
$$
have a finite value? | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P8976",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P8976",
"citation": null,
"doi": null,
"arxiv_id": null,
"license": "Selected packets label their content CC0-1.0; original references retain their own terms",
"notes": "Selected problem... | theorem flint_hills_series_converges :
answer(sorry) ↔
Summable (fun n : ℕ =>
1 / ((((n + 1) : ℝ)^3) * (Real.sin (n + 1)^2))) | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem flint_hills_series_converges :\n answer(sorry) ↔\n Summable (fun n : ℕ =>\n 1 / ((((n + 1) : ℝ)^3) * (Real.sin (n + 1)^2)))",
"target": null,
"source_url": "https://theoremdb.org/statements/P8976",
"format": "declaration-or-fragment",
"... | [] | TheoremDB P8976 · Candidate awaiting source review | null | Source-reported assessment, retrieved 8 September 2026. Frontier Science has not independently reviewed this status.
Open. No signed Lean verification is attached to this problem or its selected resolution.
This record is in the source backlog; its novelty and unresolved status have not passed source review. | null | published | incremental | hard | [
"source-candidate",
"theoremdb"
] | null | null | null | null | [
{
"kind": "website",
"citation": "TheoremDB P8976",
"url": "https://theoremdb.org/statements/P8976",
"note": null,
"sort_order": 0
},
{
"kind": "formalization",
"citation": "Source formal statement (not locally verified)",
"url": "https://theoremdb.org/statements/P8976",
"not... | {
"source_record": "P8976",
"imported_at": "2026-09-08",
"status_as_of": "2026-09-08",
"evidence_label": "Source candidate · not reviewed",
"note": "Full public directory statement. Source lane: backlog; source resolution: open. Source verification claims are attributed only; no local Lean verification has be... | {"created_at":"2026-08-17T19:44:09+00:00","display_standard":{"editorial_review":"submitter_supplied","formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":8976,"public_status":{"evidence":{"hasFormalization":false,"hasInformalProof":false,"hasRecordedWork":false,"solutionRecord":null},"lean":{... | [
"formalization-contains-placeholder",
"formalization-not-compiled-during-export",
"current-status-unassessed",
"source-full-text-not-present"
] | [] | {
"created_at": "2026-09-08T03:32:15.935Z",
"updated_at": "2026-09-08T03:32:15.935Z",
"published_at": "2026-09-08T03:32:52.677Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"b1b14217-0ab6-435b-b07b-fb7fae82fc2b","format_issues":null,"number":3912} |
1.0.0 | problem:theoremdb-p8978 | problem | theoremdb-p8978 | Infinitude of betrothed number pairs | {
"main": "math",
"sub": "number theory",
"slug": "number-theory",
"name": "Number theory",
"original_discipline": "mathematics"
} | [] | [
{
"code": "11",
"basis": "community-map",
"evidence": "Inferred from community field: number-theory"
}
] | For natural numbers $m$ and $n$, let $\sigma(m)$ denote the sum of all positive divisors of $m$. Two natural numbers $m$ and $n$ are called betrothed (or quasi-amicable) if $\sigma(m) = \sigma(n) = m + n + 1$. Equivalently, the sum of the proper divisors of $m$ equals $n + 1$, and the sum of the proper divisors of $n$ ... | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P8978",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P8978",
"citation": null,
"doi": null,
"arxiv_id": null,
"license": "Selected packets label their content CC0-1.0; original references retain their own terms",
"notes": "Selected problem... | theorem infinitely_many_betrothed :
answer(sorry) ↔ {p : ℕ × ℕ | p.1 < p.2 ∧ IsBetrothed p.1 p.2}.Infinite | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem infinitely_many_betrothed :\n answer(sorry) ↔ {p : ℕ × ℕ | p.1 < p.2 ∧ IsBetrothed p.1 p.2}.Infinite",
"target": null,
"source_url": "https://theoremdb.org/statements/P8978",
"format": "declaration-or-fragment",
"sha256": "6e8666dee67a899fef6322bcf... | [] | TheoremDB P8978 · Candidate awaiting source review | null | Source-reported assessment, retrieved 8 September 2026. Frontier Science has not independently reviewed this status.
Open. No signed Lean verification is attached to this problem or its selected resolution.
This record is in the source backlog; its novelty and unresolved status have not passed source review. | null | published | incremental | hard | [
"source-candidate",
"theoremdb"
] | null | null | null | null | [
{
"kind": "website",
"citation": "TheoremDB P8978",
"url": "https://theoremdb.org/statements/P8978",
"note": null,
"sort_order": 0
},
{
"kind": "formalization",
"citation": "Source formal statement (not locally verified)",
"url": "https://theoremdb.org/statements/P8978",
"not... | {
"source_record": "P8978",
"imported_at": "2026-09-08",
"status_as_of": "2026-09-08",
"evidence_label": "Source candidate · not reviewed",
"note": "Full public directory statement. Source lane: backlog; source resolution: open. Source verification claims are attributed only; no local Lean verification has be... | {"created_at":"2026-08-17T19:44:12+00:00","display_standard":{"editorial_review":"submitter_supplied","formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":8978,"public_status":{"evidence":{"hasFormalization":false,"hasInformalProof":false,"hasRecordedWork":false,"solutionRecord":null},"lean":{... | [
"formalization-contains-placeholder",
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"current-status-unassessed",
"source-full-text-not-present"
] | [] | {
"created_at": "2026-09-08T03:32:15.935Z",
"updated_at": "2026-09-08T03:32:15.935Z",
"published_at": "2026-09-08T03:33:01.099Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"9259a37e-0054-4d0e-8016-4f737ca336e2","format_issues":null,"number":5657} |
1.0.0 | problem:theoremdb-p8980 | problem | theoremdb-p8980 | Achievability of a logarithmic upper bound for additive complements to the 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 set $A \subseteq \mathbb{N}$ is called an \emph{additive complement to the primes} if every sufficiently large natural number can be written as $p + a$ for some prime $p$ and some element $a \in A$. For a set $A \subseteq \mathbb{N}$ and a positive integer $N$, let $|A \cap \{1, \ldots, N\}|$ denote the cardinality o... | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P8980",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P8980",
"citation": null,
"doi": null,
"arxiv_id": null,
"license": "Selected packets label their content CC0-1.0; original references retain their own terms",
"notes": "Selected problem... | theorem erdos_32.variants.log_bound : answer(sorry) ↔ ∃ A : Set ℕ,
IsAdditiveComplementToPrimes A ∧
(fun N => (((Finset.Icc 1 N).filter (· ∈ A)).card : ℝ)) =O[atTop]
fun N => Real.log N | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem erdos_32.variants.log_bound : answer(sorry) ↔ ∃ A : Set ℕ,\n IsAdditiveComplementToPrimes A ∧\n (fun N => (((Finset.Icc 1 N).filter (· ∈ A)).card : ℝ)) =O[atTop]\n fun N => Real.log N",
"target": null,
"source_url": "https://theoremdb.org/statement... | [] | TheoremDB P8980 · Candidate awaiting source review | null | Source-reported assessment, retrieved 8 September 2026. Frontier Science has not independently reviewed this status.
Open. No signed Lean verification is attached to this problem or its selected resolution.
This record is in the source backlog; its novelty and unresolved status have not passed source review. | null | published | incremental | hard | [
"source-candidate",
"theoremdb"
] | null | null | null | null | [
{
"kind": "website",
"citation": "TheoremDB P8980",
"url": "https://theoremdb.org/statements/P8980",
"note": null,
"sort_order": 0
},
{
"kind": "formalization",
"citation": "Source formal statement (not locally verified)",
"url": "https://theoremdb.org/statements/P8980",
"not... | {
"source_record": "P8980",
"imported_at": "2026-09-08",
"status_as_of": "2026-09-08",
"evidence_label": "Source candidate · not reviewed",
"note": "Full public directory statement. Source lane: backlog; source resolution: open. Source verification claims are attributed only; no local Lean verification has be... | {"created_at":"2026-08-17T19:44:15+00:00","display_standard":{"editorial_review":"submitter_supplied","formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":8980,"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": "2026-09-08T03:32:49.462Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"7b0d4bbc-dd10-48d9-80e8-66a645a25012","format_issues":null,"number":3213} |
1.0.0 | problem:theoremdb-p8982 | problem | theoremdb-p8982 | Ramsey size linearity of sparse graphs | {
"main": "math",
"sub": "combinatorics",
"slug": "combinatorics",
"name": "Combinatorics",
"original_discipline": "mathematics"
} | [] | [
{
"code": "05",
"basis": "community-map",
"evidence": "Inferred from community field: combinatorics"
}
] | A finite simple graph $G$ on $p$ vertices is called sparse if every induced subgraph on $k \geq 2$ vertices has at most $2k - 3$ edges. The size Ramsey number $\hat{r}(G, H)$ of two graphs $G$ and $H$ is the minimum number of edges in a graph $F$ such that every red-blue coloring of the edges of $F$ contains either a r... | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P8982",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P8982",
"citation": null,
"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_566 : answer(sorry) ↔
∀ (p : ℕ) (G : SimpleGraph (Fin p)),
-- G is sparse: every induced subgraph on k ≥ 2 vertices has ≤ 2k - 3 edges
(∀ S : Finset (Fin p), 2 ≤ S.card → (G.induce S).edgeSet.ncard ≤ 2 * S.card - 3) →
-- Then G is Ramsey size linear
∃ c > (0 : ℝ), ∀ (n : ℕ) (H ... | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem erdos_566 : answer(sorry) ↔\n ∀ (p : ℕ) (G : SimpleGraph (Fin p)),\n -- G is sparse: every induced subgraph on k ≥ 2 vertices has ≤ 2k - 3 edges\n (∀ S : Finset (Fin p), 2 ≤ S.card → (G.induce S).edgeSet.ncard ≤ 2 * S.card - 3) →\n -- Then G is Rams... | [] | TheoremDB P8982 · Candidate awaiting source review | null | Source-reported assessment, retrieved 8 September 2026. Frontier Science has not independently reviewed this status.
Open. No signed Lean verification is attached to this problem or its selected resolution.
This record is in the source backlog; its novelty and unresolved status have not passed source review. | null | published | incremental | hard | [
"source-candidate",
"theoremdb"
] | null | null | null | null | [
{
"kind": "website",
"citation": "TheoremDB P8982",
"url": "https://theoremdb.org/statements/P8982",
"note": null,
"sort_order": 0
},
{
"kind": "formalization",
"citation": "Source formal statement (not locally verified)",
"url": "https://theoremdb.org/statements/P8982",
"not... | {
"source_record": "P8982",
"imported_at": "2026-09-08",
"status_as_of": "2026-09-08",
"evidence_label": "Source candidate · not reviewed",
"note": "Full public directory statement. Source lane: backlog; source resolution: open. Source verification claims are attributed only; no local Lean verification has be... | {"created_at":"2026-08-17T19:44:18+00:00","display_standard":{"editorial_review":"submitter_supplied","formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":8982,"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": "2026-09-08T03:33:05.385Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"b37670a1-cad1-4121-baa3-1e10a8829de8","format_issues":null,"number":6516} |
1.0.0 | problem:theoremdb-p8984 | problem | theoremdb-p8984 | Existence of a Lindelöf space with Gδ singletons and cardinality exceeding the continuum | {
"main": "math",
"sub": "geometry and topology",
"slug": "geometry-and-topology",
"name": "Geometry and topology",
"original_discipline": "mathematics"
} | [] | [] | Let $\mathfrak{c} = 2^{\aleph_0}$ denote the cardinality of the continuum. Determine whether there exists a topological space $X$ such that: (i) every singleton subset of $X$ is a $G_\delta$ set, (ii) $X$ is a Lindelöf space, and (iii) the cardinality of $X$ satisfies $\mathfrak{c} < |X|$. | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P8984",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P8984",
"citation": null,
"doi": null,
"arxiv_id": null,
"license": "Selected packets label their content CC0-1.0; original references retain their own terms",
"notes": "Selected problem... | theorem HasGδSingletons.lindelof_card :
∃ (X : Type) (_ : TopologicalSpace X), HasGδSingletons X ∧ LindelofSpace X ∧ 𝔠 < #X | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem HasGδSingletons.lindelof_card :\n ∃ (X : Type) (_ : TopologicalSpace X), HasGδSingletons X ∧ LindelofSpace X ∧ 𝔠 < #X",
"target": null,
"source_url": "https://theoremdb.org/statements/P8984",
"format": "declaration-or-fragment",
"sha256": "4086d1b... | [] | TheoremDB P8984 · Candidate awaiting source review | null | Source-reported assessment, retrieved 8 September 2026. Frontier Science has not independently reviewed this status.
Open. No signed Lean verification is attached to this problem or its selected resolution.
This record is in the source backlog; its novelty and unresolved status have not passed source review. | null | published | incremental | hard | [
"source-candidate",
"theoremdb"
] | null | null | null | null | [
{
"kind": "website",
"citation": "TheoremDB P8984",
"url": "https://theoremdb.org/statements/P8984",
"note": null,
"sort_order": 0
},
{
"kind": "formalization",
"citation": "Source formal statement (not locally verified)",
"url": "https://theoremdb.org/statements/P8984",
"not... | {
"source_record": "P8984",
"imported_at": "2026-09-08",
"status_as_of": "2026-09-08",
"evidence_label": "Source candidate · not reviewed",
"note": "Full public directory statement. Source lane: backlog; source resolution: open. Source verification claims are attributed only; no local Lean verification has be... | {"created_at":"2026-08-17T19:44:20+00:00","display_standard":{"editorial_review":"submitter_supplied","formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":8984,"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": "2026-09-08T03:32:58.459Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"947a0358-53da-4892-884d-311b15566199","format_issues":null,"number":5187} |
1.0.0 | problem:theoremdb-p8986 | problem | theoremdb-p8986 | Zagier's conjecture on multiple zeta 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 natural number $n$, let the weight of a list $(s_1, \ldots, s_k)$ of positive integers be $s_1 + \cdots + s_k$. An admissible index is either the empty list or a list $(s_1, \ldots, s_k)$ with $s_1 \geq 2$ and all $s_i > 0$. The multiple zeta value $\zeta(s_1, \ldots, s_k)$ is defined by
$$
\zeta(s_1, \ldots... | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P8986",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P8986",
"citation": null,
"doi": null,
"arxiv_id": null,
"license": "Selected packets label their content CC0-1.0; original references retain their own terms",
"notes": "Selected problem... | theorem zagier_conjecture :
answer(sorry) ↔ ∀ n : ℕ, Module.finrank ℚ (mzvSpanOfWeight n) = zagierDim n | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem zagier_conjecture :\n answer(sorry) ↔ ∀ n : ℕ, Module.finrank ℚ (mzvSpanOfWeight n) = zagierDim n",
"target": null,
"source_url": "https://theoremdb.org/statements/P8986",
"format": "declaration-or-fragment",
"sha256": "3a08874bee3bf6a38cc7e0fe9e9c... | [] | TheoremDB P8986 · Candidate awaiting source review | null | Source-reported assessment, retrieved 8 September 2026. Frontier Science has not independently reviewed this status.
Open. No signed Lean verification is attached to this problem or its selected resolution.
This record is in the source backlog; its novelty and unresolved status have not passed source review. | null | published | incremental | hard | [
"source-candidate",
"theoremdb"
] | null | null | null | null | [
{
"kind": "website",
"citation": "TheoremDB P8986",
"url": "https://theoremdb.org/statements/P8986",
"note": null,
"sort_order": 0
},
{
"kind": "formalization",
"citation": "Source formal statement (not locally verified)",
"url": "https://theoremdb.org/statements/P8986",
"not... | {
"source_record": "P8986",
"imported_at": "2026-09-08",
"status_as_of": "2026-09-08",
"evidence_label": "Source candidate · not reviewed",
"note": "Full public directory statement. Source lane: backlog; source resolution: open. Source verification claims are attributed only; no local Lean verification has be... | {"created_at":"2026-08-17T19:44:23+00:00","display_standard":{"editorial_review":"submitter_supplied","formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":8986,"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": "2026-09-08T03:33:07.595Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"ebb2e605-c240-4477-b4b5-1796cbc05815","format_issues":null,"number":6994} |
1.0.0 | problem:theoremdb-p8988 | problem | theoremdb-p8988 | Brocard's Problem | {
"main": "math",
"sub": "number theory",
"slug": "number-theory",
"name": "Number theory",
"original_discipline": "mathematics"
} | [] | [
{
"code": "11",
"basis": "community-map",
"evidence": "Inferred from community field: number-theory"
}
] | Let $n$ and $m$ be positive integers. Consider the Diophantine equation $n! + 1 = m^2$, where $n!$ denotes the factorial of $n$. It is known that this equation has solutions when $n = 4$ (since $4! + 1 = 25 = 5^2$), when $n = 5$ (since $5! + 1 = 121 = 11^2$), and when $n = 7$ (since $7! + 1 = 5041 = 71^2$). Determine w... | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P8988",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P8988",
"citation": null,
"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_398 : answer(sorry) ↔ {n | ∃ m, n ! + 1 = m ^ 2} = {4, 5, 7} | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem erdos_398 : answer(sorry) ↔ {n | ∃ m, n ! + 1 = m ^ 2} = {4, 5, 7}",
"target": null,
"source_url": "https://theoremdb.org/statements/P8988",
"format": "declaration-or-fragment",
"sha256": "71c5228ef1b5fb9eb68fdb489ebff96500c0c7f73d49be328895d6b948d731... | [] | TheoremDB P8988 · Candidate awaiting source review | null | Source-reported assessment, retrieved 8 September 2026. Frontier Science has not independently reviewed this status.
Open. No signed Lean verification is attached to this problem or its selected resolution.
This record is in the source backlog; its novelty and unresolved status have not passed source review. | null | published | incremental | hard | [
"source-candidate",
"theoremdb"
] | null | null | null | null | [
{
"kind": "website",
"citation": "TheoremDB P8988",
"url": "https://theoremdb.org/statements/P8988",
"note": null,
"sort_order": 0
},
{
"kind": "formalization",
"citation": "Source formal statement (not locally verified)",
"url": "https://theoremdb.org/statements/P8988",
"not... | {
"source_record": "P8988",
"imported_at": "2026-09-08",
"status_as_of": "2026-09-08",
"evidence_label": "Source candidate · not reviewed",
"note": "Full public directory statement. Source lane: backlog; source resolution: open. Source verification claims are attributed only; no local Lean verification has be... | {"created_at":"2026-08-17T19:44:25+00:00","display_standard":{"editorial_review":"submitter_supplied","formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":8988,"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": "2026-09-08T03:32:50.791Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"c2447cd9-4f98-42e5-8f14-7718184b179e","format_issues":null,"number":3493} |
1.0.0 | problem:theoremdb-p8990 | problem | theoremdb-p8990 | Egyptian Fractions with Semiprime Denominators | {
"main": "math",
"sub": "number theory",
"slug": "number-theory",
"name": "Number theory",
"original_discipline": "mathematics"
} | [] | [
{
"code": "11",
"basis": "community-map",
"evidence": "Inferred from community field: number-theory"
}
] | For a positive rational number $q$, let $q = \frac{a}{b}$ in lowest terms with $b > 0$. A positive integer is called a semiprime if it is the product of two distinct prime numbers; equivalently, its number of distinct prime factors $\omega(n)$ equals $2$ and its total number of prime factors counted with multiplicity $... | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P8990",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P8990",
"citation": null,
"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_306 : answer(sorry) ↔ ∀ (q : ℚ), 0 < q → Squarefree q.den →
∃ k : ℕ, ∃ (n : Fin (k + 1) → ℕ), n 0 = 1 ∧ StrictMono n ∧
(∀ i ∈ Finset.Icc 1 (Fin.last k), ω (n i) = 2 ∧ Ω (n i) = 2) ∧
q = ∑ i ∈ Finset.Icc 1 (Fin.last k), (1 : ℚ) / (n i) | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem erdos_306 : answer(sorry) ↔ ∀ (q : ℚ), 0 < q → Squarefree q.den →\n ∃ k : ℕ, ∃ (n : Fin (k + 1) → ℕ), n 0 = 1 ∧ StrictMono n ∧\n (∀ i ∈ Finset.Icc 1 (Fin.last k), ω (n i) = 2 ∧ Ω (n i) = 2) ∧\n q = ∑ i ∈ Finset.Icc 1 (Fin.last k), (1 : ℚ) / (n i)",
"targ... | [] | TheoremDB P8990 · Candidate awaiting source review | null | Source-reported assessment, retrieved 8 September 2026. Frontier Science has not independently reviewed this status.
Open. No signed Lean verification is attached to this problem or its selected resolution.
This record is in the source backlog; its novelty and unresolved status have not passed source review. | null | published | incremental | hard | [
"source-candidate",
"theoremdb"
] | null | null | null | null | [
{
"kind": "website",
"citation": "TheoremDB P8990",
"url": "https://theoremdb.org/statements/P8990",
"note": null,
"sort_order": 0
},
{
"kind": "formalization",
"citation": "Source formal statement (not locally verified)",
"url": "https://theoremdb.org/statements/P8990",
"not... | {
"source_record": "P8990",
"imported_at": "2026-09-08",
"status_as_of": "2026-09-08",
"evidence_label": "Source candidate · not reviewed",
"note": "Full public directory statement. Source lane: backlog; source resolution: open. Source verification claims are attributed only; no local Lean verification has be... | {"created_at":"2026-08-17T19:44:27+00:00","display_standard":{"formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":8990,"public_status":{"evidence":{"hasFormalization":false,"hasInformalProof":false,"hasRecordedWork":false,"solutionRecord":null},"lean":{"kernelAccepted":false,"label":"","reaso... | [
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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": "2026-09-08T03:32:54.727Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"be5f33ea-ebea-4ce7-8e5d-eb43f9b52141","format_issues":null,"number":4365} |
1.0.0 | problem:theoremdb-p8992 | problem | theoremdb-p8992 | Polynomial lower bound for large structured subsets of approximate groups | {
"main": "math",
"sub": "algebra",
"slug": "algebra",
"name": "Algebra",
"original_discipline": "mathematics"
} | [] | [] | Let $G$ be a group and let $A \subseteq G$ be a finite subset. For a real number $K \geq 1$, we say that $A$ is a $K$-approximate subgroup if there exists a symmetric subset $X \subseteq G$ with $|X| \leq K$ such that $A^2 \subseteq XA$, where $A^2 = \{a_1 a_2 : a_1, a_2 \in A\}$ denotes the product set and $XA = \{xa ... | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P8992",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P8992",
"citation": null,
"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_29 :
answer(sorry) ↔
∃ C c : ℝ, 0 < C ∧ 0 < c ∧
∀ {G : Type*} [Group G] [DecidableEq G] (K : ℝ) (A : Finset G),
1 ≤ K → IsApproximateSubgroup K (A : Set G) →
∃ S ⊆ A, C * K ^ (-c) * (A.card : ℝ) ≤ (S.card : ℝ) ∧
S ^ 8 ⊆ A ^ 4 | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem green_29 :\n answer(sorry) ↔\n ∃ C c : ℝ, 0 < C ∧ 0 < c ∧\n ∀ {G : Type*} [Group G] [DecidableEq G] (K : ℝ) (A : Finset G),\n 1 ≤ K → IsApproximateSubgroup K (A : Set G) →\n ∃ S ⊆ A, C * K ^ (-c) * (A.card : ℝ) ≤ (S.card : ℝ) ∧\n ... | [] | TheoremDB P8992 · Candidate awaiting source review | null | Source-reported assessment, retrieved 8 September 2026. Frontier Science has not independently reviewed this status.
Open. No signed Lean verification is attached to this problem or its selected resolution.
This record is in the source backlog; its novelty and unresolved status have not passed source review. | null | published | incremental | hard | [
"source-candidate",
"theoremdb"
] | null | null | null | null | [
{
"kind": "website",
"citation": "TheoremDB P8992",
"url": "https://theoremdb.org/statements/P8992",
"note": null,
"sort_order": 0
},
{
"kind": "formalization",
"citation": "Source formal statement (not locally verified)",
"url": "https://theoremdb.org/statements/P8992",
"not... | {
"source_record": "P8992",
"imported_at": "2026-09-08",
"status_as_of": "2026-09-08",
"evidence_label": "Source candidate · not reviewed",
"note": "Full public directory statement. Source lane: backlog; source resolution: open. Source verification claims are attributed only; no local Lean verification has be... | {"created_at":"2026-08-17T19:44:30+00:00","display_standard":{"formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":8992,"public_status":{"evidence":{"hasFormalization":false,"hasInformalProof":false,"hasRecordedWork":false,"solutionRecord":null},"lean":{"kernelAccepted":false,"label":"","reaso... | [
"formalization-contains-placeholder",
"formalization-not-compiled-during-export",
"current-status-unassessed",
"source-full-text-not-present"
] | [] | {
"created_at": "2026-09-08T03:32:15.935Z",
"updated_at": "2026-09-08T03:32:15.935Z",
"published_at": "2026-09-08T03:33:04.898Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"8895aad5-1daf-4468-ac45-b98457b3131b","format_issues":null,"number":6410} |
1.0.0 | problem:theoremdb-p8994 | problem | theoremdb-p8994 | Infinitude of amicable pairs | {
"main": "math",
"sub": "number theory",
"slug": "number-theory",
"name": "Number theory",
"original_discipline": "mathematics"
} | [] | [
{
"code": "11",
"basis": "community-map",
"evidence": "Inferred from community field: number-theory"
}
] | For a positive integer $n$, let $\sigma(n)$ denote the sum of all positive divisors of $n$. Two distinct positive integers $a$ and $b$ form an amicable pair if $\sigma(a) = a + b$ and $\sigma(b) = a + b$, or equivalently, if each equals the sum of the proper divisors of the other. Are there infinitely many pairs of ami... | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P8994",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P8994",
"citation": null,
"doi": null,
"arxiv_id": null,
"license": "Selected packets label their content CC0-1.0; original references retain their own terms",
"notes": "Selected problem... | theorem infinitely_many_amicable : type_of% Erdos830.erdos_830.parts.i | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem infinitely_many_amicable : type_of% Erdos830.erdos_830.parts.i",
"target": null,
"source_url": "https://theoremdb.org/statements/P8994",
"format": "declaration-or-fragment",
"sha256": "5d771e87979ff1e30474bfeda9dbbef5d3ccbffc77b161e8ca5668a53e72dccf",... | [] | TheoremDB P8994 · Candidate awaiting source review | null | Source-reported assessment, retrieved 8 September 2026. Frontier Science has not independently reviewed this status.
Open. No signed Lean verification is attached to this problem or its selected resolution.
This record is in the source backlog; its novelty and unresolved status have not passed source review. | null | published | incremental | hard | [
"source-candidate",
"theoremdb"
] | null | null | null | null | [
{
"kind": "website",
"citation": "TheoremDB P8994",
"url": "https://theoremdb.org/statements/P8994",
"note": null,
"sort_order": 0
},
{
"kind": "formalization",
"citation": "Source formal statement (not locally verified)",
"url": "https://theoremdb.org/statements/P8994",
"not... | {
"source_record": "P8994",
"imported_at": "2026-09-08",
"status_as_of": "2026-09-08",
"evidence_label": "Source candidate · not reviewed",
"note": "Full public directory statement. Source lane: backlog; source resolution: open. Source verification claims are attributed only; no local Lean verification has be... | {"created_at":"2026-08-17T19:44:33+00:00","display_standard":{"editorial_review":"submitter_supplied","formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":8994,"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": "2026-09-08T03:33:01.077Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"798dcc5a-2cd2-4b79-8006-a69e7d7d093a","format_issues":null,"number":5652} |
1.0.0 | problem:theoremdb-p8996 | problem | theoremdb-p8996 | Density of infinite sequences with distinct consecutive sums | {
"main": "math",
"sub": "number theory",
"slug": "number-theory",
"name": "Number theory",
"original_discipline": "mathematics"
} | [] | [
{
"code": "11",
"basis": "community-map",
"evidence": "Inferred from community field: number-theory"
}
] | A sequence of natural numbers is said to have distinct consecutive sums if, for every pair of finite intervals of indices, the sums of the corresponding consecutive terms are equal only when the intervals themselves are equal. Formally, a sequence $A : \mathbb{N} \to \mathbb{N}$ has distinct consecutive sums when the m... | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P8996",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P8996",
"citation": null,
"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.infinite_set_density (A : ℕ → ℕ) (hA : StrictMono A)
(hA : HasDistinctSums A) :
(Set.range A).HasDensity 0 | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem erdos_357.variants.infinite_set_density (A : ℕ → ℕ) (hA : StrictMono A)\n (hA : HasDistinctSums A) :\n (Set.range A).HasDensity 0",
"target": null,
"source_url": "https://theoremdb.org/statements/P8996",
"format": "declaration-or-fragment",
"sha... | [] | TheoremDB P8996 · Candidate awaiting source review | null | Source-reported assessment, retrieved 8 September 2026. Frontier Science has not independently reviewed this status.
Open. No signed Lean verification is attached to this problem or its selected resolution.
This record is in the source backlog; its novelty and unresolved status have not passed source review. | null | published | incremental | hard | [
"source-candidate",
"theoremdb"
] | null | null | null | null | [
{
"kind": "website",
"citation": "TheoremDB P8996",
"url": "https://theoremdb.org/statements/P8996",
"note": null,
"sort_order": 0
},
{
"kind": "formalization",
"citation": "Source formal statement (not locally verified)",
"url": "https://theoremdb.org/statements/P8996",
"not... | {
"source_record": "P8996",
"imported_at": "2026-09-08",
"status_as_of": "2026-09-08",
"evidence_label": "Source candidate · not reviewed",
"note": "Full public directory statement. Source lane: backlog; source resolution: open. Source verification claims are attributed only; no local Lean verification has be... | {"created_at":"2026-08-17T19:44:36+00:00","display_standard":{"editorial_review":"submitter_supplied","formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":8996,"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": "2026-09-08T03:32:53.080Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"442ad165-5b2c-4759-9e54-ef53dabd9433","format_issues":null,"number":3999} |
1.0.0 | problem:theoremdb-p8998 | problem | theoremdb-p8998 | Lower bound for solution-free subsets under the equation $x + 3y = 2z + 2w$ | {
"main": "math",
"sub": "combinatorics",
"slug": "combinatorics",
"name": "Combinatorics",
"original_discipline": "mathematics"
} | [] | [
{
"code": "05",
"basis": "community-map",
"evidence": "Inferred from community field: combinatorics"
}
] | For a positive integer $N$, let $[N]$ denote the set $\{1, 2, \ldots, N\}$. A finite set $A \subseteq \mathbb{N}$ is called solution-free (for the equation $x + 3y = 2z + 2w$) if there do not exist distinct elements $x, y, z, w \in A$ such that $x + 3y = 2z + 2w$. Define $f(N)$ to be the maximum cardinality of a soluti... | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P8998",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P8998",
"citation": null,
"doi": null,
"arxiv_id": null,
"license": "Selected packets label their content CC0-1.0; original references retain their own terms",
"notes": "Selected problem... | theorem green_16_lower_bound :
(fun N ↦ (N : ℝ) ^ (1 / 2 : ℝ)) ≪ fun N ↦ (f N : ℝ) | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem green_16_lower_bound :\n (fun N ↦ (N : ℝ) ^ (1 / 2 : ℝ)) ≪ fun N ↦ (f N : ℝ)",
"target": null,
"source_url": "https://theoremdb.org/statements/P8998",
"format": "declaration-or-fragment",
"sha256": "ed0f32127c0cf1dcf84ea1c4c0b2ab902939169b6d57457be... | [] | TheoremDB P8998 · Candidate awaiting source review | null | Source-reported assessment, retrieved 8 September 2026. Frontier Science has not independently reviewed this status.
Open. No signed Lean verification is attached to this problem or its selected resolution.
This record is in the source backlog; its novelty and unresolved status have not passed source review. | null | published | incremental | hard | [
"source-candidate",
"theoremdb"
] | null | null | null | null | [
{
"kind": "website",
"citation": "TheoremDB P8998",
"url": "https://theoremdb.org/statements/P8998",
"note": null,
"sort_order": 0
},
{
"kind": "formalization",
"citation": "Source formal statement (not locally verified)",
"url": "https://theoremdb.org/statements/P8998",
"not... | {
"source_record": "P8998",
"imported_at": "2026-09-08",
"status_as_of": "2026-09-08",
"evidence_label": "Source candidate · not reviewed",
"note": "Full public directory statement. Source lane: backlog; source resolution: open. Source verification claims are attributed only; no local Lean verification has be... | {"created_at":"2026-08-17T19:44:39+00:00","display_standard":{"editorial_review":"submitter_supplied","formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":8998,"public_status":{"evidence":{"hasFormalization":false,"hasInformalProof":false,"hasRecordedWork":false,"solutionRecord":null},"lean":{... | [
"formalization-not-compiled-during-export",
"current-status-unassessed",
"source-full-text-not-present"
] | [] | {
"created_at": "2026-09-08T03:32:15.935Z",
"updated_at": "2026-09-08T03:32:15.935Z",
"published_at": "2026-09-08T03:33:03.415Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"d25851d2-7fed-497d-894e-7d9c179502cb","format_issues":null,"number":6091} |
1.0.0 | problem:theoremdb-p9 | problem | theoremdb-p9 | Hilbert's sixteenth problem, second part | {
"main": "math",
"sub": "dynamical systems",
"slug": "dynamical-systems",
"name": "Dynamical systems",
"original_discipline": "mathematics"
} | [] | [
{
"code": "37",
"basis": "community-map",
"evidence": "Inferred from community field: dynamical-systems"
}
] | For every positive integer $n$, there exists a finite number $H(n)$ such that each planar polynomial vector field of degree at most $n$ has at most $H(n)$ limit cycles. | open | {
"key": "theoremdb",
"name": "TheoremDB · P9",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P9",
"citation": null,
"doi": null,
"arxiv_id": null,
"license": "Selected packets label their content CC0-1.0; original references retain their own terms",
"notes": "Selected problem packe... | null | null | null | not-recorded | not-recorded | null | [] | [] | TheoremDB P9 · dynamical systems | null | Source-reported assessment, retrieved 8 September 2026. Frontier Science has not independently reviewed this status.
Open. No signed Lean verification is attached to this problem or its selected resolution. | null | published | incremental | hard | [
"source-curated",
"theoremdb"
] | null | null | null | null | [
{
"kind": "website",
"citation": "TheoremDB P9",
"url": "https://theoremdb.org/statements/P9",
"note": null,
"sort_order": 0
},
{
"kind": "website",
"citation": "arXiv:2411.09594, opening formulation and analysis of a claimed solution",
"url": "https://arxiv.org/abs/2411.09594",
... | {
"source_record": "P9",
"imported_at": "2026-09-08",
"status_as_of": "2026-09-08",
"evidence_label": "Source catalogue assessment",
"note": "Full public directory statement. Source lane: curated; source resolution: open. Source verification claims are attributed only; no local Lean verification has been perf... | {"created_at":"2026-07-22T21:49:47+00:00","display_standard":{"editorial_review":"source_backed","formal_correspondence":"not_applicable"},"lean_verified":false,"problem_number":9,"public_status":{"evidence":{"hasFormalization":false,"hasInformalProof":false,"hasRecordedWork":true,"solutionRecord":null},"lean":{"kernel... | [
"source-full-text-not-present"
] | [] | {
"created_at": "2026-09-08T03:32:15.935Z",
"updated_at": "2026-09-08T03:32:15.935Z",
"published_at": "2026-09-08T03:32:59.613Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"104605ed-f5d0-46df-aa76-bad6823a1d9c","format_issues":null,"number":5433} |
1.0.0 | problem:theoremdb-p9000 | problem | theoremdb-p9000 | Lower Bound on the Largest Induced Bipartite Subgraph of a Connected Graph | {
"main": "math",
"sub": "combinatorics",
"slug": "combinatorics",
"name": "Combinatorics",
"original_discipline": "mathematics"
} | [] | [
{
"code": "05",
"basis": "community-map",
"evidence": "Inferred from community field: combinatorics"
}
] | Let $G$ be a connected simple graph on a finite set $V$ of vertices with $|V| \geq 2$. For a vertex $v \in V$, the eccentricity $\operatorname{ecc}(v)$ is the maximum distance from $v$ to any other vertex of $G$. The average eccentricity of $G$ is $$\frac{1}{|V|} \sum_{v \in V} \operatorname{ecc}(v).$$ For a set $S \su... | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P9000",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P9000",
"citation": null,
"doi": null,
"arxiv_id": null,
"license": "Selected packets label their content CC0-1.0; original references retain their own terms",
"notes": "Selected problem... | theorem conjecture19 (G : SimpleGraph α) [Nontrivial α] (h_conn : G.Connected) :
⌊(∑ v ∈ Finset.univ, ((G.eccent v).toNat : ℝ)) / (Fintype.card α : ℝ) +
sSup (Set.range (indepNeighbors G))⌋ ≤ b G | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem conjecture19 (G : SimpleGraph α) [Nontrivial α] (h_conn : G.Connected) :\n ⌊(∑ v ∈ Finset.univ, ((G.eccent v).toNat : ℝ)) / (Fintype.card α : ℝ) +\n sSup (Set.range (indepNeighbors G))⌋ ≤ b G",
"target": null,
"source_url": "https://theoremdb.org/stat... | [] | TheoremDB P9000 · Candidate awaiting source review | null | Source-reported assessment, retrieved 8 September 2026. Frontier Science has not independently reviewed this status.
Open. No signed Lean verification is attached to this problem or its selected resolution.
This record is in the source backlog; its novelty and unresolved status have not passed source review. | null | published | incremental | hard | [
"source-candidate",
"theoremdb"
] | null | null | null | null | [
{
"kind": "website",
"citation": "TheoremDB P9000",
"url": "https://theoremdb.org/statements/P9000",
"note": null,
"sort_order": 0
},
{
"kind": "formalization",
"citation": "Source formal statement (not locally verified)",
"url": "https://theoremdb.org/statements/P9000",
"not... | {
"source_record": "P9000",
"imported_at": "2026-09-08",
"status_as_of": "2026-09-08",
"evidence_label": "Source candidate · not reviewed",
"note": "Full public directory statement. Source lane: backlog; source resolution: open. Source verification claims are attributed only; no local Lean verification has be... | {"created_at":"2026-08-17T19:44:41+00:00","display_standard":{"formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":9000,"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": "2026-09-08T03:33:03.570Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"094129e7-ccef-4048-a52b-16f345f098b3","format_issues":null,"number":6123} |
1.0.0 | problem:theoremdb-p9002 | problem | theoremdb-p9002 | Erdős's problem on high-girth, high-chromatic-number subgraphs | {
"main": "math",
"sub": "combinatorics",
"slug": "combinatorics",
"name": "Combinatorics",
"original_discipline": "mathematics"
} | [] | [
{
"code": "05",
"basis": "community-map",
"evidence": "Inferred from community field: combinatorics"
}
] | For every integer $r \geq 4$ and every integer $k \geq 2$, does there exist a finite integer $f(k,r)$ such that every nonempty simple graph $G$ with chromatic number at least $f(k,r)$ contains a subgraph $H$ whose girth is at least $r$ and whose chromatic number is at least $k$? | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P9002",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P9002",
"citation": null,
"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_108 :
answer(sorry) ↔ ∀ r ≥ 4, ∀ k ≥ (2 : ℕ), ∃ (f : ℕ),
∀ (V : Type u) (G : SimpleGraph V) (_ : Nonempty V)
(hchro : f ≤ SimpleGraph.chromaticNumber G),
∃ (H : G.Subgraph), (SimpleGraph.girth H.coe ≥ r) ∧
(SimpleGraph.chromaticNumber H.coe ≥ k) | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem erdos_108 :\n answer(sorry) ↔ ∀ r ≥ 4, ∀ k ≥ (2 : ℕ), ∃ (f : ℕ),\n ∀ (V : Type u) (G : SimpleGraph V) (_ : Nonempty V)\n (hchro : f ≤ SimpleGraph.chromaticNumber G),\n ∃ (H : G.Subgraph), (SimpleGraph.girth H.coe ≥ r) ∧\n (SimpleGraph.chromaticNumber... | [] | TheoremDB P9002 · Candidate awaiting source review | null | Source-reported assessment, retrieved 8 September 2026. Frontier Science has not independently reviewed this status.
Open. No signed Lean verification is attached to this problem or its selected resolution.
This record is in the source backlog; its novelty and unresolved status have not passed source review. | null | published | incremental | hard | [
"source-candidate",
"theoremdb"
] | null | null | null | null | [
{
"kind": "website",
"citation": "TheoremDB P9002",
"url": "https://theoremdb.org/statements/P9002",
"note": null,
"sort_order": 0
},
{
"kind": "formalization",
"citation": "Source formal statement (not locally verified)",
"url": "https://theoremdb.org/statements/P9002",
"not... | {
"source_record": "P9002",
"imported_at": "2026-09-08",
"status_as_of": "2026-09-08",
"evidence_label": "Source candidate · not reviewed",
"note": "Full public directory statement. Source lane: backlog; source resolution: open. Source verification claims are attributed only; no local Lean verification has be... | {"created_at":"2026-08-17T19:44:43+00:00","display_standard":{"editorial_review":"submitter_supplied","formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":9002,"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": "2026-09-08T03:32:56.469Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"a0351d6b-7ff2-48b7-b034-49f2e99bb15e","format_issues":null,"number":4747} |
1.0.0 | problem:theoremdb-p9004 | problem | theoremdb-p9004 | Affine copies of a geometric progression in measurable sets of positive measure | {
"main": "math",
"sub": "analysis",
"slug": "analysis",
"name": "Analysis",
"original_discipline": "mathematics"
} | [] | [] | Let $A \subset \mathbb{R}$ be a measurable set with positive Lebesgue measure. An affine copy of the set $\{1, \frac{1}{2}, \frac{1}{4}, \ldots\}$ in $A$ is a set of the form $\{a \cdot \frac{1}{2^n} + b : n \in \mathbb{N}\}$ for some real numbers $a, b$ with $a \neq 0$. Does every such set $A$ contain an affine copy o... | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P9004",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P9004",
"citation": null,
"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_94 :
answer(sorry) ↔ ∀ A : Set ℝ,
MeasurableSet A ∧ volume A > 0 →
∃ a b : ℝ, a ≠ 0 ∧ ∀ n : ℕ, a * (1 / 2^n) + b ∈ A | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem green_94 :\n answer(sorry) ↔ ∀ A : Set ℝ,\n MeasurableSet A ∧ volume A > 0 →\n ∃ a b : ℝ, a ≠ 0 ∧ ∀ n : ℕ, a * (1 / 2^n) + b ∈ A",
"target": null,
"source_url": "https://theoremdb.org/statements/P9004",
"format": "declaration-or-fragment",
"sha2... | [] | TheoremDB P9004 · Candidate awaiting source review | null | Source-reported assessment, retrieved 8 September 2026. Frontier Science has not independently reviewed this status.
Open. No signed Lean verification is attached to this problem or its selected resolution.
This record is in the source backlog; its novelty and unresolved status have not passed source review. | null | published | incremental | hard | [
"source-candidate",
"theoremdb"
] | null | null | null | null | [
{
"kind": "website",
"citation": "TheoremDB P9004",
"url": "https://theoremdb.org/statements/P9004",
"note": null,
"sort_order": 0
},
{
"kind": "formalization",
"citation": "Source formal statement (not locally verified)",
"url": "https://theoremdb.org/statements/P9004",
"not... | {
"source_record": "P9004",
"imported_at": "2026-09-08",
"status_as_of": "2026-09-08",
"evidence_label": "Source candidate · not reviewed",
"note": "Full public directory statement. Source lane: backlog; source resolution: open. Source verification claims are attributed only; no local Lean verification has be... | {"created_at":"2026-08-17T19:44:45+00:00","display_standard":{"editorial_review":"submitter_supplied","formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":9004,"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": "2026-09-08T03:32:49.491Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"4782575d-d54d-43e2-9586-9e9d602b3630","format_issues":null,"number":3219} |
1.0.0 | problem:theoremdb-p9006 | problem | theoremdb-p9006 | Minimum iterations of the sum-of-divisors-minus-one map to reach a prime | {
"main": "math",
"sub": "number theory",
"slug": "number-theory",
"name": "Number theory",
"original_discipline": "mathematics"
} | [] | [
{
"code": "11",
"basis": "community-map",
"evidence": "Inferred from community field: number-theory"
}
] | For each integer $n > 1$, define the arithmetic function $\sigma(n)$ to be the sum of all positive divisors of $n$. Consider the iteration of the map $f(n) = \sigma(n) - 1$. For a given integer $n > 1$, let $c(n)$ denote the minimum non-negative integer $i$ such that the $i$-fold iterate $f^{(i)}(n)$ is a prime number.... | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P9006",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P9006",
"citation": null,
"doi": null,
"arxiv_id": null,
"license": "Selected packets label their content CC0-1.0; original references retain their own terms",
"notes": "Selected problem... | theorem erdos_409.variants.sigma (n : ℕ) (hn : n > 1) :
IsLeast { i | (σ 1 · - 1)^[i] n |>.Prime } answer(sorry) | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem erdos_409.variants.sigma (n : ℕ) (hn : n > 1) :\n IsLeast { i | (σ 1 · - 1)^[i] n |>.Prime } answer(sorry)",
"target": null,
"source_url": "https://theoremdb.org/statements/P9006",
"format": "declaration-or-fragment",
"sha256": "49d1c783bd81d72066d... | [] | TheoremDB P9006 · Candidate awaiting source review | null | Source-reported assessment, retrieved 8 September 2026. Frontier Science has not independently reviewed this status.
Open. No signed Lean verification is attached to this problem or its selected resolution.
This record is in the source backlog; its novelty and unresolved status have not passed source review. | null | published | incremental | hard | [
"source-candidate",
"theoremdb"
] | null | null | null | null | [
{
"kind": "website",
"citation": "TheoremDB P9006",
"url": "https://theoremdb.org/statements/P9006",
"note": null,
"sort_order": 0
},
{
"kind": "formalization",
"citation": "Source formal statement (not locally verified)",
"url": "https://theoremdb.org/statements/P9006",
"not... | {
"source_record": "P9006",
"imported_at": "2026-09-08",
"status_as_of": "2026-09-08",
"evidence_label": "Source candidate · not reviewed",
"note": "Full public directory statement. Source lane: backlog; source resolution: open. Source verification claims are attributed only; no local Lean verification has be... | {"created_at":"2026-08-17T19:44:48+00:00","display_standard":{"editorial_review":"submitter_supplied","formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":9006,"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": "2026-09-08T03:33:04.017Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"d01bab6d-e6cb-4d2f-96fc-931b669a1ad3","format_issues":null,"number":6221} |
1.0.0 | problem:theoremdb-p9008 | problem | theoremdb-p9008 | Growth ratio of consecutive diagonal Ramsey numbers | {
"main": "math",
"sub": "combinatorics",
"slug": "combinatorics",
"name": "Combinatorics",
"original_discipline": "mathematics"
} | [] | [
{
"code": "05",
"basis": "community-map",
"evidence": "Inferred from community field: combinatorics"
}
] | Let $R(n)$ denote the diagonal Ramsey number $R(n,n)$, which is the smallest integer $N$ such that every red-blue coloring of the edges of the complete graph $K_N$ contains either a red $K_n$ or a blue $K_n$. Determine whether there exists a constant $c > 0$ such that the ratio of consecutive diagonal Ramsey numbers sa... | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P9008",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P9008",
"citation": null,
"doi": null,
"arxiv_id": null,
"license": "Selected packets label their content CC0-1.0; original references retain their own terms",
"notes": "Selected problem... | theorem erdos_812.parts.i :
answer(sorry) ↔ ∃ c > 0, ∀ᶠ n in atTop, (R (n + 1) : ℝ) / (R n : ℝ) ≥ 1 + c | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem erdos_812.parts.i :\n answer(sorry) ↔ ∃ c > 0, ∀ᶠ n in atTop, (R (n + 1) : ℝ) / (R n : ℝ) ≥ 1 + c",
"target": null,
"source_url": "https://theoremdb.org/statements/P9008",
"format": "declaration-or-fragment",
"sha256": "80685f50a5ee3175d35cf1cb7d7a... | [] | TheoremDB P9008 · Candidate awaiting source review | null | Source-reported assessment, retrieved 8 September 2026. Frontier Science has not independently reviewed this status.
Open. No signed Lean verification is attached to this problem or its selected resolution.
This record is in the source backlog; its novelty and unresolved status have not passed source review. | null | published | incremental | hard | [
"source-candidate",
"theoremdb"
] | null | null | null | null | [
{
"kind": "website",
"citation": "TheoremDB P9008",
"url": "https://theoremdb.org/statements/P9008",
"note": null,
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},
{
"kind": "formalization",
"citation": "Source formal statement (not locally verified)",
"url": "https://theoremdb.org/statements/P9008",
"not... | {
"source_record": "P9008",
"imported_at": "2026-09-08",
"status_as_of": "2026-09-08",
"evidence_label": "Source candidate · not reviewed",
"note": "Full public directory statement. Source lane: backlog; source resolution: open. Source verification claims are attributed only; no local Lean verification has be... | {"created_at":"2026-08-17T19:44:52+00:00","display_standard":{"editorial_review":"submitter_supplied","formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":9008,"public_status":{"evidence":{"hasFormalization":false,"hasInformalProof":false,"hasRecordedWork":false,"solutionRecord":null},"lean":{... | [
"formalization-contains-placeholder",
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"current-status-unassessed",
"source-full-text-not-present"
] | [] | {
"created_at": "2026-09-08T03:32:15.935Z",
"updated_at": "2026-09-08T03:32:15.935Z",
"published_at": "2026-09-08T03:32:59.527Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"d84f7c81-8c55-4cf1-80d4-f341da431ba2","format_issues":null,"number":5414} |
1.0.0 | problem:theoremdb-p9010 | problem | theoremdb-p9010 | Least radius for packing three unit squares in a circle | {
"main": "math",
"sub": "geometry and topology",
"slug": "geometry-and-topology",
"name": "Geometry and topology",
"original_discipline": "mathematics"
} | [] | [] | For a positive real number $r$, let $C(r)$ denote the open disk $\{(x,y) \in \mathbb{R}^2 : x^2 + y^2 < r^2\}$ of radius $r$ centered at the origin. Let $U$ denote the open unit square $\{(x,y) \in \mathbb{R}^2 : 0 < x < 1,\; 0 < y < 1\}$. A packing of three copies of $U$ into $C(r)$ consists of three isometric embeddi... | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P9010",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P9010",
"citation": null,
"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_three_square_packing_in_circle :
IsLeast {r : ℝ≥0 | Nonempty (Packing 3 UnitSquare (Circle r))} answer(sorry) | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem least_three_square_packing_in_circle :\n IsLeast {r : ℝ≥0 | Nonempty (Packing 3 UnitSquare (Circle r))} answer(sorry)",
"target": null,
"source_url": "https://theoremdb.org/statements/P9010",
"format": "declaration-or-fragment",
"sha256": "885f67ad... | [] | TheoremDB P9010 · Candidate awaiting source review | null | Source-reported assessment, retrieved 8 September 2026. Frontier Science has not independently reviewed this status.
Open. No signed Lean verification is attached to this problem or its selected resolution.
This record is in the source backlog; its novelty and unresolved status have not passed source review. | null | published | incremental | hard | [
"source-candidate",
"theoremdb"
] | null | null | null | null | [
{
"kind": "website",
"citation": "TheoremDB P9010",
"url": "https://theoremdb.org/statements/P9010",
"note": null,
"sort_order": 0
},
{
"kind": "formalization",
"citation": "Source formal statement (not locally verified)",
"url": "https://theoremdb.org/statements/P9010",
"not... | {
"source_record": "P9010",
"imported_at": "2026-09-08",
"status_as_of": "2026-09-08",
"evidence_label": "Source candidate · not reviewed",
"note": "Full public directory statement. Source lane: backlog; source resolution: open. Source verification claims are attributed only; no local Lean verification has be... | {"created_at":"2026-08-17T19:44:56+00:00","display_standard":{"editorial_review":"submitter_supplied","formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":9010,"public_status":{"evidence":{"hasFormalization":false,"hasInformalProof":false,"hasRecordedWork":false,"solutionRecord":null},"lean":{... | [
"formalization-contains-placeholder",
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"source-full-text-not-present"
] | [] | {
"created_at": "2026-09-08T03:32:15.935Z",
"updated_at": "2026-09-08T03:32:15.935Z",
"published_at": "2026-09-08T03:33:03.111Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"448f791b-04e6-4e76-9040-8be31d77bfb5","format_issues":null,"number":6025} |
1.0.0 | problem:theoremdb-p9012 | problem | theoremdb-p9012 | The Invariant Subspace Problem for Separable Hilbert Spaces | {
"main": "math",
"sub": "analysis",
"slug": "analysis",
"name": "Analysis",
"original_discipline": "mathematics"
} | [] | [] | Let $H$ be a separable Hilbert space over the complex numbers with dimension at least $2$, and let $T : H \to H$ be a bounded linear operator. Does there exist a closed linear subspace $W \subseteq H$ such that $W \neq \{0\}$, $W \neq H$, and $T(W) \subseteq W$? | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P9012",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P9012",
"citation": null,
"doi": null,
"arxiv_id": null,
"license": "Selected packets label their content CC0-1.0; original references retain their own terms",
"notes": "Selected problem... | theorem Invariant_subspace_problem [InnerProductSpace ℂ H] [TopologicalSpace.SeparableSpace H]
[CompleteSpace H] (hdim : 2 ≤ Module.rank ℂ H) (T : H →L[ℂ] H) :
Nonempty (ClosedInvariantSubspace T) | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem Invariant_subspace_problem [InnerProductSpace ℂ H] [TopologicalSpace.SeparableSpace H]\n [CompleteSpace H] (hdim : 2 ≤ Module.rank ℂ H) (T : H →L[ℂ] H) :\n Nonempty (ClosedInvariantSubspace T)",
"target": null,
"source_url": "https://theoremdb.org/state... | [] | TheoremDB P9012 · Candidate awaiting source review | null | Source-reported assessment, retrieved 8 September 2026. Frontier Science has not independently reviewed this status.
Open. No signed Lean verification is attached to this problem or its selected resolution.
This record is in the source backlog; its novelty and unresolved status have not passed source review. | null | published | incremental | hard | [
"source-candidate",
"theoremdb"
] | null | null | null | null | [
{
"kind": "website",
"citation": "TheoremDB P9012",
"url": "https://theoremdb.org/statements/P9012",
"note": null,
"sort_order": 0
},
{
"kind": "formalization",
"citation": "Source formal statement (not locally verified)",
"url": "https://theoremdb.org/statements/P9012",
"not... | {
"source_record": "P9012",
"imported_at": "2026-09-08",
"status_as_of": "2026-09-08",
"evidence_label": "Source candidate · not reviewed",
"note": "Full public directory statement. Source lane: backlog; source resolution: open. Source verification claims are attributed only; no local Lean verification has be... | {"created_at":"2026-08-17T19:45:00+00:00","display_standard":{"formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":9012,"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": "2026-09-08T03:33:06.561Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"9d722fb3-7224-4bfe-9db9-d40b8ea7db10","format_issues":null,"number":6769} |
1.0.0 | problem:theoremdb-p9014 | problem | theoremdb-p9014 | Irreducibility of the Third Cuboid Polynomial | {
"main": "math",
"sub": "algebra",
"slug": "algebra",
"name": "Algebra",
"original_discipline": "mathematics"
} | [] | [] | For integers $a$, $b$, and $c$, define the third Cuboid polynomial $P_{a,b,c}(X)$ to be
$$
X^{12} + (6c^2 - 2a^2 - 2b^2)X^{10} + (c^4 + b^4 + a^4 + 4a^2c^2 + 4b^2c^2 - 12b^2a^2)X^8
$$
$$
+ (6a^4c^2 + 6c^2b^4 - 8a^2b^2c^2 - 2c^4a^2 - 2c^4b^2 - 2a^4b^2 - 2b^4a^2)X^6
$$
$$
+ (4c^2b^4a^2 + 4a^4c^2b^2 - 12c^4a^2b^2 + c... | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P9014",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P9014",
"citation": null,
"doi": null,
"arxiv_id": null,
"license": "Selected packets label their content CC0-1.0; original references retain their own terms",
"notes": "Selected problem... | theorem cuboidThree : CuboidThree | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem cuboidThree : CuboidThree",
"target": null,
"source_url": "https://theoremdb.org/statements/P9014",
"format": "declaration-or-fragment",
"sha256": "36f73e8060157c2a5fb298ee53aadf332603f3d6795cb4c368559e5042e93b83",
"license": "Selected packets lab... | [] | TheoremDB P9014 · Candidate awaiting source review | null | Source-reported assessment, retrieved 8 September 2026. Frontier Science has not independently reviewed this status.
Open. No signed Lean verification is attached to this problem or its selected resolution.
This record is in the source backlog; its novelty and unresolved status have not passed source review. | null | published | incremental | hard | [
"source-candidate",
"theoremdb"
] | null | null | null | null | [
{
"kind": "website",
"citation": "TheoremDB P9014",
"url": "https://theoremdb.org/statements/P9014",
"note": null,
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},
{
"kind": "formalization",
"citation": "Source formal statement (not locally verified)",
"url": "https://theoremdb.org/statements/P9014",
"not... | {
"source_record": "P9014",
"imported_at": "2026-09-08",
"status_as_of": "2026-09-08",
"evidence_label": "Source candidate · not reviewed",
"note": "Full public directory statement. Source lane: backlog; source resolution: open. Source verification claims are attributed only; no local Lean verification has be... | {"created_at":"2026-08-17T19:45:04+00:00","display_standard":{"editorial_review":"submitter_supplied","formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":9014,"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": "2026-09-08T03:33:01.854Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"0bce58d7-4a31-4519-ba2f-818abb19097e","format_issues":null,"number":5749} |
1.0.0 | problem:theoremdb-p9016 | problem | theoremdb-p9016 | Upper bound on the maximal length of consecutive integers with distinct divisor counts | {
"main": "math",
"sub": "number theory",
"slug": "number-theory",
"name": "Number theory",
"original_discipline": "mathematics"
} | [] | [
{
"code": "11",
"basis": "community-map",
"evidence": "Inferred from community field: number-theory"
}
] | For a positive integer $m$, let $\tau(m)$ denote the number of positive divisors of $m$. For a real number $x \geq 1$, define $F(x)$ to be the maximum integer $k$ such that there exists a natural number $n$ with $n + k \leq x$ and the values $\tau(n+1), \tau(n+2), \ldots, \tau(n+k)$ are all distinct. Is it true that th... | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P9016",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P9016",
"citation": null,
"doi": null,
"arxiv_id": null,
"license": "Selected packets label their content CC0-1.0; original references retain their own terms",
"notes": "Selected problem... | theorem erdos_945 : answer(sorry) ↔ Erdos945Prop | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem erdos_945 : answer(sorry) ↔ Erdos945Prop",
"target": null,
"source_url": "https://theoremdb.org/statements/P9016",
"format": "declaration-or-fragment",
"sha256": "3b360090ed73543de299ad00e089b65df6a770c3be2a64d7e267881dc2225114",
"license": "Selec... | [] | TheoremDB P9016 · Candidate awaiting source review | null | Source-reported assessment, retrieved 8 September 2026. Frontier Science has not independently reviewed this status.
Open. No signed Lean verification is attached to this problem or its selected resolution.
This record is in the source backlog; its novelty and unresolved status have not passed source review. | null | published | incremental | hard | [
"source-candidate",
"theoremdb"
] | null | null | null | null | [
{
"kind": "website",
"citation": "TheoremDB P9016",
"url": "https://theoremdb.org/statements/P9016",
"note": null,
"sort_order": 0
},
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"citation": "Source formal statement (not locally verified)",
"url": "https://theoremdb.org/statements/P9016",
"not... | {
"source_record": "P9016",
"imported_at": "2026-09-08",
"status_as_of": "2026-09-08",
"evidence_label": "Source candidate · not reviewed",
"note": "Full public directory statement. Source lane: backlog; source resolution: open. Source verification claims are attributed only; no local Lean verification has be... | {"created_at":"2026-08-17T19:45:08+00:00","display_standard":{"editorial_review":"submitter_supplied","formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":9016,"public_status":{"evidence":{"hasFormalization":false,"hasInformalProof":false,"hasRecordedWork":false,"solutionRecord":null},"lean":{... | [
"formalization-contains-placeholder",
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"source-full-text-not-present"
] | [] | {
"created_at": "2026-09-08T03:32:15.935Z",
"updated_at": "2026-09-08T03:32:15.935Z",
"published_at": "2026-09-08T03:33:07.395Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"924d12d8-09b0-47ca-92da-e613444b0377","format_issues":null,"number":6950} |
1.0.0 | problem:theoremdb-p9018 | problem | theoremdb-p9018 | Erdős's Problem on Sets with Prime Differences and Positive Prime Density | {
"main": "math",
"sub": "number theory",
"slug": "number-theory",
"name": "Number theory",
"original_discipline": "mathematics"
} | [] | [
{
"code": "11",
"basis": "community-map",
"evidence": "Inferred from community field: number-theory"
}
] | Let $\pi(n)$ denote the prime counting function, which gives the number of primes less than or equal to $n$. For a set $A \subseteq \mathbb{N}$, define the prime density ratio of $A$ up to $n$ as
$$
\frac{|A \cap [1,n]|}{\pi(n)},
$$
where $|A \cap [1,n]|$ denotes the cardinality of the finite set $A \cap [1,n]$. Do... | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P9018",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P9018",
"citation": null,
"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_428 :
answer(sorry) ↔ ∃ A : Set ℕ,
(∃ᶠ n in atTop, ∀ a ∈ A, 0 < a → a < n → (n - a).Prime) ∧
liminf (fun n ↦ primeDensityRatio A n) atTop > 0 | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem erdos_428 :\n answer(sorry) ↔ ∃ A : Set ℕ,\n (∃ᶠ n in atTop, ∀ a ∈ A, 0 < a → a < n → (n - a).Prime) ∧\n liminf (fun n ↦ primeDensityRatio A n) atTop > 0",
"target": null,
"source_url": "https://theoremdb.org/statements/P9018",
"format": "dec... | [] | TheoremDB P9018 · Candidate awaiting source review | null | Source-reported assessment, retrieved 8 September 2026. Frontier Science has not independently reviewed this status.
Open. No signed Lean verification is attached to this problem or its selected resolution.
This record is in the source backlog; its novelty and unresolved status have not passed source review. | null | published | incremental | hard | [
"source-candidate",
"theoremdb"
] | null | null | null | null | [
{
"kind": "website",
"citation": "TheoremDB P9018",
"url": "https://theoremdb.org/statements/P9018",
"note": null,
"sort_order": 0
},
{
"kind": "formalization",
"citation": "Source formal statement (not locally verified)",
"url": "https://theoremdb.org/statements/P9018",
"not... | {
"source_record": "P9018",
"imported_at": "2026-09-08",
"status_as_of": "2026-09-08",
"evidence_label": "Source candidate · not reviewed",
"note": "Full public directory statement. Source lane: backlog; source resolution: open. Source verification claims are attributed only; no local Lean verification has be... | {"created_at":"2026-08-17T19:45:13+00:00","display_standard":{"editorial_review":"submitter_supplied","formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":9018,"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": "2026-09-08T03:32:56.535Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"a00d96d5-8b3d-4d23-b0d6-3400c41f6eea","format_issues":null,"number":4761} |
1.0.0 | problem:theoremdb-p9020 | problem | theoremdb-p9020 | Monotonicity of the minimum degree forcing a 4-cycle | {
"main": "math",
"sub": "combinatorics",
"slug": "combinatorics",
"name": "Combinatorics",
"original_discipline": "mathematics"
} | [] | [
{
"code": "05",
"basis": "community-map",
"evidence": "Inferred from community field: combinatorics"
}
] | For each positive integer $n$, let $f(n)$ denote the smallest integer $k$ such that every simple graph on $n$ vertices with minimum degree at least $k$ contains a cycle of length $4$ as a subgraph. Determine whether it is true that $f(n) \leq f(n+1)$ holds for all sufficiently large integers $n$. | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P9020",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P9020",
"citation": null,
"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_85 : answer(sorry) ↔ ∀ᶠ n in atTop, f n ≤ f (n + 1) | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem erdos_85 : answer(sorry) ↔ ∀ᶠ n in atTop, f n ≤ f (n + 1)",
"target": null,
"source_url": "https://theoremdb.org/statements/P9020",
"format": "declaration-or-fragment",
"sha256": "d17d98a3c9a12bc9d54be2c287b14365b9e84e7e62279408a593bade10655fbd",
... | [] | TheoremDB P9020 · Candidate awaiting source review | null | Source-reported assessment, retrieved 8 September 2026. Frontier Science has not independently reviewed this status.
Open. No signed Lean verification is attached to this problem or its selected resolution.
This record is in the source backlog; its novelty and unresolved status have not passed source review. | null | published | incremental | hard | [
"source-candidate",
"theoremdb"
] | null | null | null | null | [
{
"kind": "website",
"citation": "TheoremDB P9020",
"url": "https://theoremdb.org/statements/P9020",
"note": null,
"sort_order": 0
},
{
"kind": "formalization",
"citation": "Source formal statement (not locally verified)",
"url": "https://theoremdb.org/statements/P9020",
"not... | {
"source_record": "P9020",
"imported_at": "2026-09-08",
"status_as_of": "2026-09-08",
"evidence_label": "Source candidate · not reviewed",
"note": "Full public directory statement. Source lane: backlog; source resolution: open. Source verification claims are attributed only; no local Lean verification has be... | {"created_at":"2026-08-17T19:45:16+00:00","display_standard":{"editorial_review":"submitter_supplied","formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":9020,"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": "2026-09-08T03:33:04.109Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"b3685640-532a-4787-8bb3-0f77e526fa1a","format_issues":null,"number":6240} |
1.0.0 | problem:theoremdb-p9022 | problem | theoremdb-p9022 | Infinitude of Lucas–Wieferich primes with prescribed fundamental discriminant | {
"main": "math",
"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 fundamental discriminant is an integer $D$ that arises as the discriminant of a quadratic number field; equivalently, either $D = 4m$ where $m \equiv 2$ or $3 \pmod{4}$ is squarefree, or $D \equiv 1 \pmod{4}$ is squarefree and $D \neq 1$. For integers $a$ and $b$, the discriminant of the pair $(a,b)$ is $a^2 - 4b$. F... | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P9022",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P9022",
"citation": null,
"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_of_disc_eq {D : ℤ} (hD : IsFundamentalDiscr D)
(hD₁ : D ≠ 1) :
{p : ℕ | ∃ a b, a ^ 2 - 4 * b = D ∧ IsLucasWieferichPrime a b p}.Infinite | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem infinite_isWallSunSunPrime_of_disc_eq {D : ℤ} (hD : IsFundamentalDiscr D)\n (hD₁ : D ≠ 1) :\n {p : ℕ | ∃ a b, a ^ 2 - 4 * b = D ∧ IsLucasWieferichPrime a b p}.Infinite",
"target": null,
"source_url": "https://theoremdb.org/statements/P9022",
"format... | [] | TheoremDB P9022 · Candidate awaiting source review | null | Source-reported assessment, retrieved 8 September 2026. Frontier Science has not independently reviewed this status.
Open. No signed Lean verification is attached to this problem or its selected resolution.
This record is in the source backlog; its novelty and unresolved status have not passed source review. | null | published | incremental | hard | [
"source-candidate",
"theoremdb"
] | null | null | null | null | [
{
"kind": "website",
"citation": "TheoremDB P9022",
"url": "https://theoremdb.org/statements/P9022",
"note": null,
"sort_order": 0
},
{
"kind": "formalization",
"citation": "Source formal statement (not locally verified)",
"url": "https://theoremdb.org/statements/P9022",
"not... | {
"source_record": "P9022",
"imported_at": "2026-09-08",
"status_as_of": "2026-09-08",
"evidence_label": "Source candidate · not reviewed",
"note": "Full public directory statement. Source lane: backlog; source resolution: open. Source verification claims are attributed only; no local Lean verification has be... | {"created_at":"2026-08-17T19:45:20+00:00","display_standard":{"editorial_review":"submitter_supplied","formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":9022,"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": "2026-09-08T03:33:01.168Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"97b4248b-5e5e-4392-8584-a8936cf63d47","format_issues":null,"number":5672} |
1.0.0 | problem:theoremdb-p9024 | problem | theoremdb-p9024 | Existence of a universal area threshold for unit-area triangles in measurable plane sets | {
"main": "math",
"sub": "geometry and topology",
"slug": "geometry-and-topology",
"name": "Geometry and topology",
"original_discipline": "mathematics"
} | [] | [] | Let $\mathbb{R}^2$ denote the Euclidean plane. For a measurable set $A \subseteq \mathbb{R}^2$, let $\mathbb{P}(A)$ denote its Lebesgue measure. A triangle in $\mathbb{R}^2$ is determined by three non-collinear points, and its Euclidean area is denoted $\operatorname{area}(\triangle)$. Does there exist a constant $c > ... | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P9024",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P9024",
"citation": null,
"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_352 :
answer(sorry) ↔ ∃ c > (0: ℝ), ∀ A : Set ℝ², MeasurableSet A → ℙ A ≥ c.toEReal
→ (∃ t : Affine.Triangle ℝ ℝ²,
(∀ p : Fin 3, t.points p ∈ A) ∧
EuclideanGeometry.triangle_area (t.points 0) (t.points 1) (t.points 2) = 1) | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem erdos_352 :\n answer(sorry) ↔ ∃ c > (0: ℝ), ∀ A : Set ℝ², MeasurableSet A → ℙ A ≥ c.toEReal\n → (∃ t : Affine.Triangle ℝ ℝ²,\n (∀ p : Fin 3, t.points p ∈ A) ∧\n EuclideanGeometry.triangle_area (t.points 0) (t.points 1) (t.points 2) = 1)",... | [] | TheoremDB P9024 · Candidate awaiting source review | null | Source-reported assessment, retrieved 8 September 2026. Frontier Science has not independently reviewed this status.
Open. No signed Lean verification is attached to this problem or its selected resolution.
This record is in the source backlog; its novelty and unresolved status have not passed source review. | null | published | incremental | hard | [
"source-candidate",
"theoremdb"
] | null | null | null | null | [
{
"kind": "website",
"citation": "TheoremDB P9024",
"url": "https://theoremdb.org/statements/P9024",
"note": null,
"sort_order": 0
},
{
"kind": "formalization",
"citation": "Source formal statement (not locally verified)",
"url": "https://theoremdb.org/statements/P9024",
"not... | {
"source_record": "P9024",
"imported_at": "2026-09-08",
"status_as_of": "2026-09-08",
"evidence_label": "Source candidate · not reviewed",
"note": "Full public directory statement. Source lane: backlog; source resolution: open. Source verification claims are attributed only; no local Lean verification has be... | {"created_at":"2026-08-17T19:45:23+00:00","display_standard":{"editorial_review":"submitter_supplied","formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":9024,"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": "2026-09-08T03:32:58.560Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"6e016b27-8421-454c-834d-20e358508076","format_issues":null,"number":5208} |
1.0.0 | problem:theoremdb-p9026 | problem | theoremdb-p9026 | Value Distribution of Entire Functions with Fabry Gaps | {
"main": "math",
"sub": "analysis",
"slug": "analysis",
"name": "Analysis",
"original_discipline": "mathematics"
} | [] | [] | An entire function is a complex-valued function that is holomorphic on the entire complex plane. A sequence of natural numbers $n : \mathbb{N} \to \mathbb{N}$ is said to have Fabry gaps if $\displaystyle\lim_{k \to \infty} \frac{n_k}{k} = \infty$. Let $f : \mathbb{C} \to \mathbb{C}$ be an entire function with power ser... | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P9026",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P9026",
"citation": null,
"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_517 : answer(sorry) ↔ ∀ {f : ℂ → ℂ} {n : ℕ → ℕ} (hn : HasFabryGaps n)
{a : ℕ → ℂ} (ha : ∀ k, a k ≠ 0) (hf : ∀ z, HasSum (fun k => a k * z ^ n k) (f z)) (z : ℂ),
{x : ℂ | f x = z}.Infinite | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem erdos_517 : answer(sorry) ↔ ∀ {f : ℂ → ℂ} {n : ℕ → ℕ} (hn : HasFabryGaps n)\n {a : ℕ → ℂ} (ha : ∀ k, a k ≠ 0) (hf : ∀ z, HasSum (fun k => a k * z ^ n k) (f z)) (z : ℂ),\n {x : ℂ | f x = z}.Infinite",
"target": null,
"source_url": "https://theoremdb.org/... | [] | TheoremDB P9026 · Candidate awaiting source review | null | Source-reported assessment, retrieved 8 September 2026. Frontier Science has not independently reviewed this status.
Open. No signed Lean verification is attached to this problem or its selected resolution.
This record is in the source backlog; its novelty and unresolved status have not passed source review. | null | published | incremental | hard | [
"source-candidate",
"theoremdb"
] | null | null | null | null | [
{
"kind": "website",
"citation": "TheoremDB P9026",
"url": "https://theoremdb.org/statements/P9026",
"note": null,
"sort_order": 0
},
{
"kind": "formalization",
"citation": "Source formal statement (not locally verified)",
"url": "https://theoremdb.org/statements/P9026",
"not... | {
"source_record": "P9026",
"imported_at": "2026-09-08",
"status_as_of": "2026-09-08",
"evidence_label": "Source candidate · not reviewed",
"note": "Full public directory statement. Source lane: backlog; source resolution: open. Source verification claims are attributed only; no local Lean verification has be... | {"created_at":"2026-08-17T19:45:27+00:00","display_standard":{"editorial_review":"submitter_supplied","formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":9026,"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": "2026-09-08T03:33:07.413Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"a72dc5df-6c99-43f8-8635-e7b6a4ecc523","format_issues":null,"number":6954} |
1.0.0 | problem:theoremdb-p9028 | problem | theoremdb-p9028 | Linear Growth Lower Bound for the Arithmetic Sum Associated with the Classical Theta Function | {
"main": "math",
"sub": "number theory",
"slug": "number-theory",
"name": "Number theory",
"original_discipline": "mathematics"
} | [] | [
{
"code": "11",
"basis": "community-map",
"evidence": "Inferred from community field: number-theory"
}
] | For each fixed natural number $n$, determine whether there exists a threshold $K$ (depending on $n$) such that for every odd prime $k > K$, the inequality $n k < S(k)$ holds, where the arithmetic sum $S(k)$ associated with the classical theta function is defined as follows. For positive integers $h$ and $k$, define
$$... | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P9028",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P9028",
"citation": null,
"doi": null,
"arxiv_id": null,
"license": "Selected packets label their content CC0-1.0; original references retain their own terms",
"notes": "Selected problem... | theorem conjecture_4_4 (n : ℕ) : ∀ᶠ (k : ℕ) in Filter.atTop, k.Prime → Odd k → n * k < S k | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem conjecture_4_4 (n : ℕ) : ∀ᶠ (k : ℕ) in Filter.atTop, k.Prime → Odd k → n * k < S k",
"target": null,
"source_url": "https://theoremdb.org/statements/P9028",
"format": "declaration-or-fragment",
"sha256": "1b520dfed871d9e8b82af5a8c22d8de864249c87e919eb... | [] | TheoremDB P9028 · Candidate awaiting source review | null | Source-reported assessment, retrieved 8 September 2026. Frontier Science has not independently reviewed this status.
Open. No signed Lean verification is attached to this problem or its selected resolution.
This record is in the source backlog; its novelty and unresolved status have not passed source review. | null | published | incremental | hard | [
"source-candidate",
"theoremdb"
] | null | null | null | null | [
{
"kind": "website",
"citation": "TheoremDB P9028",
"url": "https://theoremdb.org/statements/P9028",
"note": null,
"sort_order": 0
},
{
"kind": "formalization",
"citation": "Source formal statement (not locally verified)",
"url": "https://theoremdb.org/statements/P9028",
"not... | {
"source_record": "P9028",
"imported_at": "2026-09-08",
"status_as_of": "2026-09-08",
"evidence_label": "Source candidate · not reviewed",
"note": "Full public directory statement. Source lane: backlog; source resolution: open. Source verification claims are attributed only; no local Lean verification has be... | {"created_at":"2026-08-17T19:45:31+00:00","display_standard":{"editorial_review":"submitter_supplied","formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":9028,"public_status":{"evidence":{"hasFormalization":false,"hasInformalProof":false,"hasRecordedWork":false,"solutionRecord":null},"lean":{... | [
"formalization-not-compiled-during-export",
"current-status-unassessed",
"source-full-text-not-present"
] | [] | {
"created_at": "2026-09-08T03:32:15.935Z",
"updated_at": "2026-09-08T03:32:15.935Z",
"published_at": "2026-09-08T03:33:03.236Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"be2a0b17-4964-486c-909a-a5cee3e33701","format_issues":null,"number":6052} |
1.0.0 | problem:theoremdb-p9030 | problem | theoremdb-p9030 | Infinitely many consecutive integers with distinct prime factorization exponents | {
"main": "math",
"sub": "number theory",
"slug": "number-theory",
"name": "Number theory",
"original_discipline": "mathematics"
} | [] | [
{
"code": "11",
"basis": "community-map",
"evidence": "Inferred from community field: number-theory"
}
] | For a positive integer $m$, let the prime factorization of $m$ be written as $m = \prod_i p_i^{k_i}$ where the $p_i$ are distinct primes and the $k_i$ are positive integers. We say that $m$ has distinct exponents if all the exponents $k_i$ in its prime factorization are distinct. Determine whether there are infinitely ... | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P9030",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P9030",
"citation": null,
"doi": null,
"arxiv_id": null,
"license": "Selected packets label their content CC0-1.0; original references retain their own terms",
"notes": "Selected problem... | theorem erdos_913 : answer(sorry) ↔
{ n | Set.InjOn (n * (n + 1)).factorization (n * (n + 1)).primeFactors }.Infinite | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem erdos_913 : answer(sorry) ↔\n { n | Set.InjOn (n * (n + 1)).factorization (n * (n + 1)).primeFactors }.Infinite",
"target": null,
"source_url": "https://theoremdb.org/statements/P9030",
"format": "declaration-or-fragment",
"sha256": "bbbeeaaa0db801... | [] | TheoremDB P9030 · Candidate awaiting source review | null | Source-reported assessment, retrieved 8 September 2026. Frontier Science has not independently reviewed this status.
Open. No signed Lean verification is attached to this problem or its selected resolution.
This record is in the source backlog; its novelty and unresolved status have not passed source review. | null | published | incremental | hard | [
"source-candidate",
"theoremdb"
] | null | null | null | null | [
{
"kind": "website",
"citation": "TheoremDB P9030",
"url": "https://theoremdb.org/statements/P9030",
"note": null,
"sort_order": 0
},
{
"kind": "formalization",
"citation": "Source formal statement (not locally verified)",
"url": "https://theoremdb.org/statements/P9030",
"not... | {
"source_record": "P9030",
"imported_at": "2026-09-08",
"status_as_of": "2026-09-08",
"evidence_label": "Source candidate · not reviewed",
"note": "Full public directory statement. Source lane: backlog; source resolution: open. Source verification claims are attributed only; no local Lean verification has be... | {"created_at":"2026-08-17T19:45:34+00:00","display_standard":{"editorial_review":"submitter_supplied","formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":9030,"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": "2026-09-08T03:33:00.843Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"3a4ca650-f9e1-4d3b-a113-d34cff50339c","format_issues":null,"number":5602} |
1.0.0 | problem:theoremdb-p9032 | problem | theoremdb-p9032 | Density of hyperbolicity in the Mandelbrot set | {
"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 complex parameter $c$, let $f_c : \mathbb{C} \to \mathbb{C}$ be the quadratic polynomial defined by $f_c(z) = z^2 + c$. A point $z \in \mathbb{C}$ is said to be part of an attracting cycle of period $m$ of $f_c$ if $m > 0$, $f_c^{[m]}(z) = z$ (where $f_c^{[m]}$ denotes the $m$-fold iterate of $f_c$), the itera... | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P9032",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P9032",
"citation": null,
"doi": null,
"arxiv_id": null,
"license": "Selected packets label their content CC0-1.0; original references retain their own terms",
"notes": "Selected problem... | theorem density_of_hyperbolicity :
mandelbrotSet ⊆ closure {c | ∃ m z, IsAttractingCycle (fun z ↦ z ^ 2 + c) m z} | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem density_of_hyperbolicity :\n mandelbrotSet ⊆ closure {c | ∃ m z, IsAttractingCycle (fun z ↦ z ^ 2 + c) m z}",
"target": null,
"source_url": "https://theoremdb.org/statements/P9032",
"format": "declaration-or-fragment",
"sha256": "b592311d59d1dc0dc1... | [] | TheoremDB P9032 · Candidate awaiting source review | null | Source-reported assessment, retrieved 8 September 2026. Frontier Science has not independently reviewed this status.
Open. No signed Lean verification is attached to this problem or its selected resolution.
This record is in the source backlog; its novelty and unresolved status have not passed source review. | null | published | incremental | hard | [
"source-candidate",
"theoremdb"
] | null | null | null | null | [
{
"kind": "website",
"citation": "TheoremDB P9032",
"url": "https://theoremdb.org/statements/P9032",
"note": null,
"sort_order": 0
},
{
"kind": "formalization",
"citation": "Source formal statement (not locally verified)",
"url": "https://theoremdb.org/statements/P9032",
"not... | {
"source_record": "P9032",
"imported_at": "2026-09-08",
"status_as_of": "2026-09-08",
"evidence_label": "Source candidate · not reviewed",
"note": "Full public directory statement. Source lane: backlog; source resolution: open. Source verification claims are attributed only; no local Lean verification has be... | {"created_at":"2026-08-17T19:45:38+00:00","display_standard":{"editorial_review":"submitter_supplied","formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":9032,"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": "2026-09-08T03:32:53.074Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"0be73c2d-c4b8-44fa-b6d7-9ff7b59ddbc1","format_issues":null,"number":3998} |
1.0.0 | problem:theoremdb-p9034 | problem | theoremdb-p9034 | Large Gap Dilates for Square-Root Sized Sets in Cyclic Groups of Prime Order | {
"main": "math",
"sub": "combinatorics",
"slug": "combinatorics",
"name": "Combinatorics",
"original_discipline": "mathematics"
} | [] | [
{
"code": "05",
"basis": "community-map",
"evidence": "Inferred from community field: combinatorics"
}
] | For a prime $p$, a subset $A \subseteq \mathbb{Z}/p\mathbb{Z}$, and a unit $c \in (\mathbb{Z}/p\mathbb{Z})^\times$, the dilate of $A$ by $c$ is the set $c \cdot A = \{ca : a \in A\}$. A set $A$ has a gap of length $L$ if there exists $x \in \mathbb{Z}/p\mathbb{Z}$ such that none of $x, x+1, \ldots, x+L-1$ belong to $A$... | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P9034",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P9034",
"citation": null,
"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_32 :
answer(sorry) ↔ HasLargeGapDilate (fun p ↦ Real.sqrt p) | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem green_32 :\n answer(sorry) ↔ HasLargeGapDilate (fun p ↦ Real.sqrt p)",
"target": null,
"source_url": "https://theoremdb.org/statements/P9034",
"format": "declaration-or-fragment",
"sha256": "b74c2a23fe0865c10fa17cea794e4dd5127cdd0172e329fa210f18b12... | [] | TheoremDB P9034 · Candidate awaiting source review | null | Source-reported assessment, retrieved 8 September 2026. Frontier Science has not independently reviewed this status.
Open. No signed Lean verification is attached to this problem or its selected resolution.
This record is in the source backlog; its novelty and unresolved status have not passed source review. | null | published | incremental | hard | [
"source-candidate",
"theoremdb"
] | null | null | null | null | [
{
"kind": "website",
"citation": "TheoremDB P9034",
"url": "https://theoremdb.org/statements/P9034",
"note": null,
"sort_order": 0
},
{
"kind": "formalization",
"citation": "Source formal statement (not locally verified)",
"url": "https://theoremdb.org/statements/P9034",
"not... | {
"source_record": "P9034",
"imported_at": "2026-09-08",
"status_as_of": "2026-09-08",
"evidence_label": "Source candidate · not reviewed",
"note": "Full public directory statement. Source lane: backlog; source resolution: open. Source verification claims are attributed only; no local Lean verification has be... | {"created_at":"2026-08-17T19:45:42+00:00","display_standard":{"formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":9034,"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": "2026-09-08T03:33:02.900Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"45d52a8d-b10b-4782-a314-2c1b25943d8f","format_issues":null,"number":5979} |
1.0.0 | problem:theoremdb-p9036 | problem | theoremdb-p9036 | The Salzer–Levine bound for non-representable integers | {
"main": "math",
"sub": "number theory",
"slug": "number-theory",
"name": "Number theory",
"original_discipline": "mathematics"
} | [] | [
{
"code": "11",
"basis": "community-map",
"evidence": "Inferred from community field: number-theory"
}
] | For each natural number $n$, the $n$-th tetrahedral number is defined as
$$
T_n = \frac{n(n+1)(n+2)}{6}.
$$
A natural number $N$ is said to belong to the set $\mathcal{E}$ if there do not exist natural numbers $a_1, a_2, a_3, a_4$ such that
$$
N = T_{a_1} + T_{a_2} + T_{a_3} + T_{a_4}.
$$
Determine whether $343... | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P9036",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P9036",
"citation": null,
"doi": null,
"arxiv_id": null,
"license": "Selected packets label their content CC0-1.0; original references retain their own terms",
"notes": "Selected problem... | theorem pollock_tetrahedral.salzer_levine :
IsGreatest NotSumOfFourTetrahedral 343867 | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem pollock_tetrahedral.salzer_levine :\n IsGreatest NotSumOfFourTetrahedral 343867",
"target": null,
"source_url": "https://theoremdb.org/statements/P9036",
"format": "declaration-or-fragment",
"sha256": "11d78c5e5c3a5c009bccb69fa2c405148cbb59b137e85a... | [] | TheoremDB P9036 · Candidate awaiting source review | null | Source-reported assessment, retrieved 8 September 2026. Frontier Science has not independently reviewed this status.
Open. No signed Lean verification is attached to this problem or its selected resolution.
This record is in the source backlog; its novelty and unresolved status have not passed source review. | null | published | incremental | hard | [
"source-candidate",
"theoremdb"
] | null | null | null | null | [
{
"kind": "website",
"citation": "TheoremDB P9036",
"url": "https://theoremdb.org/statements/P9036",
"note": null,
"sort_order": 0
},
{
"kind": "formalization",
"citation": "Source formal statement (not locally verified)",
"url": "https://theoremdb.org/statements/P9036",
"not... | {
"source_record": "P9036",
"imported_at": "2026-09-08",
"status_as_of": "2026-09-08",
"evidence_label": "Source candidate · not reviewed",
"note": "Full public directory statement. Source lane: backlog; source resolution: open. Source verification claims are attributed only; no local Lean verification has be... | {"created_at":"2026-08-17T19:45:47+00:00","display_standard":{"editorial_review":"submitter_supplied","formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":9036,"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": "2026-09-08T03:33:06.750Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"54424a81-9c8e-4ef4-94fc-58ff379225b5","format_issues":null,"number":6809} |
1.0.0 | problem:theoremdb-p9038 | problem | theoremdb-p9038 | Asymptotic order of the subset sum 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$, define $h(n)$ to be the largest integer $m$ such that for every set $A$ of $n$ integers, there exists a subset $B \subseteq A$ with $|B| \geq m$ having the following property: if two nonempty subsets of $B$ have equal sums, then they have equal cardinalities. Equivalently, all subset sums... | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P9038",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P9038",
"citation": null,
"doi": null,
"arxiv_id": null,
"license": "Selected packets label their content CC0-1.0; original references retain their own terms",
"notes": "Selected problem... | theorem erdos_789.variants.sq :
(fun n ↦ (subsetSumThreshold n : ℝ)) =Θ[atTop] fun n ↦ √n | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem erdos_789.variants.sq :\n (fun n ↦ (subsetSumThreshold n : ℝ)) =Θ[atTop] fun n ↦ √n",
"target": null,
"source_url": "https://theoremdb.org/statements/P9038",
"format": "declaration-or-fragment",
"sha256": "60c694f0cf8fc382d32b456e1bbceb82b573ad6432... | [] | TheoremDB P9038 · Candidate awaiting source review | null | Source-reported assessment, retrieved 8 September 2026. Frontier Science has not independently reviewed this status.
Open. No signed Lean verification is attached to this problem or its selected resolution.
This record is in the source backlog; its novelty and unresolved status have not passed source review. | null | published | incremental | hard | [
"source-candidate",
"theoremdb"
] | null | null | null | null | [
{
"kind": "website",
"citation": "TheoremDB P9038",
"url": "https://theoremdb.org/statements/P9038",
"note": null,
"sort_order": 0
},
{
"kind": "formalization",
"citation": "Source formal statement (not locally verified)",
"url": "https://theoremdb.org/statements/P9038",
"not... | {
"source_record": "P9038",
"imported_at": "2026-09-08",
"status_as_of": "2026-09-08",
"evidence_label": "Source candidate · not reviewed",
"note": "Full public directory statement. Source lane: backlog; source resolution: open. Source verification claims are attributed only; no local Lean verification has be... | {"created_at":"2026-08-17T19:45:51+00:00","display_standard":{"editorial_review":"submitter_supplied","formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":9038,"public_status":{"evidence":{"hasFormalization":false,"hasInformalProof":false,"hasRecordedWork":false,"solutionRecord":null},"lean":{... | [
"formalization-not-compiled-during-export",
"current-status-unassessed",
"source-full-text-not-present"
] | [] | {
"created_at": "2026-09-08T03:32:15.935Z",
"updated_at": "2026-09-08T03:32:15.935Z",
"published_at": "2026-09-08T03:32:50.374Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"b20cfdee-8928-4291-b3a9-cfdeedf41d7d","format_issues":null,"number":3406} |
1.0.0 | problem:theoremdb-p9040 | problem | theoremdb-p9040 | Lower bound for the two-color van der Waerden number W(3, 28) | {
"main": "math",
"sub": "combinatorics",
"slug": "combinatorics",
"name": "Combinatorics",
"original_discipline": "mathematics"
} | [] | [
{
"code": "05",
"basis": "community-map",
"evidence": "Inferred from community field: combinatorics"
}
] | For positive integers $k$ and $r$, the two-color van der Waerden number $W(k, r)$ is defined as the least positive integer $N$ such that every coloring of the set $\{1, 2, \ldots, N\}$ with two colors contains either a monochromatic arithmetic progression of length $k$ in the first color or a monochromatic arithmetic p... | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P9040",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P9040",
"citation": null,
"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_28_lower : answer(sorry) ↔ W 3 28 ≥ 827 | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem W_3_28_lower : answer(sorry) ↔ W 3 28 ≥ 827",
"target": null,
"source_url": "https://theoremdb.org/statements/P9040",
"format": "declaration-or-fragment",
"sha256": "86f61cbdf860f51c608293879af02f05da5bc00e682444e81cf64f6a89574c91",
"license": "Se... | [] | TheoremDB P9040 · Candidate awaiting source review | null | Source-reported assessment, retrieved 8 September 2026. Frontier Science has not independently reviewed this status.
Open. No signed Lean verification is attached to this problem or its selected resolution.
This record is in the source backlog; its novelty and unresolved status have not passed source review. | null | published | incremental | hard | [
"source-candidate",
"theoremdb"
] | null | null | null | null | [
{
"kind": "website",
"citation": "TheoremDB P9040",
"url": "https://theoremdb.org/statements/P9040",
"note": null,
"sort_order": 0
},
{
"kind": "formalization",
"citation": "Source formal statement (not locally verified)",
"url": "https://theoremdb.org/statements/P9040",
"not... | {
"source_record": "P9040",
"imported_at": "2026-09-08",
"status_as_of": "2026-09-08",
"evidence_label": "Source candidate · not reviewed",
"note": "Full public directory statement. Source lane: backlog; source resolution: open. Source verification claims are attributed only; no local Lean verification has be... | {"created_at":"2026-08-17T19:45:55+00:00","display_standard":{"formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":9040,"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": "2026-09-08T03:33:03.518Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"f88a7baf-2818-4395-aa4d-82c59eb2cc5e","format_issues":null,"number":6112} |
1.0.0 | problem:theoremdb-p9042 | problem | theoremdb-p9042 | Infinitude of composite integers with large minimal GCD | {
"main": "math",
"sub": "number theory",
"slug": "number-theory",
"name": "Number theory",
"original_discipline": "mathematics"
} | [] | [
{
"code": "11",
"basis": "community-map",
"evidence": "Inferred from community field: number-theory"
}
] | For a positive integer $n$, define
$$
f(n)=\min_{1<k\le n/2}\gcd\!\Bigl(n,\binom{n}{k}\Bigr),
$$
where the minimum is taken over all integers $k$ satisfying $1<k\le n/2$. (If no such $k$ exists, i.e. if $n<4$, the infimum is $0$.)
Determine whether there exist infinitely many composite integers $n>1$ such that $f(n)... | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P9042",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P9042",
"citation": null,
"doi": null,
"arxiv_id": null,
"license": "Selected packets label their content CC0-1.0; original references retain their own terms",
"notes": "Selected problem... | theorem erdos_700.parts.ii :
answer(sorry) ↔ {n : ℕ | ¬ n.Prime ∧ 1 < n ∧ (f n) ^ 2 > n}.Infinite | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem erdos_700.parts.ii :\n answer(sorry) ↔ {n : ℕ | ¬ n.Prime ∧ 1 < n ∧ (f n) ^ 2 > n}.Infinite",
"target": null,
"source_url": "https://theoremdb.org/statements/P9042",
"format": "declaration-or-fragment",
"sha256": "dff317654cb8b5aa641370f58e00c4d84f... | [] | TheoremDB P9042 · Candidate awaiting source review | null | Source-reported assessment, retrieved 8 September 2026. Frontier Science has not independently reviewed this status.
Open. No signed Lean verification is attached to this problem or its selected resolution.
This record is in the source backlog; its novelty and unresolved status have not passed source review. | null | published | incremental | hard | [
"source-candidate",
"theoremdb"
] | null | null | null | null | [
{
"kind": "website",
"citation": "TheoremDB P9042",
"url": "https://theoremdb.org/statements/P9042",
"note": null,
"sort_order": 0
},
{
"kind": "formalization",
"citation": "Source formal statement (not locally verified)",
"url": "https://theoremdb.org/statements/P9042",
"not... | {
"source_record": "P9042",
"imported_at": "2026-09-08",
"status_as_of": "2026-09-08",
"evidence_label": "Source candidate · not reviewed",
"note": "Full public directory statement. Source lane: backlog; source resolution: open. Source verification claims are attributed only; no local Lean verification has be... | {"created_at":"2026-08-17T19:45:58+00:00","display_standard":{"editorial_review":"submitter_supplied","formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":9042,"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": "2026-09-08T03:33:01.123Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"593dae9a-0239-4fdc-9cc1-3272af5293c0","format_issues":null,"number":5662} |
1.0.0 | problem:theoremdb-p9044 | problem | theoremdb-p9044 | Erdős's Conjecture on Divergent Reciprocal Sums and 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"
}
] | Let $A$ be a set of natural numbers. If the sum of reciprocals $\sum_{n \in A} \frac{1}{n}$ diverges, must $A$ contain arbitrarily long arithmetic progressions? That is, does there exist, for every positive integer $k$, a subset $S \subseteq A$ that forms an arithmetic progression of length $k$? | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P9044",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P9044",
"citation": null,
"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_3 : answer(sorry) ↔ ∀ A : Set ℕ,
(¬ Summable fun a : A ↦ 1 / (a : ℝ)) →
∃ᶠ (k : ℕ) in Filter.atTop, ∃ S ⊆ A, S.IsAPOfLength k | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem erdos_3 : answer(sorry) ↔ ∀ A : Set ℕ,\n (¬ Summable fun a : A ↦ 1 / (a : ℝ)) →\n ∃ᶠ (k : ℕ) in Filter.atTop, ∃ S ⊆ A, S.IsAPOfLength k",
"target": null,
"source_url": "https://theoremdb.org/statements/P9044",
"format": "declaration-or-fragment",
... | [] | TheoremDB P9044 · Candidate awaiting source review | null | Source-reported assessment, retrieved 8 September 2026. Frontier Science has not independently reviewed this status.
Open. No signed Lean verification is attached to this problem or its selected resolution.
This record is in the source backlog; its novelty and unresolved status have not passed source review. | null | published | incremental | hard | [
"source-candidate",
"theoremdb"
] | null | null | null | null | [
{
"kind": "website",
"citation": "TheoremDB P9044",
"url": "https://theoremdb.org/statements/P9044",
"note": null,
"sort_order": 0
},
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"citation": "Source formal statement (not locally verified)",
"url": "https://theoremdb.org/statements/P9044",
"not... | {
"source_record": "P9044",
"imported_at": "2026-09-08",
"status_as_of": "2026-09-08",
"evidence_label": "Source candidate · not reviewed",
"note": "Full public directory statement. Source lane: backlog; source resolution: open. Source verification claims are attributed only; no local Lean verification has be... | {"created_at":"2026-08-17T19:46:03+00:00","display_standard":{"editorial_review":"submitter_supplied","formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":9044,"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": "2026-09-08T03:32:56.334Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"94b6548d-ba14-4450-9344-1250dba2bbd9","format_issues":null,"number":4718} |
1.0.0 | problem:theoremdb-p9046 | problem | theoremdb-p9046 | Irrationality of the partial LCM reciprocal series for finite sets 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$ be a finite set of primes with at least two elements. Let $\{a_1 < a_2 < \dots\}$ be the strictly increasing sequence of all positive integers whose prime factors all belong to $P$, where by convention $1$ is included since it has no prime divisors. For each $n \geq 1$, let $[a_1, \ldots, a_n]$ denote the least... | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P9046",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P9046",
"citation": null,
"doi": null,
"arxiv_id": null,
"license": "Selected packets label their content CC0-1.0; original references retain their own terms",
"notes": "Selected problem... | theorem erdos_269.variants.irrational : answer(sorry) ↔
∀ᵉ (P : Finset ℕ) (h : ∀ p ∈ P, p.Prime) (h_card : P.card ≥ 2),
Irrational (series (P : Set ℕ)) | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem erdos_269.variants.irrational : answer(sorry) ↔\n ∀ᵉ (P : Finset ℕ) (h : ∀ p ∈ P, p.Prime) (h_card : P.card ≥ 2),\n Irrational (series (P : Set ℕ))",
"target": null,
"source_url": "https://theoremdb.org/statements/P9046",
"format": "declaration-or-f... | [] | TheoremDB P9046 · Candidate awaiting source review | null | Source-reported assessment, retrieved 8 September 2026. Frontier Science has not independently reviewed this status.
Open. No signed Lean verification is attached to this problem or its selected resolution.
This record is in the source backlog; its novelty and unresolved status have not passed source review. | null | published | incremental | hard | [
"source-candidate",
"theoremdb"
] | null | null | null | null | [
{
"kind": "website",
"citation": "TheoremDB P9046",
"url": "https://theoremdb.org/statements/P9046",
"note": null,
"sort_order": 0
},
{
"kind": "formalization",
"citation": "Source formal statement (not locally verified)",
"url": "https://theoremdb.org/statements/P9046",
"not... | {
"source_record": "P9046",
"imported_at": "2026-09-08",
"status_as_of": "2026-09-08",
"evidence_label": "Source candidate · not reviewed",
"note": "Full public directory statement. Source lane: backlog; source resolution: open. Source verification claims are attributed only; no local Lean verification has be... | {"created_at":"2026-08-17T19:46:07+00:00","display_standard":{"editorial_review":"submitter_supplied","formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":9046,"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": "2026-09-08T03:33:01.798Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"6de84d42-d822-4f0c-acac-fdb4082934cc","format_issues":null,"number":5737} |
1.0.0 | problem:theoremdb-p9048 | problem | theoremdb-p9048 | Erdős Problem 535: Upper bound for the first open case | {
"main": "math",
"sub": "combinatorics",
"slug": "combinatorics",
"name": "Combinatorics",
"original_discipline": "mathematics"
} | [] | [
{
"code": "05",
"basis": "community-map",
"evidence": "Inferred from community field: combinatorics"
}
] | For each positive integer $N$, let $f_3(N)$ denote the maximum cardinality of a subset $A \subseteq \{1, 2, \ldots, N\}$ such that no three-element subset of $A$ has the property that all three pairwise greatest common divisors are equal. Does there exist a constant $c > 0$ such that $f_3(N) \leq N^{c / \log\log N}$ fo... | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P9048",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P9048",
"citation": null,
"doi": null,
"arxiv_id": null,
"license": "Selected packets label their content CC0-1.0; original references retain their own terms",
"notes": "Selected problem... | theorem erdos_535.variants.first_open_case : ∃ c > (0 : ℝ),
∀ᶠ (N : ℕ) in atTop,
(f 3 N : ℝ) ≤ (N : ℝ) ^ (c / log (log (N : ℝ))) | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem erdos_535.variants.first_open_case : ∃ c > (0 : ℝ),\n ∀ᶠ (N : ℕ) in atTop,\n (f 3 N : ℝ) ≤ (N : ℝ) ^ (c / log (log (N : ℝ)))",
"target": null,
"source_url": "https://theoremdb.org/statements/P9048",
"format": "declaration-or-fragment",
"sha256... | [] | TheoremDB P9048 · Candidate awaiting source review | null | Source-reported assessment, retrieved 8 September 2026. Frontier Science has not independently reviewed this status.
Open. No signed Lean verification is attached to this problem or its selected resolution.
This record is in the source backlog; its novelty and unresolved status have not passed source review. | null | published | incremental | hard | [
"source-candidate",
"theoremdb"
] | null | null | null | null | [
{
"kind": "website",
"citation": "TheoremDB P9048",
"url": "https://theoremdb.org/statements/P9048",
"note": null,
"sort_order": 0
},
{
"kind": "formalization",
"citation": "Source formal statement (not locally verified)",
"url": "https://theoremdb.org/statements/P9048",
"not... | {
"source_record": "P9048",
"imported_at": "2026-09-08",
"status_as_of": "2026-09-08",
"evidence_label": "Source candidate · not reviewed",
"note": "Full public directory statement. Source lane: backlog; source resolution: open. Source verification claims are attributed only; no local Lean verification has be... | {"created_at":"2026-08-17T19:46:10+00:00","display_standard":{"formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":9048,"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": "2026-09-08T03:32:55.597Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"2ea1f7b4-029f-45ef-a02b-03670d2630b4","format_issues":null,"number":4554} |
1.0.0 | problem:theoremdb-p9050 | problem | theoremdb-p9050 | Erdős's sum-distinct set problem for real numbers | {
"main": "math",
"sub": "combinatorics",
"slug": "combinatorics",
"name": "Combinatorics",
"original_discipline": "mathematics"
} | [] | [
{
"code": "05",
"basis": "community-map",
"evidence": "Inferred from community field: combinatorics"
}
] | A finite set of real numbers $A$ is said to be a sum-distinct set for $N \in \mathbb{N}$ if $A \subseteq (0, N]$ and the subset sums $\sum_{a \in S} a$ differ by at least $1$ for all distinct subsets $S \subseteq A$. That is, for any two distinct subsets $S_1, S_2 \subseteq A$, we have $1 \leq |\sum_{a \in S_1} a - \su... | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P9050",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P9050",
"citation": null,
"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_1.variants.real : ∃ C > (0 : ℝ), ∀ (N : ℕ) (A : Finset ℝ)
(_ : IsSumDistinctRealSet A N), N ≠ 0 → C * 2 ^ A.card < N | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem erdos_1.variants.real : ∃ C > (0 : ℝ), ∀ (N : ℕ) (A : Finset ℝ)\n (_ : IsSumDistinctRealSet A N), N ≠ 0 → C * 2 ^ A.card < N",
"target": null,
"source_url": "https://theoremdb.org/statements/P9050",
"format": "declaration-or-fragment",
"sha256": "2... | [] | TheoremDB P9050 · Candidate awaiting source review | null | Source-reported assessment, retrieved 8 September 2026. Frontier Science has not independently reviewed this status.
Open. No signed Lean verification is attached to this problem or its selected resolution.
This record is in the source backlog; its novelty and unresolved status have not passed source review. | null | published | incremental | hard | [
"source-candidate",
"theoremdb"
] | null | null | null | null | [
{
"kind": "website",
"citation": "TheoremDB P9050",
"url": "https://theoremdb.org/statements/P9050",
"note": null,
"sort_order": 0
},
{
"kind": "formalization",
"citation": "Source formal statement (not locally verified)",
"url": "https://theoremdb.org/statements/P9050",
"not... | {
"source_record": "P9050",
"imported_at": "2026-09-08",
"status_as_of": "2026-09-08",
"evidence_label": "Source candidate · not reviewed",
"note": "Full public directory statement. Source lane: backlog; source resolution: open. Source verification claims are attributed only; no local Lean verification has be... | {"created_at":"2026-08-17T19:46:14+00:00","display_standard":{"editorial_review":"submitter_supplied","formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":9050,"public_status":{"evidence":{"hasFormalization":false,"hasInformalProof":false,"hasRecordedWork":false,"solutionRecord":null},"lean":{... | [
"formalization-not-compiled-during-export",
"current-status-unassessed",
"source-full-text-not-present"
] | [] | {
"created_at": "2026-09-08T03:32:15.935Z",
"updated_at": "2026-09-08T03:32:15.935Z",
"published_at": "2026-09-08T03:32:56.604Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"2d7cb780-cebc-42b5-8c8e-2649feeb9c25","format_issues":null,"number":4776} |
1.0.0 | problem:theoremdb-p9052 | problem | theoremdb-p9052 | Infinitely many relaxed barriers for the number of distinct prime divisors | {
"main": "math",
"sub": "number theory",
"slug": "number-theory",
"name": "Number theory",
"original_discipline": "mathematics"
} | [] | [
{
"code": "11",
"basis": "community-map",
"evidence": "Inferred from community field: number-theory"
}
] | Let $\omega(m)$ denote the number of distinct prime divisors of a natural number $m$. For a function $f:\mathbb{N}\to\mathbb{R}$, a natural number $n$ is called a *barrier for $f$* if $m + f(m) \leq n$ for all $m < n$. Determine whether there exists a real number $\varepsilon > 0$ such that infinitely many natural numb... | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P9052",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P9052",
"citation": null,
"doi": null,
"arxiv_id": null,
"license": "Selected packets label their content CC0-1.0; original references retain their own terms",
"notes": "Selected problem... | theorem erdos_413.parts.ii :
answer(sorry) ↔
(∃ ε > (0 : ℝ), { n | IsBarrier (fun n => ε * ω n) n }.Infinite) | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem erdos_413.parts.ii :\n answer(sorry) ↔\n (∃ ε > (0 : ℝ), { n | IsBarrier (fun n => ε * ω n) n }.Infinite)",
"target": null,
"source_url": "https://theoremdb.org/statements/P9052",
"format": "declaration-or-fragment",
"sha256": "c335ce4f7cd72... | [] | TheoremDB P9052 · Candidate awaiting source review | null | Source-reported assessment, retrieved 8 September 2026. Frontier Science has not independently reviewed this status.
Open. No signed Lean verification is attached to this problem or its selected resolution.
This record is in the source backlog; its novelty and unresolved status have not passed source review. | null | published | incremental | hard | [
"source-candidate",
"theoremdb"
] | null | null | null | null | [
{
"kind": "website",
"citation": "TheoremDB P9052",
"url": "https://theoremdb.org/statements/P9052",
"note": null,
"sort_order": 0
},
{
"kind": "formalization",
"citation": "Source formal statement (not locally verified)",
"url": "https://theoremdb.org/statements/P9052",
"not... | {
"source_record": "P9052",
"imported_at": "2026-09-08",
"status_as_of": "2026-09-08",
"evidence_label": "Source candidate · not reviewed",
"note": "Full public directory statement. Source lane: backlog; source resolution: open. Source verification claims are attributed only; no local Lean verification has be... | {"created_at":"2026-08-17T19:46:18+00:00","display_standard":{"editorial_review":"submitter_supplied","formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":9052,"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": "2026-09-08T03:33:01.019Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"9cc742cd-713d-461e-923c-b4749eeffe61","format_issues":null,"number":5639} |
1.0.0 | problem:theoremdb-p9054 | problem | theoremdb-p9054 | Erdős Problem 30 on 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"
}
] | Let $h(N)$ denote the maximum cardinality of a Sidon set contained in $\{1, 2, \dots, N\}$. A set $A \subseteq \mathbb{N}$ is called a Sidon set if all pairwise sums $a + b$ with $a \leq b$ and $a, b \in A$ are distinct. Determine whether, for every $\varepsilon > 0$, the asymptotic bound
$$
h(N) - \sqrt{N} = O\bigl(N... | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P9054",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P9054",
"citation": null,
"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_30 : answer(sorry) ↔
∀ᵉ (ε > 0), (fun N => h N - (N : Real).sqrt) =O[atTop] fun N => (N : ℝ)^(ε : ℝ) | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem erdos_30 : answer(sorry) ↔\n ∀ᵉ (ε > 0), (fun N => h N - (N : Real).sqrt) =O[atTop] fun N => (N : ℝ)^(ε : ℝ)",
"target": null,
"source_url": "https://theoremdb.org/statements/P9054",
"format": "declaration-or-fragment",
"sha256": "55aa874ec2672afb3... | [] | TheoremDB P9054 · Candidate awaiting source review | null | Source-reported assessment, retrieved 8 September 2026. Frontier Science has not independently reviewed this status.
Open. No signed Lean verification is attached to this problem or its selected resolution.
This record is in the source backlog; its novelty and unresolved status have not passed source review. | null | published | incremental | hard | [
"source-candidate",
"theoremdb"
] | null | null | null | null | [
{
"kind": "website",
"citation": "TheoremDB P9054",
"url": "https://theoremdb.org/statements/P9054",
"note": null,
"sort_order": 0
},
{
"kind": "formalization",
"citation": "Source formal statement (not locally verified)",
"url": "https://theoremdb.org/statements/P9054",
"not... | {
"source_record": "P9054",
"imported_at": "2026-09-08",
"status_as_of": "2026-09-08",
"evidence_label": "Source candidate · not reviewed",
"note": "Full public directory statement. Source lane: backlog; source resolution: open. Source verification claims are attributed only; no local Lean verification has be... | {"created_at":"2026-08-17T19:46:22+00:00","display_standard":{"formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":9054,"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": "2026-09-08T03:32:55.302Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"47f073b7-ee9e-4a5a-8a29-2cc972249ff6","format_issues":null,"number":4490} |
1.0.0 | problem:theoremdb-p9056 | problem | theoremdb-p9056 | Existence of an absolutely maximally entangled state for seven 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"
}
] | A pure state $\psi$ on $n$ parties of local dimension $d$ is called absolutely maximally entangled (AME) if, for every subset of at most half of the parties, the corresponding reduced density matrix is maximally mixed. Does there exist an absolutely maximally entangled pure state on $n = 7$ parties of local dimension $... | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P9056",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P9056",
"citation": null,
"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_7_10_open :
answer(sorry) ↔ ExistsAME 7 10 | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem ame_7_10_open :\n answer(sorry) ↔ ExistsAME 7 10",
"target": null,
"source_url": "https://theoremdb.org/statements/P9056",
"format": "declaration-or-fragment",
"sha256": "53f3f48449db17a1f45b9ec0e6fe209330ceeec3f972b86c51489cd2ff9d5bb8",
"licen... | [] | TheoremDB P9056 · Candidate awaiting source review | null | Source-reported assessment, retrieved 8 September 2026. Frontier Science has not independently reviewed this status.
Open. No signed Lean verification is attached to this problem or its selected resolution.
This record is in the source backlog; its novelty and unresolved status have not passed source review. | null | published | incremental | hard | [
"source-candidate",
"theoremdb"
] | null | null | null | null | [
{
"kind": "website",
"citation": "TheoremDB P9056",
"url": "https://theoremdb.org/statements/P9056",
"note": null,
"sort_order": 0
},
{
"kind": "formalization",
"citation": "Source formal statement (not locally verified)",
"url": "https://theoremdb.org/statements/P9056",
"not... | {
"source_record": "P9056",
"imported_at": "2026-09-08",
"status_as_of": "2026-09-08",
"evidence_label": "Source candidate · not reviewed",
"note": "Full public directory statement. Source lane: backlog; source resolution: open. Source verification claims are attributed only; no local Lean verification has be... | {"created_at":"2026-08-17T19:46:25+00:00","display_standard":{"editorial_review":"submitter_supplied","formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":9056,"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": "2026-09-08T03:32:58.650Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"e1ac3bf1-f94a-4f01-865f-1b09596224e8","format_issues":null,"number":5226} |
1.0.0 | problem:theoremdb-p9058 | problem | theoremdb-p9058 | Explicit lower-order function for the maximal length of a distinct consecutive-sums sequence | {
"main": "math",
"sub": "number theory",
"slug": "number-theory",
"name": "Number theory",
"original_discipline": "mathematics"
} | [] | [
{
"code": "11",
"basis": "community-map",
"evidence": "Inferred from community field: number-theory"
}
] | For each positive integer $n$, let $f(n)$ denote the largest integer $k$ for which there exist integers $1 \le a_1 < a_2 < \dots < a_k \le n$ such that all sums of the form $\sum_{i=u}^{v} a_i$ with $1 \le u \le v \le k$ are distinct. Determine an explicit function $g: \mathbb{N} \to \mathbb{R}$ such that $g(n) = o(f(n... | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P9058",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P9058",
"citation": null,
"doi": null,
"arxiv_id": null,
"license": "Selected packets label their content CC0-1.0; original references retain their own terms",
"notes": "Selected problem... | theorem erdos_357.parts.ii.littleO_version :
(answer(sorry) : ℕ → ℝ) =o[atTop] (fun n ↦ (f n : ℝ)) | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem erdos_357.parts.ii.littleO_version :\n (answer(sorry) : ℕ → ℝ) =o[atTop] (fun n ↦ (f n : ℝ))",
"target": null,
"source_url": "https://theoremdb.org/statements/P9058",
"format": "declaration-or-fragment",
"sha256": "734f5a206ff76a4dfb40fd629d00190bc... | [] | TheoremDB P9058 · Candidate awaiting source review | null | Source-reported assessment, retrieved 8 September 2026. Frontier Science has not independently reviewed this status.
Open. No signed Lean verification is attached to this problem or its selected resolution.
This record is in the source backlog; its novelty and unresolved status have not passed source review. | null | published | incremental | hard | [
"source-candidate",
"theoremdb"
] | null | null | null | null | [
{
"kind": "website",
"citation": "TheoremDB P9058",
"url": "https://theoremdb.org/statements/P9058",
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},
{
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"citation": "Source formal statement (not locally verified)",
"url": "https://theoremdb.org/statements/P9058",
"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":"2026-08-17T19:46:29+00:00","display_standard":{"editorial_review":"submitter_supplied","formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":9058,"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": "2026-09-08T03:32:58.907Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"edb2075b-3958-4b19-9798-37927f515760","format_issues":null,"number":5280} |
1.0.0 | problem:theoremdb-p9060 | problem | theoremdb-p9060 | Prime between a square and a pronic number | {
"main": "math",
"sub": "number theory",
"slug": "number-theory",
"name": "Number theory",
"original_discipline": "mathematics"
} | [] | [
{
"code": "11",
"basis": "community-map",
"evidence": "Inferred from community field: number-theory"
}
] | For every integer $x \ge 2$, prove that there exists a prime $p$ satisfying $x^2 < p < x(x+1)$. | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P9060",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P9060",
"citation": null,
"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.ii (x : ℕ) (hx : 2 ≤ x) :
∃ p ∈ Ioo (x^2) (x * (x + 1)), p.Prime | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem oppermann_conjecture.parts.ii (x : ℕ) (hx : 2 ≤ x) :\n ∃ p ∈ Ioo (x^2) (x * (x + 1)), p.Prime",
"target": null,
"source_url": "https://theoremdb.org/statements/P9060",
"format": "declaration-or-fragment",
"sha256": "73ac643aaa5cf49610cb4b0b87fdc711... | [] | TheoremDB P9060 · Candidate awaiting source review | null | Source-reported assessment, retrieved 8 September 2026. Frontier Science has not independently reviewed this status.
Open. No signed Lean verification is attached to this problem or its selected resolution.
This record is in the source backlog; its novelty and unresolved status have not passed source review. | null | published | incremental | hard | [
"source-candidate",
"theoremdb"
] | null | null | null | null | [
{
"kind": "website",
"citation": "TheoremDB P9060",
"url": "https://theoremdb.org/statements/P9060",
"note": null,
"sort_order": 0
},
{
"kind": "formalization",
"citation": "Source formal statement (not locally verified)",
"url": "https://theoremdb.org/statements/P9060",
"not... | {
"source_record": "P9060",
"imported_at": "2026-09-08",
"status_as_of": "2026-09-08",
"evidence_label": "Source candidate · not reviewed",
"note": "Full public directory statement. Source lane: backlog; source resolution: open. Source verification claims are attributed only; no local Lean verification has be... | {"created_at":"2026-08-17T19:46:31+00:00","display_standard":{"formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":9060,"public_status":{"evidence":{"hasFormalization":false,"hasInformalProof":false,"hasRecordedWork":false,"solutionRecord":null},"lean":{"kernelAccepted":false,"label":"","reaso... | [
"formalization-not-compiled-during-export",
"current-status-unassessed",
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] | [] | {
"created_at": "2026-09-08T03:32:15.935Z",
"updated_at": "2026-09-08T03:32:15.935Z",
"published_at": "2026-09-08T03:33:05.062Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"fa65b204-bc05-4b41-83d6-3322df4976e3","format_issues":null,"number":6445} |
1.0.0 | problem:theoremdb-p9062 | problem | theoremdb-p9062 | Transcendence of $e + π$ | {
"main": "math",
"sub": "number theory",
"slug": "number-theory",
"name": "Number theory",
"original_discipline": "mathematics"
} | [] | [
{
"code": "11",
"basis": "community-map",
"evidence": "Inferred from community field: number-theory"
}
] | A real number $α$ is called transcendental over $ℚ$ if there is no non-zero polynomial $P$ with rational coefficients such that $P(α) = 0$. Let $e = ∑_{n=0}^{∞} \frac{1}{n!}$ denote the base of the natural logarithm, and let $π$ denote the ratio of the circumference of a circle to its diameter. Prove that the sum $e + ... | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P9062",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P9062",
"citation": null,
"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_add_pi_transcendental : Transcendental ℚ (exp 1 + π) | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem exp_add_pi_transcendental : Transcendental ℚ (exp 1 + π)",
"target": null,
"source_url": "https://theoremdb.org/statements/P9062",
"format": "declaration-or-fragment",
"sha256": "0739990860b037a344a945ddf12f1ed1225c907987f126b4eec1135e215d7fde",
"... | [] | TheoremDB P9062 · Candidate awaiting source review | null | Source-reported assessment, retrieved 8 September 2026. Frontier Science has not independently reviewed this status.
Open. No signed Lean verification is attached to this problem or its selected resolution.
This record is in the source backlog; its novelty and unresolved status have not passed source review. | null | published | incremental | hard | [
"source-candidate",
"theoremdb"
] | null | null | null | null | [
{
"kind": "website",
"citation": "TheoremDB P9062",
"url": "https://theoremdb.org/statements/P9062",
"note": null,
"sort_order": 0
},
{
"kind": "formalization",
"citation": "Source formal statement (not locally verified)",
"url": "https://theoremdb.org/statements/P9062",
"not... | {
"source_record": "P9062",
"imported_at": "2026-09-08",
"status_as_of": "2026-09-08",
"evidence_label": "Source candidate · not reviewed",
"note": "Full public directory statement. Source lane: backlog; source resolution: open. Source verification claims are attributed only; no local Lean verification has be... | {"created_at":"2026-08-17T19:46:35+00:00","display_standard":{"editorial_review":"submitter_supplied","formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":9062,"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": "2026-09-08T03:33:06.968Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"f79bf357-208a-4c05-ab8c-bdccf3e1f646","format_issues":null,"number":6857} |
1.0.0 | problem:theoremdb-p9064 | problem | theoremdb-p9064 | Erdős's Problem on Binomial Coefficients as Products of Consecutive Primes | {
"main": "math",
"sub": "number theory",
"slug": "number-theory",
"name": "Number theory",
"original_discipline": "mathematics"
} | [] | [
{
"code": "11",
"basis": "community-map",
"evidence": "Inferred from community field: number-theory"
}
] | Determine whether there exists an integer $k \geq 2$ such that the binomial coefficient $\binom{n}{k}$ equals the product of consecutive prime numbers for infinitely many integers $n$ with $k \leq n - 2$. More precisely, does there exist $k \geq 2$ such that for infinitely many $n \geq k + 2$, there exist natural numbe... | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P9064",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P9064",
"citation": null,
"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 :
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 :\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/P9064",
"format": "declaration-or-fragment",
"sha256... | [] | TheoremDB P9064 · Candidate awaiting source review | null | Source-reported assessment, retrieved 8 September 2026. Frontier Science has not independently reviewed this status.
Open. No signed Lean verification is attached to this problem or its selected resolution.
This record is in the source backlog; its novelty and unresolved status have not passed source review. | null | published | incremental | hard | [
"source-candidate",
"theoremdb"
] | null | null | null | null | [
{
"kind": "website",
"citation": "TheoremDB P9064",
"url": "https://theoremdb.org/statements/P9064",
"note": null,
"sort_order": 0
},
{
"kind": "formalization",
"citation": "Source formal statement (not locally verified)",
"url": "https://theoremdb.org/statements/P9064",
"not... | {
"source_record": "P9064",
"imported_at": "2026-09-08",
"status_as_of": "2026-09-08",
"evidence_label": "Source candidate · not reviewed",
"note": "Full public directory statement. Source lane: backlog; source resolution: open. Source verification claims are attributed only; no local Lean verification has be... | {"created_at":"2026-08-17T19:46:38+00:00","display_standard":{"editorial_review":"submitter_supplied","formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":9064,"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": "2026-09-08T03:32:56.447Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"c5542afd-b39c-44e0-9c99-dc15bce43b68","format_issues":null,"number":4742} |
1.0.0 | problem:theoremdb-p9066 | problem | theoremdb-p9066 | Selfridge's conjecture on the smallest Sierpiński number | {
"main": "math",
"sub": "number theory",
"slug": "number-theory",
"name": "Number theory",
"original_discipline": "mathematics"
} | [] | [
{
"code": "11",
"basis": "community-map",
"evidence": "Inferred from community field: number-theory"
}
] | A positive odd integer $k$ is called a *Sierpiński number* if $k \cdot 2^n + 1$ is composite for every natural number $n$. In 1960, Sierpiński proved that infinitely many such numbers exist. John Selfridge proved in 1962 that $78557$ is a Sierpiński number by exhibiting the covering set $\{3, 5, 7, 13, 19, 37, 73\}$: f... | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P9066",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P9066",
"citation": null,
"doi": null,
"arxiv_id": null,
"license": "Selected packets label their content CC0-1.0; original references retain their own terms",
"notes": "Selected problem... | theorem selfridge_conjecture :
answer(sorry) ↔ IsLeast {k | k.IsSierpinskiNumber} 78557 | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem selfridge_conjecture :\n answer(sorry) ↔ IsLeast {k | k.IsSierpinskiNumber} 78557",
"target": null,
"source_url": "https://theoremdb.org/statements/P9066",
"format": "declaration-or-fragment",
"sha256": "f2504fe4be9e1e5c40b888294344606fa22298a35f44... | [] | TheoremDB P9066 · Candidate awaiting source review | null | Source-reported assessment, retrieved 8 September 2026. Frontier Science has not independently reviewed this status.
Open. No signed Lean verification is attached to this problem or its selected resolution.
This record is in the source backlog; its novelty and unresolved status have not passed source review. | null | published | incremental | hard | [
"source-candidate",
"theoremdb"
] | null | null | null | null | [
{
"kind": "website",
"citation": "TheoremDB P9066",
"url": "https://theoremdb.org/statements/P9066",
"note": null,
"sort_order": 0
},
{
"kind": "formalization",
"citation": "Source formal statement (not locally verified)",
"url": "https://theoremdb.org/statements/P9066",
"not... | {
"source_record": "P9066",
"imported_at": "2026-09-08",
"status_as_of": "2026-09-08",
"evidence_label": "Source candidate · not reviewed",
"note": "Full public directory statement. Source lane: backlog; source resolution: open. Source verification claims are attributed only; no local Lean verification has be... | {"created_at":"2026-08-17T19:46:41+00:00","display_standard":{"editorial_review":"submitter_supplied","formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":9066,"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": "2026-09-08T03:33:05.765Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"71fb03e2-749e-420a-bc14-f2278ea3f533","format_issues":null,"number":6598} |
1.0.0 | problem:theoremdb-p9068 | problem | theoremdb-p9068 | Characterisation of Erdős–Hajnal exceptional graph pairs | {
"main": "math",
"sub": "combinatorics",
"slug": "combinatorics",
"name": "Combinatorics",
"original_discipline": "mathematics"
} | [] | [
{
"code": "05",
"basis": "community-map",
"evidence": "Inferred from community field: combinatorics"
}
] | Let $G_1$ and $G_2$ be simple graphs. We say that the pair $(G_1, G_2)$ is Erdős–Hajnal exceptional if it satisfies the following two conditions: (1) for every integer $n \geq 1$, there exists a graph $H$ containing no copy of $G_1$ such that every $n$-colouring of the edges of $H$ contains a monochromatic copy of $G_2... | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P9068",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P9068",
"citation": null,
"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_596 :
∀ {U₁ U₂ : Type} (G₁ : SimpleGraph U₁) (G₂ : SimpleGraph U₂),
IsErdosHajnalExceptional G₁ G₂ ↔
(answer(sorry) : ∀ {U₁ U₂ : Type}, SimpleGraph U₁ → SimpleGraph U₂ → Prop) G₁ G₂ | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem erdos_596 :\n ∀ {U₁ U₂ : Type} (G₁ : SimpleGraph U₁) (G₂ : SimpleGraph U₂),\n IsErdosHajnalExceptional G₁ G₂ ↔\n (answer(sorry) : ∀ {U₁ U₂ : Type}, SimpleGraph U₁ → SimpleGraph U₂ → Prop) G₁ G₂",
"target": null,
"source_url": "https://theoremdb.o... | [] | TheoremDB P9068 · Candidate awaiting source review | null | Source-reported assessment, retrieved 8 September 2026. Frontier Science has not independently reviewed this status.
Open. No signed Lean verification is attached to this problem or its selected resolution.
This record is in the source backlog; its novelty and unresolved status have not passed source review. | null | published | incremental | hard | [
"source-candidate",
"theoremdb"
] | null | null | null | null | [
{
"kind": "website",
"citation": "TheoremDB P9068",
"url": "https://theoremdb.org/statements/P9068",
"note": null,
"sort_order": 0
},
{
"kind": "formalization",
"citation": "Source formal statement (not locally verified)",
"url": "https://theoremdb.org/statements/P9068",
"not... | {
"source_record": "P9068",
"imported_at": "2026-09-08",
"status_as_of": "2026-09-08",
"evidence_label": "Source candidate · not reviewed",
"note": "Full public directory statement. Source lane: backlog; source resolution: open. Source verification claims are attributed only; no local Lean verification has be... | {"created_at":"2026-08-17T19:46:44+00:00","display_standard":{"formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":9068,"public_status":{"evidence":{"hasFormalization":false,"hasInformalProof":false,"hasRecordedWork":false,"solutionRecord":null},"lean":{"kernelAccepted":false,"label":"","reaso... | [
"formalization-contains-placeholder",
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"current-status-unassessed",
"source-full-text-not-present"
] | [] | {
"created_at": "2026-09-08T03:32:15.935Z",
"updated_at": "2026-09-08T03:32:15.935Z",
"published_at": "2026-09-08T03:32:51.044Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"c97a6e77-db1b-4cdc-b57b-eaa08637bad7","format_issues":null,"number":3549} |
1.0.0 | problem:theoremdb-p9070 | problem | theoremdb-p9070 | Finiteness of interval pairs with integer harmonic sum | {
"main": "math",
"sub": "number theory",
"slug": "number-theory",
"name": "Number theory",
"original_discipline": "mathematics"
} | [] | [
{
"code": "11",
"basis": "community-map",
"evidence": "Inferred from community field: number-theory"
}
] | A positive interval is a set of the form $\{a, a+1, \dots, b\}$ where $a, b \in \mathbb{N}^+$ with $a \leq b$. Consider pairs of positive intervals $(I_1, I_2)$. For each such pair, define the rational number
$$
H(I_1, I_2) = \sum_{n_1 \in I_1} \frac{1}{n_1} + \sum_{n_2 \in I_2} \frac{1}{n_2}.
$$
Determine whether th... | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P9070",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P9070",
"citation": null,
"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_288 : answer(sorry) ↔ Set.Finite { I : Fin 2 → ℕ+ × ℕ+ |
∀ j, (I j).1 ≤ (I j).2 ∧
∃ n : ℕ+, (∑ j : Fin 2, ∑ nⱼ ∈ Set.Icc (I j).1 (I j).2, (nⱼ⁻¹ : ℚ)) = n } | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem erdos_288 : answer(sorry) ↔ Set.Finite { I : Fin 2 → ℕ+ × ℕ+ |\n ∀ j, (I j).1 ≤ (I j).2 ∧\n ∃ n : ℕ+, (∑ j : Fin 2, ∑ nⱼ ∈ Set.Icc (I j).1 (I j).2, (nⱼ⁻¹ : ℚ)) = n }",
"target": null,
"source_url": "https://theoremdb.org/statements/P9070",
"format... | [] | TheoremDB P9070 · Candidate awaiting source review | null | Source-reported assessment, retrieved 8 September 2026. Frontier Science has not independently reviewed this status.
Open. No signed Lean verification is attached to this problem or its selected resolution.
This record is in the source backlog; its novelty and unresolved status have not passed source review. | null | published | incremental | hard | [
"source-candidate",
"theoremdb"
] | null | null | null | null | [
{
"kind": "website",
"citation": "TheoremDB P9070",
"url": "https://theoremdb.org/statements/P9070",
"note": null,
"sort_order": 0
},
{
"kind": "formalization",
"citation": "Source formal statement (not locally verified)",
"url": "https://theoremdb.org/statements/P9070",
"not... | {
"source_record": "P9070",
"imported_at": "2026-09-08",
"status_as_of": "2026-09-08",
"evidence_label": "Source candidate · not reviewed",
"note": "Full public directory statement. Source lane: backlog; source resolution: open. Source verification claims are attributed only; no local Lean verification has be... | {"created_at":"2026-08-17T19:46:48+00:00","display_standard":{"editorial_review":"submitter_supplied","formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":9070,"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": "2026-09-08T03:32:59.058Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"46b36cc8-5127-4bae-ae39-0cf655bb747e","format_issues":null,"number":5313} |
1.0.0 | problem:theoremdb-p9072 | problem | theoremdb-p9072 | Irrationality of the Euler–Mascheroni Constant | {
"main": "math",
"sub": "analysis",
"slug": "analysis",
"name": "Analysis",
"original_discipline": "mathematics"
} | [] | [] | Let $\gamma$ denote the Euler–Mascheroni constant, defined as the limiting difference between the harmonic series and the natural logarithm:
$$
\gamma = \lim_{n \to \infty} \left( \sum_{k=1}^{n} \frac{1}{k} - \ln n \right).
$$
Determine whether $\gamma$ is irrational. | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P9072",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P9072",
"citation": null,
"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_eulerMascheroniConstant :
answer(sorry) ↔ Irrational eulerMascheroniConstant | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem irrational_eulerMascheroniConstant :\n answer(sorry) ↔ Irrational eulerMascheroniConstant",
"target": null,
"source_url": "https://theoremdb.org/statements/P9072",
"format": "declaration-or-fragment",
"sha256": "85be43cb8f9015053edfc006d64d8ff15457... | [] | TheoremDB P9072 · Candidate awaiting source review | null | Source-reported assessment, retrieved 8 September 2026. Frontier Science has not independently reviewed this status.
Open. No signed Lean verification is attached to this problem or its selected resolution.
This record is in the source backlog; its novelty and unresolved status have not passed source review. | null | published | incremental | hard | [
"source-candidate",
"theoremdb"
] | null | null | null | null | [
{
"kind": "website",
"citation": "TheoremDB P9072",
"url": "https://theoremdb.org/statements/P9072",
"note": null,
"sort_order": 0
},
{
"kind": "formalization",
"citation": "Source formal statement (not locally verified)",
"url": "https://theoremdb.org/statements/P9072",
"not... | {
"source_record": "P9072",
"imported_at": "2026-09-08",
"status_as_of": "2026-09-08",
"evidence_label": "Source candidate · not reviewed",
"note": "Full public directory statement. Source lane: backlog; source resolution: open. Source verification claims are attributed only; no local Lean verification has be... | {"created_at":"2026-08-17T19:46:51+00:00","display_standard":{"editorial_review":"submitter_supplied","formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":9072,"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": "2026-09-08T03:33:01.765Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"ccebfc33-b72e-4ed8-8e4d-6fe8f062ddd7","format_issues":null,"number":5730} |
1.0.0 | problem:theoremdb-p9074 | problem | theoremdb-p9074 | Asymptotic order of the greedy Sidon sequence | {
"main": "math",
"sub": "combinatorics",
"slug": "combinatorics",
"name": "Combinatorics",
"original_discipline": "mathematics"
} | [] | [
{
"code": "05",
"basis": "community-map",
"evidence": "Inferred from community field: combinatorics"
}
] | Let $A = \{1, 2, 4, 8, 13, 21, 31, 45, 66, 81, 97, \ldots\}$ be the greedy Sidon sequence, constructed by beginning with $1$ and iteratively including the next smallest integer that preserves the Sidon property: there are no non-trivial solutions to $a + b = c + d$ with $a, b, c, d \in A$. For each positive integer $n$... | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P9074",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P9074",
"citation": null,
"doi": null,
"arxiv_id": null,
"license": "Selected packets label their content CC0-1.0; original references retain their own terms",
"notes": "Selected problem... | theorem erdos_340.variants.isTheta (ε : ℝ) (hε : ε > 0) :
(fun n : ℕ ↦ ((Set.range greedySidon ∩ Set.Icc 1 n).ncard : ℝ)) =Θ[atTop]
(answer(sorry) : ℕ → ℝ) | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem erdos_340.variants.isTheta (ε : ℝ) (hε : ε > 0) :\n (fun n : ℕ ↦ ((Set.range greedySidon ∩ Set.Icc 1 n).ncard : ℝ)) =Θ[atTop]\n (answer(sorry) : ℕ → ℝ)",
"target": null,
"source_url": "https://theoremdb.org/statements/P9074",
"format": "declaratio... | [] | TheoremDB P9074 · Candidate awaiting source review | null | Source-reported assessment, retrieved 8 September 2026. Frontier Science has not independently reviewed this status.
Open. No signed Lean verification is attached to this problem or its selected resolution.
This record is in the source backlog; its novelty and unresolved status have not passed source review. | null | published | incremental | hard | [
"source-candidate",
"theoremdb"
] | null | null | null | null | [
{
"kind": "website",
"citation": "TheoremDB P9074",
"url": "https://theoremdb.org/statements/P9074",
"note": null,
"sort_order": 0
},
{
"kind": "formalization",
"citation": "Source formal statement (not locally verified)",
"url": "https://theoremdb.org/statements/P9074",
"not... | {
"source_record": "P9074",
"imported_at": "2026-09-08",
"status_as_of": "2026-09-08",
"evidence_label": "Source candidate · not reviewed",
"note": "Full public directory statement. Source lane: backlog; source resolution: open. Source verification claims are attributed only; no local Lean verification has be... | {"created_at":"2026-08-17T19:46:55+00:00","display_standard":{"editorial_review":"submitter_supplied","formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":9074,"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": "2026-09-08T03:32:50.359Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"d0b45c2f-b8ee-48ba-980e-80cc8365c3aa","format_issues":null,"number":3403} |
1.0.0 | problem:theoremdb-p9076 | problem | theoremdb-p9076 | Lower Bound for Sumsets of Squares | {
"main": "math",
"sub": "number theory",
"slug": "number-theory",
"name": "Number theory",
"original_discipline": "mathematics"
} | [] | [
{
"code": "11",
"basis": "community-map",
"evidence": "Inferred from community field: number-theory"
}
] | Is there an absolute constant $c > 0$ such that, for every finite set $A \subseteq \mathbb{N}$ consisting entirely of perfect squares with $|A| \geq 2$, the sumset $A + A = \{a + b : a, b \in A\}$ satisfies $|A + A| \geq |A|^{1 + c}$? | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P9076",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P9076",
"citation": null,
"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_60 :
answer(sorry) ↔ ∃ c > (0 : ℝ),
∀ (A : Finset ℕ),
(∀ a ∈ A, IsSquare a) →
2 ≤ A.card →
((A + A).card : ℝ) ≥ (A.card : ℝ) ^ (1 + c) | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem green_60 :\n answer(sorry) ↔ ∃ c > (0 : ℝ),\n ∀ (A : Finset ℕ),\n (∀ a ∈ A, IsSquare a) →\n 2 ≤ A.card →\n ((A + A).card : ℝ) ≥ (A.card : ℝ) ^ (1 + c)",
"target": null,
"source_url": "https://theoremdb.org/statements/P9076",
... | [] | TheoremDB P9076 · Candidate awaiting source review | null | Source-reported assessment, retrieved 8 September 2026. Frontier Science has not independently reviewed this status.
Open. No signed Lean verification is attached to this problem or its selected resolution.
This record is in the source backlog; its novelty and unresolved status have not passed source review. | null | published | incremental | hard | [
"source-candidate",
"theoremdb"
] | null | null | null | null | [
{
"kind": "website",
"citation": "TheoremDB P9076",
"url": "https://theoremdb.org/statements/P9076",
"note": null,
"sort_order": 0
},
{
"kind": "formalization",
"citation": "Source formal statement (not locally verified)",
"url": "https://theoremdb.org/statements/P9076",
"not... | {
"source_record": "P9076",
"imported_at": "2026-09-08",
"status_as_of": "2026-09-08",
"evidence_label": "Source candidate · not reviewed",
"note": "Full public directory statement. Source lane: backlog; source resolution: open. Source verification claims are attributed only; no local Lean verification has be... | {"created_at":"2026-08-17T19:46:59+00:00","display_standard":{"editorial_review":"submitter_supplied","formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":9076,"public_status":{"evidence":{"hasFormalization":false,"hasInformalProof":false,"hasRecordedWork":false,"solutionRecord":null},"lean":{... | [
"formalization-contains-placeholder",
"formalization-not-compiled-during-export",
"current-status-unassessed",
"source-full-text-not-present"
] | [] | {
"created_at": "2026-09-08T03:32:15.935Z",
"updated_at": "2026-09-08T03:32:15.935Z",
"published_at": "2026-09-08T03:33:03.420Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"446d523a-28c4-4afc-a672-84364f13aefe","format_issues":null,"number":6092} |
1.0.0 | problem:theoremdb-p9078 | problem | theoremdb-p9078 | Lemoine Conjecture Extension | {
"main": "math",
"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 with $n > 8$. An odd prime is a prime number different from $2$. Determine whether there always exist odd primes $p, q, r, s$ and natural numbers $a, b$ such that $p + 2q = n$, $2 + pq = 2^a + r$, and $2p + q = 2^b + s$. | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P9078",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P9078",
"citation": null,
"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_extension (n : ℕ) (hn : 8 < n) (odd : Odd n) :
∃ (p q r s a b : ℕ), OddPrime p ∧ OddPrime q ∧ OddPrime r ∧ OddPrime s ∧
p + 2 * q = n ∧ 2 + p * q = 2 ^ a + r ∧ 2 * p + q = 2 ^ b + s | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem lemoine_conjecture_extension (n : ℕ) (hn : 8 < n) (odd : Odd n) :\n ∃ (p q r s a b : ℕ), OddPrime p ∧ OddPrime q ∧ OddPrime r ∧ OddPrime s ∧\n p + 2 * q = n ∧ 2 + p * q = 2 ^ a + r ∧ 2 * p + q = 2 ^ b + s",
"target": null,
"source_url": "https://theorem... | [] | TheoremDB P9078 · Candidate awaiting source review | null | Source-reported assessment, retrieved 8 September 2026. Frontier Science has not independently reviewed this status.
Open. No signed Lean verification is attached to this problem or its selected resolution.
This record is in the source backlog; its novelty and unresolved status have not passed source review. | null | published | incremental | hard | [
"source-candidate",
"theoremdb"
] | null | null | null | null | [
{
"kind": "website",
"citation": "TheoremDB P9078",
"url": "https://theoremdb.org/statements/P9078",
"note": null,
"sort_order": 0
},
{
"kind": "formalization",
"citation": "Source formal statement (not locally verified)",
"url": "https://theoremdb.org/statements/P9078",
"not... | {
"source_record": "P9078",
"imported_at": "2026-09-08",
"status_as_of": "2026-09-08",
"evidence_label": "Source candidate · not reviewed",
"note": "Full public directory statement. Source lane: backlog; source resolution: open. Source verification claims are attributed only; no local Lean verification has be... | {"created_at":"2026-08-17T19:47:02+00:00","display_standard":{"editorial_review":"submitter_supplied","formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":9078,"public_status":{"evidence":{"hasFormalization":false,"hasInformalProof":false,"hasRecordedWork":false,"solutionRecord":null},"lean":{... | [
"formalization-not-compiled-during-export",
"current-status-unassessed",
"source-full-text-not-present"
] | [] | {
"created_at": "2026-09-08T03:32:15.935Z",
"updated_at": "2026-09-08T03:32:15.935Z",
"published_at": "2026-09-08T03:33:03.179Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"a609d639-82ac-4ade-8dee-9156487d5079","format_issues":null,"number":6040} |
1.0.0 | problem:theoremdb-p9080 | problem | theoremdb-p9080 | Congruence modulo 216 for members of 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 $n$. A divisor $d$ of $n$ is called unitary if $\gcd(d, n/d) = 1$, and $\text{usigma}(n)$ denotes the sum of all unitary divisors of $n$. A positive integer $n$ is said to belong to sequence A063880 if $\sigma(n) = 2 \cdot \text{usig... | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P9080",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P9080",
"citation": null,
"doi": null,
"arxiv_id": null,
"license": "Selected packets label their content CC0-1.0; original references retain their own terms",
"notes": "Selected problem... | theorem mod_216_of_a {n : ℕ} (h : A n) : n % 216 = 108 | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem mod_216_of_a {n : ℕ} (h : A n) : n % 216 = 108",
"target": null,
"source_url": "https://theoremdb.org/statements/P9080",
"format": "declaration-or-fragment",
"sha256": "8c3f47785b9b6048ef5f99e80423e50ed1c792269a0669856297190603b5e560",
"license": ... | [] | TheoremDB P9080 · Candidate awaiting source review | null | Source-reported assessment, retrieved 8 September 2026. Frontier Science has not independently reviewed this status.
Open. No signed Lean verification is attached to this problem or its selected resolution.
This record is in the source backlog; its novelty and unresolved status have not passed source review. | null | published | incremental | hard | [
"source-candidate",
"theoremdb"
] | null | null | null | null | [
{
"kind": "website",
"citation": "TheoremDB P9080",
"url": "https://theoremdb.org/statements/P9080",
"note": null,
"sort_order": 0
},
{
"kind": "formalization",
"citation": "Source formal statement (not locally verified)",
"url": "https://theoremdb.org/statements/P9080",
"not... | {
"source_record": "P9080",
"imported_at": "2026-09-08",
"status_as_of": "2026-09-08",
"evidence_label": "Source candidate · not reviewed",
"note": "Full public directory statement. Source lane: backlog; source resolution: open. Source verification claims are attributed only; no local Lean verification has be... | {"created_at":"2026-08-17T19:47:06+00:00","display_standard":{"editorial_review":"submitter_supplied","formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":9080,"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": "2026-09-08T03:32:52.556Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"2e4f21e5-5512-4eb1-8597-e97d1003215f","format_issues":null,"number":3886} |
1.0.0 | problem:theoremdb-p9082 | problem | theoremdb-p9082 | The Agoh–Giuga conjecture on a Bernoulli congruence for 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 $B_n$ denote the $n$th Bernoulli number, written in lowest terms as $B_n = \frac{\mathrm{num}(B_n)}{\mathrm{den}(B_n)}$ with $\mathrm{num}(B_n) \in \mathbb{Z}$ and $\mathrm{den}(B_n) \in \mathbb{Z}_{>0}$. Prove or disprove: for every integer $p \geq 2$, the number $p$ is prime if and only if there exists an integer... | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P9082",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P9082",
"citation": null,
"doi": null,
"arxiv_id": null,
"license": "Selected packets label their content CC0-1.0; original references retain their own terms",
"notes": "Selected problem... | theorem agoh_giuga : AgohGiugaCongr | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem agoh_giuga : AgohGiugaCongr",
"target": null,
"source_url": "https://theoremdb.org/statements/P9082",
"format": "declaration-or-fragment",
"sha256": "eec173c715b98999be49e257733ae0357211c065113300fa768baa132c785a36",
"license": "Selected packets l... | [] | TheoremDB P9082 · Candidate awaiting source review | null | Source-reported assessment, retrieved 8 September 2026. Frontier Science has not independently reviewed this status.
Open. No signed Lean verification is attached to this problem or its selected resolution.
This record is in the source backlog; its novelty and unresolved status have not passed source review. | null | published | incremental | hard | [
"source-candidate",
"theoremdb"
] | null | null | null | null | [
{
"kind": "website",
"citation": "TheoremDB P9082",
"url": "https://theoremdb.org/statements/P9082",
"note": null,
"sort_order": 0
},
{
"kind": "formalization",
"citation": "Source formal statement (not locally verified)",
"url": "https://theoremdb.org/statements/P9082",
"not... | {
"source_record": "P9082",
"imported_at": "2026-09-08",
"status_as_of": "2026-09-08",
"evidence_label": "Source candidate · not reviewed",
"note": "Full public directory statement. Source lane: backlog; source resolution: open. Source verification claims are attributed only; no local Lean verification has be... | {"created_at":"2026-08-17T19:47:09+00:00","display_standard":{"formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":9082,"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": "2026-09-08T03:33:06.361Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"a42bdf42-0662-4f1d-adf3-ba536c291dc2","format_issues":null,"number":6726} |
1.0.0 | problem:theoremdb-p9084 | problem | theoremdb-p9084 | Asymptotic density of rank 0 and rank 1 elliptic curves over the rationals | {
"main": "math",
"sub": "number theory",
"slug": "number-theory",
"name": "Number theory",
"original_discipline": "mathematics"
} | [] | [
{
"code": "11",
"basis": "community-map",
"evidence": "Inferred from community field: number-theory"
}
] | Let $E$ be an elliptic curve over $\mathbb{Q}$. By the Mordell–Weil theorem, the group $E(\mathbb{Q})$ of rational points is finitely generated, so its rank $\operatorname{rank}(E)$ is a well-defined natural number. Every elliptic curve over $\mathbb{Q}$ is isomorphic to a unique curve in short Weierstrass form $y^2 = ... | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P9084",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P9084",
"citation": null,
"doi": null,
"arxiv_id": null,
"license": "Selected packets label their content CC0-1.0; original references retain their own terms",
"notes": "Selected problem... | theorem half_rank_zero_and_half_rank_one (r : ℕ) (hr : r = 0 ∨ r = 1) :
atTop.Tendsto
(fun H ↦ ({E ∈ heightLE H | E.rank = r}.ncard / (heightLE H).ncard : ℝ)) (𝓝 (1 / 2)) | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem half_rank_zero_and_half_rank_one (r : ℕ) (hr : r = 0 ∨ r = 1) :\n atTop.Tendsto\n (fun H ↦ ({E ∈ heightLE H | E.rank = r}.ncard / (heightLE H).ncard : ℝ)) (𝓝 (1 / 2))",
"target": null,
"source_url": "https://theoremdb.org/statements/P9084",
"form... | [] | TheoremDB P9084 · Candidate awaiting source review | null | Source-reported assessment, retrieved 8 September 2026. Frontier Science has not independently reviewed this status.
Open. No signed Lean verification is attached to this problem or its selected resolution.
This record is in the source backlog; its novelty and unresolved status have not passed source review. | null | published | incremental | hard | [
"source-candidate",
"theoremdb"
] | null | null | null | null | [
{
"kind": "website",
"citation": "TheoremDB P9084",
"url": "https://theoremdb.org/statements/P9084",
"note": null,
"sort_order": 0
},
{
"kind": "formalization",
"citation": "Source formal statement (not locally verified)",
"url": "https://theoremdb.org/statements/P9084",
"not... | {
"source_record": "P9084",
"imported_at": "2026-09-08",
"status_as_of": "2026-09-08",
"evidence_label": "Source candidate · not reviewed",
"note": "Full public directory statement. Source lane: backlog; source resolution: open. Source verification claims are attributed only; no local Lean verification has be... | {"created_at":"2026-08-17T19:47:12+00:00","display_standard":{"editorial_review":"submitter_supplied","formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":9084,"public_status":{"evidence":{"hasFormalization":false,"hasInformalProof":false,"hasRecordedWork":false,"solutionRecord":null},"lean":{... | [
"formalization-not-compiled-during-export",
"current-status-unassessed",
"source-full-text-not-present"
] | [] | {
"created_at": "2026-09-08T03:32:15.935Z",
"updated_at": "2026-09-08T03:32:15.935Z",
"published_at": "2026-09-08T03:32:50.015Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"69f9f753-5c23-408e-848d-0cc87a71c49b","format_issues":null,"number":3333} |
1.0.0 | problem:theoremdb-p9086 | problem | theoremdb-p9086 | Existence of an uncountably chromatic graph with all small subgraphs countably chromatic | {
"main": "math",
"sub": "combinatorics",
"slug": "combinatorics",
"name": "Combinatorics",
"original_discipline": "mathematics"
} | [] | [
{
"code": "05",
"basis": "community-map",
"evidence": "Inferred from community field: combinatorics"
}
] | Does there exist a graph with exactly $\aleph_2$ vertices and chromatic number $\aleph_2$ such that every subgraph on exactly $\aleph_1$ vertices has chromatic number at most $\aleph_0$?
More precisely, determine whether there exist a set $V$ of cardinality $\aleph_2$ and a simple graph $G$ on vertex set $V$ such that... | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P9086",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P9086",
"citation": null,
"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_918.variants.all_subgraphs.parts.i :
answer(sorry) ↔ ∃ (V : Type u) (G : SimpleGraph V), #V = ℵ_ 2 ∧ G.chromaticCardinal = ℵ_ 2 ∧
∀ (H : G.Subgraph) (_ : #H.verts = ℵ₁), H.coe.chromaticCardinal ≤ ℵ₀ | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem erdos_918.variants.all_subgraphs.parts.i :\n answer(sorry) ↔ ∃ (V : Type u) (G : SimpleGraph V), #V = ℵ_ 2 ∧ G.chromaticCardinal = ℵ_ 2 ∧\n ∀ (H : G.Subgraph) (_ : #H.verts = ℵ₁), H.coe.chromaticCardinal ≤ ℵ₀",
"target": null,
"source_url": "https://t... | [] | TheoremDB P9086 · Candidate awaiting source review | null | Source-reported assessment, retrieved 8 September 2026. Frontier Science has not independently reviewed this status.
Open. No signed Lean verification is attached to this problem or its selected resolution.
This record is in the source backlog; its novelty and unresolved status have not passed source review. | null | published | incremental | hard | [
"source-candidate",
"theoremdb"
] | null | null | null | null | [
{
"kind": "website",
"citation": "TheoremDB P9086",
"url": "https://theoremdb.org/statements/P9086",
"note": null,
"sort_order": 0
},
{
"kind": "formalization",
"citation": "Source formal statement (not locally verified)",
"url": "https://theoremdb.org/statements/P9086",
"not... | {
"source_record": "P9086",
"imported_at": "2026-09-08",
"status_as_of": "2026-09-08",
"evidence_label": "Source candidate · not reviewed",
"note": "Full public directory statement. Source lane: backlog; source resolution: open. Source verification claims are attributed only; no local Lean verification has be... | {"created_at":"2026-08-17T19:47:15+00:00","display_standard":{"editorial_review":"submitter_supplied","formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":9086,"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": "2026-09-08T03:32:58.702Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"6ca161c1-7d1b-4d2a-b3f8-b8d861b199e9","format_issues":null,"number":5237} |
1.0.0 | problem:theoremdb-p9088 | problem | theoremdb-p9088 | Bounded divisors near the square root | {
"main": "math",
"sub": "number theory",
"slug": "number-theory",
"name": "Number theory",
"original_discipline": "mathematics"
} | [] | [
{
"code": "11",
"basis": "community-map",
"evidence": "Inferred from community field: number-theory"
}
] | Does there exist an absolute constant $K$ such that, for every real number $C > 0$, if $n$ is a sufficiently large positive integer then $n$ has at most $K$ divisors in the open interval $(\sqrt{n}, \sqrt{n} + C n^{1/4})$? | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P9088",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P9088",
"citation": null,
"doi": null,
"arxiv_id": null,
"license": "Selected packets label their content CC0-1.0; original references retain their own terms",
"notes": "Selected problem... | theorem erdos_887.parts.ii : ∃ K, ∀ C > (0 : ℝ), ∀ᶠ n in atTop,
#{ d ∈ Ioo ⌊√n⌋₊ ⌈√n + C * n^((1 : ℝ) / 4)⌉₊ | d ∣ n } ≤ K | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem erdos_887.parts.ii : ∃ K, ∀ C > (0 : ℝ), ∀ᶠ n in atTop,\n #{ d ∈ Ioo ⌊√n⌋₊ ⌈√n + C * n^((1 : ℝ) / 4)⌉₊ | d ∣ n } ≤ K",
"target": null,
"source_url": "https://theoremdb.org/statements/P9088",
"format": "declaration-or-fragment",
"sha256": "27ad1e249... | [] | TheoremDB P9088 · Candidate awaiting source review | null | Source-reported assessment, retrieved 8 September 2026. Frontier Science has not independently reviewed this status.
Open. No signed Lean verification is attached to this problem or its selected resolution.
This record is in the source backlog; its novelty and unresolved status have not passed source review. | null | published | incremental | hard | [
"source-candidate",
"theoremdb"
] | null | null | null | null | [
{
"kind": "website",
"citation": "TheoremDB P9088",
"url": "https://theoremdb.org/statements/P9088",
"note": null,
"sort_order": 0
},
{
"kind": "formalization",
"citation": "Source formal statement (not locally verified)",
"url": "https://theoremdb.org/statements/P9088",
"not... | {
"source_record": "P9088",
"imported_at": "2026-09-08",
"status_as_of": "2026-09-08",
"evidence_label": "Source candidate · not reviewed",
"note": "Full public directory statement. Source lane: backlog; source resolution: open. Source verification claims are attributed only; no local Lean verification has be... | {"created_at":"2026-08-17T19:47:18+00:00","display_standard":{"editorial_review":"submitter_supplied","formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":9088,"public_status":{"evidence":{"hasFormalization":false,"hasInformalProof":false,"hasRecordedWork":false,"solutionRecord":null},"lean":{... | [
"formalization-not-compiled-during-export",
"current-status-unassessed",
"source-full-text-not-present"
] | [] | {
"created_at": "2026-09-08T03:32:15.935Z",
"updated_at": "2026-09-08T03:32:15.935Z",
"published_at": "2026-09-08T03:32:50.705Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"4cbb6587-c657-42eb-a7c8-3b352bf3e34b","format_issues":null,"number":3475} |
1.0.0 | problem:theoremdb-p9090 | problem | theoremdb-p9090 | Zhi-Wei Sun's 2-4-6-8 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 positive integer $n$, there exist nonnegative integers $w$, $x$, $y$, and $z$ such that
$$
n = \binom{w+2}{2} + \binom{x+3}{4} + \binom{y+5}{6} + \binom{z+7}{8}.
$$
Prove that this representation always exists. | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P9090",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P9090",
"citation": null,
"doi": null,
"arxiv_id": null,
"license": "Selected packets label their content CC0-1.0; original references retain their own terms",
"notes": "Selected problem... | theorem conjecture (n : ℕ) (hn : 0 < n) : A n | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem conjecture (n : ℕ) (hn : 0 < n) : A n",
"target": null,
"source_url": "https://theoremdb.org/statements/P9090",
"format": "declaration-or-fragment",
"sha256": "61e76f441c5d6adff942a7ccd2e2a0b60eed9e305d336d71d210639e3a3b5baa",
"license": "Selected... | [] | TheoremDB P9090 · Candidate awaiting source review | null | Source-reported assessment, retrieved 8 September 2026. Frontier Science has not independently reviewed this status.
Open. No signed Lean verification is attached to this problem or its selected resolution.
This record is in the source backlog; its novelty and unresolved status have not passed source review. | null | published | incremental | hard | [
"source-candidate",
"theoremdb"
] | null | null | null | null | [
{
"kind": "website",
"citation": "TheoremDB P9090",
"url": "https://theoremdb.org/statements/P9090",
"note": null,
"sort_order": 0
},
{
"kind": "formalization",
"citation": "Source formal statement (not locally verified)",
"url": "https://theoremdb.org/statements/P9090",
"not... | {
"source_record": "P9090",
"imported_at": "2026-09-08",
"status_as_of": "2026-09-08",
"evidence_label": "Source candidate · not reviewed",
"note": "Full public directory statement. Source lane: backlog; source resolution: open. Source verification claims are attributed only; no local Lean verification has be... | {"created_at":"2026-08-17T19:47:21+00:00","display_standard":{"formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":9090,"public_status":{"evidence":{"hasFormalization":false,"hasInformalProof":false,"hasRecordedWork":false,"solutionRecord":null},"lean":{"kernelAccepted":false,"label":"","reaso... | [
"formalization-not-compiled-during-export",
"current-status-unassessed",
"source-full-text-not-present"
] | [] | {
"created_at": "2026-09-08T03:32:15.935Z",
"updated_at": "2026-09-08T03:32:15.935Z",
"published_at": "2026-09-08T03:33:07.645Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"b51068f0-1f65-4a28-98e0-9deeedecd52c","format_issues":null,"number":7004} |
1.0.0 | problem:theoremdb-p9092 | problem | theoremdb-p9092 | Existence of a fifth-power taxicab number with two representations | {
"main": "math",
"sub": "number theory",
"slug": "number-theory",
"name": "Number theory",
"original_discipline": "mathematics"
} | [] | [
{
"code": "11",
"basis": "community-map",
"evidence": "Inferred from community field: number-theory"
}
] | For natural numbers $k$, $m$, and $n$, a *taxicab number* is the smallest positive integer $x$ that can be expressed as a sum of $m$ positive $k$th powers in at least $n$ distinct ways, where the representations share no common terms. Specifically, $x$ is a candidate for being a taxicab number for $k$, $m$, $n$ if ther... | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P9092",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P9092",
"citation": null,
"doi": null,
"arxiv_id": null,
"license": "Selected packets label their content CC0-1.0; original references retain their own terms",
"notes": "Selected problem... | theorem taxicab_for_5_2_2 : answer(sorry) ↔ ∃ x : ℕ, IsTaxicabFor 5 2 2 x | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem taxicab_for_5_2_2 : answer(sorry) ↔ ∃ x : ℕ, IsTaxicabFor 5 2 2 x",
"target": null,
"source_url": "https://theoremdb.org/statements/P9092",
"format": "declaration-or-fragment",
"sha256": "0a09613c9ff3e359eed1600f3dd0f379c5e753d02dae45d94609e596be25a9d... | [] | TheoremDB P9092 · Candidate awaiting source review | null | Source-reported assessment, retrieved 8 September 2026. Frontier Science has not independently reviewed this status.
Open. No signed Lean verification is attached to this problem or its selected resolution.
This record is in the source backlog; its novelty and unresolved status have not passed source review. | null | published | incremental | hard | [
"source-candidate",
"theoremdb"
] | null | null | null | null | [
{
"kind": "website",
"citation": "TheoremDB P9092",
"url": "https://theoremdb.org/statements/P9092",
"note": null,
"sort_order": 0
},
{
"kind": "formalization",
"citation": "Source formal statement (not locally verified)",
"url": "https://theoremdb.org/statements/P9092",
"not... | {
"source_record": "P9092",
"imported_at": "2026-09-08",
"status_as_of": "2026-09-08",
"evidence_label": "Source candidate · not reviewed",
"note": "Full public directory statement. Source lane: backlog; source resolution: open. Source verification claims are attributed only; no local Lean verification has be... | {"created_at":"2026-08-17T19:47:24+00:00","display_standard":{"editorial_review":"submitter_supplied","formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":9092,"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": "2026-09-08T03:32:58.431Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"93a15ff2-fe63-4200-ac5f-054b2948ed3c","format_issues":null,"number":5181} |
1.0.0 | problem:theoremdb-p9094 | problem | theoremdb-p9094 | Asymptotic equivalence of the sum of squared prime values of Erdős's prime gap function to 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 each positive integer $n$, define
$$
f(n) = \sum_{\substack{p < n \\ p \text{ prime}}} \frac{1}{n - p},
$$
where the sum is taken over all prime numbers $p$ strictly less than $n$. Let $\pi(x)$ denote the prime counting function, which gives the number of primes not exceeding $x$. Determine whether
$$
\sum_{\... | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P9094",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P9094",
"citation": null,
"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.sum_primes :
answer(sorry) ↔
(fun x : ℕ ↦ ∑ p ∈ (Finset.range x).filter Prime, (f p) ^ 2) ~[atTop]
fun x ↦ (π x : ℝ) | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem erdos_950.variants.sum_primes :\n answer(sorry) ↔\n (fun x : ℕ ↦ ∑ p ∈ (Finset.range x).filter Prime, (f p) ^ 2) ~[atTop]\n fun x ↦ (π x : ℝ)",
"target": null,
"source_url": "https://theoremdb.org/statements/P9094",
"format": "declaration-o... | [] | TheoremDB P9094 · Candidate awaiting source review | null | Source-reported assessment, retrieved 8 September 2026. Frontier Science has not independently reviewed this status.
Open. No signed Lean verification is attached to this problem or its selected resolution.
This record is in the source backlog; its novelty and unresolved status have not passed source review. | null | published | incremental | hard | [
"source-candidate",
"theoremdb"
] | null | null | null | null | [
{
"kind": "website",
"citation": "TheoremDB P9094",
"url": "https://theoremdb.org/statements/P9094",
"note": null,
"sort_order": 0
},
{
"kind": "formalization",
"citation": "Source formal statement (not locally verified)",
"url": "https://theoremdb.org/statements/P9094",
"not... | {
"source_record": "P9094",
"imported_at": "2026-09-08",
"status_as_of": "2026-09-08",
"evidence_label": "Source candidate · not reviewed",
"note": "Full public directory statement. Source lane: backlog; source resolution: open. Source verification claims are attributed only; no local Lean verification has be... | {"created_at":"2026-08-17T19:47:26+00:00","display_standard":{"editorial_review":"submitter_supplied","formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":9094,"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": "2026-09-08T03:32:50.029Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"7c81f197-5dd2-44f4-bf63-b90c2bd378a7","format_issues":null,"number":3336} |
1.0.0 | problem:theoremdb-p9096 | problem | theoremdb-p9096 | Asymptotic growth of the supremal covering density for linear codes | {
"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}_2^n$ denote the $n$-dimensional vector space over the field with two elements. For a non-negative integer $r$, the Hamming ball $H_n(r)$ is the set of all vectors in $\mathbb{F}_2^n$ with Hamming norm at most $r$, i.e., the set of all vectors with at most $r$ nonzero coord... | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P9096",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P9096",
"citation": null,
"doi": null,
"arxiv_id": null,
"license": "Selected packets label their content CC0-1.0; original references retain their own terms",
"notes": "Selected problem... | theorem green_40.variants.all_n : answer(sorry) ↔ Tendsto f_all atTop (𝓝 ⊤) | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem green_40.variants.all_n : answer(sorry) ↔ Tendsto f_all atTop (𝓝 ⊤)",
"target": null,
"source_url": "https://theoremdb.org/statements/P9096",
"format": "declaration-or-fragment",
"sha256": "05b25ffff57d6eb5f9ef11ba53a55d069be9000732aedbe129addb858e7a... | [] | TheoremDB P9096 · Candidate awaiting source review | null | Source-reported assessment, retrieved 8 September 2026. Frontier Science has not independently reviewed this status.
Open. No signed Lean verification is attached to this problem or its selected resolution.
This record is in the source backlog; its novelty and unresolved status have not passed source review. | null | published | incremental | hard | [
"source-candidate",
"theoremdb"
] | null | null | null | null | [
{
"kind": "website",
"citation": "TheoremDB P9096",
"url": "https://theoremdb.org/statements/P9096",
"note": null,
"sort_order": 0
},
{
"kind": "formalization",
"citation": "Source formal statement (not locally verified)",
"url": "https://theoremdb.org/statements/P9096",
"not... | {
"source_record": "P9096",
"imported_at": "2026-09-08",
"status_as_of": "2026-09-08",
"evidence_label": "Source candidate · not reviewed",
"note": "Full public directory statement. Source lane: backlog; source resolution: open. Source verification claims are attributed only; no local Lean verification has be... | {"created_at":"2026-08-17T19:47:29+00:00","display_standard":{"editorial_review":"submitter_supplied","formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":9096,"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": "2026-09-08T03:32:50.263Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"7a06f15a-958f-439d-ad0d-43c89fe8872e","format_issues":null,"number":3384} |
1.0.0 | problem:theoremdb-p9098 | problem | theoremdb-p9098 | Lower bound for the off-diagonal van der Waerden number W(3, 24) | {
"main": "math",
"sub": "combinatorics",
"slug": "combinatorics",
"name": "Combinatorics",
"original_discipline": "mathematics"
} | [] | [
{
"code": "05",
"basis": "community-map",
"evidence": "Inferred from community field: combinatorics"
}
] | For positive integers $k$ and $r$, let $W(k, r)$ denote the least positive integer $N$ such that every 2-coloring of $\{1, 2, \ldots, N\}$ contains either a monochromatic arithmetic progression of length $k$ in the first color or a monochromatic arithmetic progression of length $r$ in the second color. Determine whethe... | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P9098",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P9098",
"citation": null,
"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_24_lower : answer(sorry) ↔ W 3 24 ≥ 593 | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem W_3_24_lower : answer(sorry) ↔ W 3 24 ≥ 593",
"target": null,
"source_url": "https://theoremdb.org/statements/P9098",
"format": "declaration-or-fragment",
"sha256": "6f756b2978efd1452c0cad6ec931f940455af8a84463eee001dbd048d75ea508",
"license": "Se... | [] | TheoremDB P9098 · Candidate awaiting source review | null | Source-reported assessment, retrieved 8 September 2026. Frontier Science has not independently reviewed this status.
Open. No signed Lean verification is attached to this problem or its selected resolution.
This record is in the source backlog; its novelty and unresolved status have not passed source review. | null | published | incremental | hard | [
"source-candidate",
"theoremdb"
] | null | null | null | null | [
{
"kind": "website",
"citation": "TheoremDB P9098",
"url": "https://theoremdb.org/statements/P9098",
"note": null,
"sort_order": 0
},
{
"kind": "formalization",
"citation": "Source formal statement (not locally verified)",
"url": "https://theoremdb.org/statements/P9098",
"not... | {
"source_record": "P9098",
"imported_at": "2026-09-08",
"status_as_of": "2026-09-08",
"evidence_label": "Source candidate · not reviewed",
"note": "Full public directory statement. Source lane: backlog; source resolution: open. Source verification claims are attributed only; no local Lean verification has be... | {"created_at":"2026-08-17T19:47:32+00:00","display_standard":{"formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":9098,"public_status":{"evidence":{"hasFormalization":false,"hasInformalProof":false,"hasRecordedWork":false,"solutionRecord":null},"lean":{"kernelAccepted":false,"label":"","reaso... | [
"formalization-contains-placeholder",
"formalization-not-compiled-during-export",
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"source-full-text-not-present"
] | [] | {
"created_at": "2026-09-08T03:32:15.935Z",
"updated_at": "2026-09-08T03:32:15.935Z",
"published_at": "2026-09-08T03:33:03.470Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"c0fc7daa-1334-4d4f-a4a1-0fe66d95d917","format_issues":null,"number":6102} |
1.0.0 | problem:theoremdb-p9100 | problem | theoremdb-p9100 | Existence of a better quantitative bound for pyjama set coverings | {
"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
$$
P_\varepsilon = \{ z \in \mathbb{C} : \exists k \in \mathbb{Z}, \; |\operatorname{Re}(z) - k| \leq \varepsilon \},
$$
or equivalently, the set of points in the plane whose $x$-coordinate is within distance $\varepsilon$ of an i... | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P9100",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P9100",
"citation": null,
"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.exists_better_bound : answer(sorry) ↔
∃ C : ℝ, C > 0 ∧ ∃ ε₀ > 0, ∀ ε ∈ Ioc 0 ε₀,
∃ ans : ℝ, (minCopies ε : ℝ) ≤ ans ∧ ans < Real.exp (Real.exp (Real.exp (ε ^ (-C)))) | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem green_41.variants.exists_better_bound : answer(sorry) ↔\n ∃ C : ℝ, C > 0 ∧ ∃ ε₀ > 0, ∀ ε ∈ Ioc 0 ε₀,\n ∃ ans : ℝ, (minCopies ε : ℝ) ≤ ans ∧ ans < Real.exp (Real.exp (Real.exp (ε ^ (-C))))",
"target": null,
"source_url": "https://theoremdb.org/statemen... | [] | TheoremDB P9100 · Candidate awaiting source review | null | Source-reported assessment, retrieved 8 September 2026. Frontier Science has not independently reviewed this status.
Open. No signed Lean verification is attached to this problem or its selected resolution.
This record is in the source backlog; its novelty and unresolved status have not passed source review. | null | published | incremental | hard | [
"source-candidate",
"theoremdb"
] | null | null | null | null | [
{
"kind": "website",
"citation": "TheoremDB P9100",
"url": "https://theoremdb.org/statements/P9100",
"note": null,
"sort_order": 0
},
{
"kind": "formalization",
"citation": "Source formal statement (not locally verified)",
"url": "https://theoremdb.org/statements/P9100",
"not... | {
"source_record": "P9100",
"imported_at": "2026-09-08",
"status_as_of": "2026-09-08",
"evidence_label": "Source candidate · not reviewed",
"note": "Full public directory statement. Source lane: backlog; source resolution: open. Source verification claims are attributed only; no local Lean verification has be... | {"created_at":"2026-08-17T19:47:34+00:00","display_standard":{"editorial_review":"submitter_supplied","formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":9100,"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": "2026-09-08T03:32:58.403Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"70a191eb-cc93-44f6-b1ca-7d587be6a727","format_issues":null,"number":5175} |
1.0.0 | problem:theoremdb-p9102 | problem | theoremdb-p9102 | Local Connectivity of the Mandelbrot Set | {
"main": "math",
"sub": "dynamical systems",
"slug": "dynamical-systems",
"name": "Dynamical systems",
"original_discipline": "mathematics"
} | [] | [
{
"code": "37",
"basis": "community-map",
"evidence": "Inferred from community field: dynamical-systems"
}
] | Let the Mandelbrot set be the set of all complex parameters $c \in \mathbb{C}$ for which the orbit of $0$ under iteration of the quadratic polynomial $z \mapsto z^2 + c$ does not escape to infinity; equivalently, it is the set of all $c \in \mathbb{C}$ such that for every non-negative integer $k$, the $k$-fold iterate ... | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P9102",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P9102",
"citation": null,
"doi": null,
"arxiv_id": null,
"license": "Selected packets label their content CC0-1.0; original references retain their own terms",
"notes": "Selected problem... | theorem MLC : LocallyConnectedSpace mandelbrotSet | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem MLC : LocallyConnectedSpace mandelbrotSet",
"target": null,
"source_url": "https://theoremdb.org/statements/P9102",
"format": "declaration-or-fragment",
"sha256": "7321685e078af9e790c70eafc402f05b1a6f2074b360a799d356b078df5a5a88",
"license": "Sele... | [] | TheoremDB P9102 · Candidate awaiting source review | null | Source-reported assessment, retrieved 8 September 2026. Frontier Science has not independently reviewed this status.
Open. No signed Lean verification is attached to this problem or its selected resolution.
This record is in the source backlog; its novelty and unresolved status have not passed source review. | null | published | incremental | hard | [
"source-candidate",
"theoremdb"
] | null | null | null | null | [
{
"kind": "website",
"citation": "TheoremDB P9102",
"url": "https://theoremdb.org/statements/P9102",
"note": null,
"sort_order": 0
},
{
"kind": "formalization",
"citation": "Source formal statement (not locally verified)",
"url": "https://theoremdb.org/statements/P9102",
"not... | {
"source_record": "P9102",
"imported_at": "2026-09-08",
"status_as_of": "2026-09-08",
"evidence_label": "Source candidate · not reviewed",
"note": "Full public directory statement. Source lane: backlog; source resolution: open. Source verification claims are attributed only; no local Lean verification has be... | {"created_at":"2026-08-17T19:47:35+00:00","display_standard":{"editorial_review":"submitter_supplied","formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":9102,"public_status":{"evidence":{"hasFormalization":false,"hasInformalProof":false,"hasRecordedWork":false,"solutionRecord":null},"lean":{... | [
"formalization-not-compiled-during-export",
"current-status-unassessed",
"source-full-text-not-present"
] | [] | {
"created_at": "2026-09-08T03:32:15.935Z",
"updated_at": "2026-09-08T03:32:15.935Z",
"published_at": "2026-09-08T03:33:03.298Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"2805b68e-5141-403d-8865-497d967f683e","format_issues":null,"number":6065} |
1.0.0 | problem:theoremdb-p9104 | problem | theoremdb-p9104 | Asymptotic growth of the minimum subset size for eventually surjective set functions | {
"main": "math",
"sub": "combinatorics",
"slug": "combinatorics",
"name": "Combinatorics",
"original_discipline": "mathematics"
} | [] | [
{
"code": "05",
"basis": "community-map",
"evidence": "Inferred from community field: combinatorics"
}
] | For each positive integer $n$, define $H(n)$ to be the minimum integer $m$ for which there exists a function $f$ from the power set of an $n$-element set $X$ to $X$ itself, such that for every subset $Y \subseteq X$ with $|Y| \geq m$, the image of the power set of $Y$ under $f$ equals all of $X$; that is, $\{f(A) : A \... | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P9104",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P9104",
"citation": null,
"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_624 :
atTop.Tendsto (fun n : ℕ => H n - Real.logb 2 (n : ℝ)) atTop | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem erdos_624 :\n atTop.Tendsto (fun n : ℕ => H n - Real.logb 2 (n : ℝ)) atTop",
"target": null,
"source_url": "https://theoremdb.org/statements/P9104",
"format": "declaration-or-fragment",
"sha256": "dde160c7756017449585322791e3c965beb843790f970023387... | [] | TheoremDB P9104 · Candidate awaiting source review | null | Source-reported assessment, retrieved 8 September 2026. Frontier Science has not independently reviewed this status.
Open. No signed Lean verification is attached to this problem or its selected resolution.
This record is in the source backlog; its novelty and unresolved status have not passed source review. | null | published | incremental | hard | [
"source-candidate",
"theoremdb"
] | null | null | null | null | [
{
"kind": "website",
"citation": "TheoremDB P9104",
"url": "https://theoremdb.org/statements/P9104",
"note": null,
"sort_order": 0
},
{
"kind": "formalization",
"citation": "Source formal statement (not locally verified)",
"url": "https://theoremdb.org/statements/P9104",
"not... | {
"source_record": "P9104",
"imported_at": "2026-09-08",
"status_as_of": "2026-09-08",
"evidence_label": "Source candidate · not reviewed",
"note": "Full public directory statement. Source lane: backlog; source resolution: open. Source verification claims are attributed only; no local Lean verification has be... | {"created_at":"2026-08-17T19:47:36+00:00","display_standard":{"editorial_review":"submitter_supplied","formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":9104,"public_status":{"evidence":{"hasFormalization":false,"hasInformalProof":false,"hasRecordedWork":false,"solutionRecord":null},"lean":{... | [
"formalization-not-compiled-during-export",
"current-status-unassessed",
"source-full-text-not-present"
] | [] | {
"created_at": "2026-09-08T03:32:15.935Z",
"updated_at": "2026-09-08T03:32:15.935Z",
"published_at": "2026-09-08T03:32:50.244Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"ad142a1b-7354-47cb-9aad-cbdd85219a98","format_issues":null,"number":3380} |
1.0.0 | problem:theoremdb-p9106 | problem | theoremdb-p9106 | Asymptotic growth of the maximal subset-sum-separating cardinality | {
"main": "math",
"sub": "combinatorics",
"slug": "combinatorics",
"name": "Combinatorics",
"original_discipline": "mathematics"
} | [] | [
{
"code": "05",
"basis": "community-map",
"evidence": "Inferred from community field: combinatorics"
}
] | For each positive integer $n$, define $h(n)$ to be the largest integer $m$ with the following property: for every set $A$ of $n$ integers, there exists a subset $B\subseteq A$ with $|B|\geq m$ such that whenever two nonempty subsets of $B$ have equal sums, they have equal cardinalities. Equivalently, all subset sums of... | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P9106",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P9106",
"citation": null,
"doi": null,
"arxiv_id": null,
"license": "Selected packets label their content CC0-1.0; original references retain their own terms",
"notes": "Selected problem... | theorem erdos_789 :
(fun n ↦ (subsetSumThreshold n : ℝ)) =Θ[atTop] (answer(sorry) : ℕ → ℝ) | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem erdos_789 :\n (fun n ↦ (subsetSumThreshold n : ℝ)) =Θ[atTop] (answer(sorry) : ℕ → ℝ)",
"target": null,
"source_url": "https://theoremdb.org/statements/P9106",
"format": "declaration-or-fragment",
"sha256": "3b899ea08193fdf01bdab14e86fbb96401760b3d3... | [] | TheoremDB P9106 · Candidate awaiting source review | null | Source-reported assessment, retrieved 8 September 2026. Frontier Science has not independently reviewed this status.
Open. No signed Lean verification is attached to this problem or its selected resolution.
This record is in the source backlog; its novelty and unresolved status have not passed source review. | null | published | incremental | hard | [
"source-candidate",
"theoremdb"
] | null | null | null | null | [
{
"kind": "website",
"citation": "TheoremDB P9106",
"url": "https://theoremdb.org/statements/P9106",
"note": null,
"sort_order": 0
},
{
"kind": "formalization",
"citation": "Source formal statement (not locally verified)",
"url": "https://theoremdb.org/statements/P9106",
"not... | {
"source_record": "P9106",
"imported_at": "2026-09-08",
"status_as_of": "2026-09-08",
"evidence_label": "Source candidate · not reviewed",
"note": "Full public directory statement. Source lane: backlog; source resolution: open. Source verification claims are attributed only; no local Lean verification has be... | {"created_at":"2026-08-17T19:47:37+00:00","display_standard":{"editorial_review":"submitter_supplied","formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":9106,"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": "2026-09-08T03:32:50.211Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"aee74bcc-a3f0-4a95-a98d-4b28b5ea0a8a","format_issues":null,"number":3374} |
1.0.0 | problem:theoremdb-p9108 | problem | theoremdb-p9108 | Strong variant of Erdős's distinct distances diameter problem | {
"main": "math",
"sub": "geometry and topology",
"slug": "geometry-and-topology",
"name": "Geometry and topology",
"original_discipline": "mathematics"
} | [] | [] | Let $\mathbb{R}^2$ denote the Euclidean plane. A finite set $A \subset \mathbb{R}^2$ is said to have separated distances if for any two pairs of points $(p_1, q_1)$ and $(p_2, q_2)$ from $A$, whenever the Euclidean distances $\operatorname{dist}(p_1, q_1)$ and $\operatorname{dist}(p_2, q_2)$ are unequal, they differ by... | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P9108",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P9108",
"citation": null,
"doi": null,
"arxiv_id": null,
"license": "Selected packets label their content CC0-1.0; original references retain their own terms",
"notes": "Selected problem... | theorem erdos_100.variants.strong :
∀ᶠ n in atTop, ∀ A : Finset ℝ²,
A.card = n →
DistancesSeparated A →
diam (A : Set ℝ²) ≥ n - 1 | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem erdos_100.variants.strong :\n ∀ᶠ n in atTop, ∀ A : Finset ℝ²,\n A.card = n →\n DistancesSeparated A →\n diam (A : Set ℝ²) ≥ n - 1",
"target": null,
"source_url": "https://theoremdb.org/statements/P9108",
"format": "declaration-or-fragmen... | [] | TheoremDB P9108 · Candidate awaiting source review | null | Source-reported assessment, retrieved 8 September 2026. Frontier Science has not independently reviewed this status.
Open. No signed Lean verification is attached to this problem or its selected resolution.
This record is in the source backlog; its novelty and unresolved status have not passed source review. | null | published | incremental | hard | [
"source-candidate",
"theoremdb"
] | null | null | null | null | [
{
"kind": "website",
"citation": "TheoremDB P9108",
"url": "https://theoremdb.org/statements/P9108",
"note": null,
"sort_order": 0
},
{
"kind": "formalization",
"citation": "Source formal statement (not locally verified)",
"url": "https://theoremdb.org/statements/P9108",
"not... | {
"source_record": "P9108",
"imported_at": "2026-09-08",
"status_as_of": "2026-09-08",
"evidence_label": "Source candidate · not reviewed",
"note": "Full public directory statement. Source lane: backlog; source resolution: open. Source verification claims are attributed only; no local Lean verification has be... | {"created_at":"2026-08-17T19:47:38+00:00","display_standard":{"editorial_review":"submitter_supplied","formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":9108,"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": "2026-09-08T03:33:06.127Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"c857293f-e5fe-46b2-8528-31ac13401e8b","format_issues":null,"number":6676} |
1.0.0 | problem:theoremdb-p9110 | problem | theoremdb-p9110 | Irrationality of Ahmes series for the factorial sequence | {
"main": "math",
"sub": "number theory",
"slug": "number-theory",
"name": "Number theory",
"original_discipline": "mathematics"
} | [] | [
{
"code": "11",
"basis": "community-map",
"evidence": "Inferred from community field: number-theory"
}
] | A sequence $a_n$ of positive integers is called an irrationality sequence if for every bounded sequence of integers $b_n$ with $a_n + b_n \neq 0$ and $b_n \neq 0$ for all $n$, the series
$$
\sum_{n=0}^{\infty} \frac{1}{a_n + b_n}
$$
is irrational. Determine whether the factorial sequence $n!$ is an irrationality se... | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P9110",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P9110",
"citation": null,
"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_264.parts.ii : answer(sorry) ↔ IsIrrationalitySequence Nat.factorial | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem erdos_264.parts.ii : answer(sorry) ↔ IsIrrationalitySequence Nat.factorial",
"target": null,
"source_url": "https://theoremdb.org/statements/P9110",
"format": "declaration-or-fragment",
"sha256": "8f2f69675c53c4fd8d54e106c8c7e4ce7af036090fca7b70dda6a2... | [] | TheoremDB P9110 · Candidate awaiting source review | null | Source-reported assessment, retrieved 8 September 2026. Frontier Science has not independently reviewed this status.
Open. No signed Lean verification is attached to this problem or its selected resolution.
This record is in the source backlog; its novelty and unresolved status have not passed source review. | null | published | incremental | hard | [
"source-candidate",
"theoremdb"
] | null | null | null | null | [
{
"kind": "website",
"citation": "TheoremDB P9110",
"url": "https://theoremdb.org/statements/P9110",
"note": null,
"sort_order": 0
},
{
"kind": "formalization",
"citation": "Source formal statement (not locally verified)",
"url": "https://theoremdb.org/statements/P9110",
"not... | {
"source_record": "P9110",
"imported_at": "2026-09-08",
"status_as_of": "2026-09-08",
"evidence_label": "Source candidate · not reviewed",
"note": "Full public directory statement. Source lane: backlog; source resolution: open. Source verification claims are attributed only; no local Lean verification has be... | {"created_at":"2026-08-17T19:47:39+00:00","display_standard":{"editorial_review":"submitter_supplied","formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":9110,"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": "2026-09-08T03:33:01.394Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"7c292de8-88bd-490e-8051-9f43494d032e","format_issues":null,"number":5721} |
1.0.0 | problem:theoremdb-p9112 | problem | theoremdb-p9112 | Asymptotic growth of iterated-logarithmic hypergraph Ramsey numbers | {
"main": "math",
"sub": "combinatorics",
"slug": "combinatorics",
"name": "Combinatorics",
"original_discipline": "mathematics"
} | [] | [
{
"code": "05",
"basis": "community-map",
"evidence": "Inferred from community field: combinatorics"
}
] | For integers $r \ge 3$ and $n \ge 1$, let $R_r(n)$ denote the $r$-uniform hypergraph Ramsey number: the minimum integer $m$ such that every $2$-colouring of the edges of the complete $r$-uniform hypergraph on $m$ vertices contains a monochromatic copy of the complete $r$-uniform hypergraph on $n$ vertices. Let $\log^{[... | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P9112",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P9112",
"citation": null,
"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_562 : answer(sorry) ↔
∀ r ≥ 3, (fun n ↦ log^[r - 1] (hypergraphRamsey r n)) =Θ[atTop] (fun n ↦ (n : ℝ)) | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem erdos_562 : answer(sorry) ↔\n ∀ r ≥ 3, (fun n ↦ log^[r - 1] (hypergraphRamsey r n)) =Θ[atTop] (fun n ↦ (n : ℝ))",
"target": null,
"source_url": "https://theoremdb.org/statements/P9112",
"format": "declaration-or-fragment",
"sha256": "aaa18e8942eb84... | [] | TheoremDB P9112 · Candidate awaiting source review | null | Source-reported assessment, retrieved 8 September 2026. Frontier Science has not independently reviewed this status.
Open. No signed Lean verification is attached to this problem or its selected resolution.
This record is in the source backlog; its novelty and unresolved status have not passed source review. | null | published | incremental | hard | [
"source-candidate",
"theoremdb"
] | null | null | null | null | [
{
"kind": "website",
"citation": "TheoremDB P9112",
"url": "https://theoremdb.org/statements/P9112",
"note": null,
"sort_order": 0
},
{
"kind": "formalization",
"citation": "Source formal statement (not locally verified)",
"url": "https://theoremdb.org/statements/P9112",
"not... | {
"source_record": "P9112",
"imported_at": "2026-09-08",
"status_as_of": "2026-09-08",
"evidence_label": "Source candidate · not reviewed",
"note": "Full public directory statement. Source lane: backlog; source resolution: open. Source verification claims are attributed only; no local Lean verification has be... | {"created_at":"2026-08-17T19:47:40+00:00","display_standard":{"editorial_review":"submitter_supplied","formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":9112,"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": "2026-09-08T03:32:50.089Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"2da6ffda-17cc-4b78-9fc3-800532faae7c","format_issues":null,"number":3348} |
1.0.0 | problem:theoremdb-p9114 | problem | theoremdb-p9114 | Strict lower bound for the van der Waerden number W(3,36) | {
"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 van der Waerden number $W(k,r)$ is defined as the least positive integer $N$ such that every 2-coloring of $\{1, 2, \ldots, N\}$ contains either a monochromatic arithmetic progression of length $k$ in the first color or a monochromatic arithmetic progression of length $r$ in the s... | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P9114",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P9114",
"citation": null,
"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_36_lower : answer(sorry) ↔ W 3 36 > 1257 | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem W_3_36_lower : answer(sorry) ↔ W 3 36 > 1257",
"target": null,
"source_url": "https://theoremdb.org/statements/P9114",
"format": "declaration-or-fragment",
"sha256": "c18f161f02125e71c5582336fccd449c8f545b475ee89c6d85032e580f5bc342",
"license": "S... | [] | TheoremDB P9114 · Candidate awaiting source review | null | Source-reported assessment, retrieved 8 September 2026. Frontier Science has not independently reviewed this status.
Open. No signed Lean verification is attached to this problem or its selected resolution.
This record is in 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/P9114",
"not... | {
"source_record": "P9114",
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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":"2026-08-17T19:47:40+00:00","display_standard":{"editorial_review":"submitter_supplied","formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":9114,"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": "2026-09-08T03:33:06.099Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"30d765c8-ab26-4594-894a-975c1ad76513","format_issues":null,"number":6670} |
1.0.0 | problem:theoremdb-p9116 | problem | theoremdb-p9116 | Asymptotic upper bound for the maximum minimal length of Egyptian fraction representations | {
"main": "math",
"sub": "number theory",
"slug": "number-theory",
"name": "Number theory",
"original_discipline": "mathematics"
} | [] | [
{
"code": "11",
"basis": "community-map",
"evidence": "Inferred from community field: number-theory"
}
] | For positive integers $a$ and $b$, let $N(a,b)$ denote the smallest integer $k$ for which there exist integers $n_1, n_2, \ldots, n_k$ with $1 < n_1 < n_2 < \cdots < n_k$ such that
$$
\frac{a}{b} = \sum_{i=1}^{k} \frac{1}{n_i}.
$$
For each positive integer $b$, define
$$
N(b) = \max_{1 \leq a < b} N(a,b).
$$
Determ... | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P9116",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P9116",
"citation": null,
"doi": null,
"arxiv_id": null,
"license": "Selected packets label their content CC0-1.0; original references retain their own terms",
"notes": "Selected problem... | theorem upper_bound : answer(sorry) ↔
(fun b : ℕ => (smallestCollectionTo b : ℝ)) =O[atTop] (fun b : ℕ => Real.log (Real.log b)) | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem upper_bound : answer(sorry) ↔\n (fun b : ℕ => (smallestCollectionTo b : ℝ)) =O[atTop] (fun b : ℕ => Real.log (Real.log b))",
"target": null,
"source_url": "https://theoremdb.org/statements/P9116",
"format": "declaration-or-fragment",
"sha256": "b1a... | [] | TheoremDB P9116 · Candidate awaiting source review | null | Source-reported assessment, retrieved 8 September 2026. Frontier Science has not independently reviewed this status.
Open. No signed Lean verification is attached to this problem or its selected resolution.
This record is in the source backlog; its novelty and unresolved status have not passed source review. | null | published | incremental | hard | [
"source-candidate",
"theoremdb"
] | null | null | null | null | [
{
"kind": "website",
"citation": "TheoremDB P9116",
"url": "https://theoremdb.org/statements/P9116",
"note": null,
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},
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"citation": "Source formal statement (not locally verified)",
"url": "https://theoremdb.org/statements/P9116",
"not... | {
"source_record": "P9116",
"imported_at": "2026-09-08",
"status_as_of": "2026-09-08",
"evidence_label": "Source candidate · not reviewed",
"note": "Full public directory statement. Source lane: backlog; source resolution: open. Source verification claims are attributed only; no local Lean verification has be... | {"created_at":"2026-08-17T19:47:40+00:00","display_standard":{"editorial_review":"submitter_supplied","formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":9116,"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": "2026-09-08T03:32:50.472Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"4942cb0d-c385-4a9b-b255-1107218ac33e","format_issues":null,"number":3425} |
1.0.0 | problem:theoremdb-p9118 | problem | theoremdb-p9118 | Coset Containment in Doubling Sets with Bounded Additive Complements | {
"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$, we write $A + A = \{a + a' : a, a' \in A\}$ for its sumset. A set $S \subseteq \mathbb{F}_2^n$ is called an additive complement of $A$ if $A + S = \mathbb{F}_2^n$, meaning every elem... | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P9118",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P9118",
"citation": null,
"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_52 :
answer(sorry) ↔ ∃ (c : ℕ → ℕ), ∀ (n K : ℕ) (A : Set (𝔽₂ n)) (S : Finset (𝔽₂ n)),
S.card = K → A + (S : Set (𝔽₂ n)) = Set.univ →
∃ (V : AffineSubspace (ZMod 2) (𝔽₂ n)), (V : Set (𝔽₂ n)) ⊆ A + A ∧
n ≤ Module.finrank (ZMod 2) V.direction + c K | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem green_52 :\n answer(sorry) ↔ ∃ (c : ℕ → ℕ), ∀ (n K : ℕ) (A : Set (𝔽₂ n)) (S : Finset (𝔽₂ n)),\n S.card = K → A + (S : Set (𝔽₂ n)) = Set.univ →\n ∃ (V : AffineSubspace (ZMod 2) (𝔽₂ n)), (V : Set (𝔽₂ n)) ⊆ A + A ∧\n n ≤ Module.finrank (ZMod 2) ... | [] | TheoremDB P9118 · Candidate awaiting source review | null | Source-reported assessment, retrieved 8 September 2026. Frontier Science has not independently reviewed this status.
Open. No signed Lean verification is attached to this problem or its selected resolution.
This record is in the source backlog; its novelty and unresolved status have not passed source review. | null | published | incremental | hard | [
"source-candidate",
"theoremdb"
] | null | null | null | null | [
{
"kind": "website",
"citation": "TheoremDB P9118",
"url": "https://theoremdb.org/statements/P9118",
"note": null,
"sort_order": 0
},
{
"kind": "formalization",
"citation": "Source formal statement (not locally verified)",
"url": "https://theoremdb.org/statements/P9118",
"not... | {
"source_record": "P9118",
"imported_at": "2026-09-08",
"status_as_of": "2026-09-08",
"evidence_label": "Source candidate · not reviewed",
"note": "Full public directory statement. Source lane: backlog; source resolution: open. Source verification claims are attributed only; no local Lean verification has be... | {"created_at":"2026-08-17T19:47:41+00:00","display_standard":{"editorial_review":"submitter_supplied","formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":9118,"public_status":{"evidence":{"hasFormalization":false,"hasInformalProof":false,"hasRecordedWork":false,"solutionRecord":null},"lean":{... | [
"formalization-contains-placeholder",
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"current-status-unassessed",
"source-full-text-not-present"
] | [] | {
"created_at": "2026-09-08T03:32:15.935Z",
"updated_at": "2026-09-08T03:32:15.935Z",
"published_at": "2026-09-08T03:32:52.741Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"683d06b9-4805-46d0-83e8-964ae2699077","format_issues":null,"number":3925} |
1.0.0 | problem:theoremdb-p9120 | problem | theoremdb-p9120 | Erdős Problem 655 with general position hypothesis | {
"main": "math",
"sub": "combinatorics",
"slug": "combinatorics",
"name": "Combinatorics",
"original_discipline": "mathematics"
} | [] | [
{
"code": "05",
"basis": "community-map",
"evidence": "Inferred from community field: combinatorics"
}
] | A finite set $X \subset \mathbb{R}^2$ is called *valid* if for every point $x \in X$ and every radius $r > 0$, the circle centered at $x$ with radius $r$ contains fewer than three other points of $X$. A finite set $X \subset \mathbb{R}^2$ is in *general position* if no three points of $X$ lie on a common line and no fo... | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P9120",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P9120",
"citation": null,
"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_655.variants.general_position :
answer(sorry) ↔ ∃ c > (0 : ℝ), ∀ᶠ n in atTop, ∀ (X : Finset ℝ²), #X = n → IsValid X →
InGeneralPosition X → (1 + c) * n / 2 ≤ distinctDistances X | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem erdos_655.variants.general_position :\n answer(sorry) ↔ ∃ c > (0 : ℝ), ∀ᶠ n in atTop, ∀ (X : Finset ℝ²), #X = n → IsValid X →\n InGeneralPosition X → (1 + c) * n / 2 ≤ distinctDistances X",
"target": null,
"source_url": "https://theoremdb.org/statemen... | [] | TheoremDB P9120 · Candidate awaiting source review | null | Source-reported assessment, retrieved 8 September 2026. Frontier Science has not independently reviewed this status.
Open. No signed Lean verification is attached to this problem or its selected resolution.
This record is in the source backlog; its novelty and unresolved status have not passed source review. | null | published | incremental | hard | [
"source-candidate",
"theoremdb"
] | null | null | null | null | [
{
"kind": "website",
"citation": "TheoremDB P9120",
"url": "https://theoremdb.org/statements/P9120",
"note": null,
"sort_order": 0
},
{
"kind": "formalization",
"citation": "Source formal statement (not locally verified)",
"url": "https://theoremdb.org/statements/P9120",
"not... | {
"source_record": "P9120",
"imported_at": "2026-09-08",
"status_as_of": "2026-09-08",
"evidence_label": "Source candidate · not reviewed",
"note": "Full public directory statement. Source lane: backlog; source resolution: open. Source verification claims are attributed only; no local Lean verification has be... | {"created_at":"2026-08-17T19:47:41+00:00","display_standard":{"formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":9120,"public_status":{"evidence":{"hasFormalization":false,"hasInformalProof":false,"hasRecordedWork":false,"solutionRecord":null},"lean":{"kernelAccepted":false,"label":"","reaso... | [
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] | [] | {
"created_at": "2026-09-08T03:32:15.935Z",
"updated_at": "2026-09-08T03:32:15.935Z",
"published_at": "2026-09-08T03:32:55.813Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"b6d6624f-2fe9-4ef5-b8fa-9229dc093064","format_issues":null,"number":4602} |
1.0.0 | problem:theoremdb-p9122 | problem | theoremdb-p9122 | Density of integers where the largest prime factor increases | {
"main": "math",
"sub": "number theory",
"slug": "number-theory",
"name": "Number theory",
"original_discipline": "mathematics"
} | [] | [
{
"code": "11",
"basis": "community-map",
"evidence": "Inferred from community field: number-theory"
}
] | For each positive integer $n$, let $P(n)$ denote the largest prime factor of $n$. Determine whether the set of positive integers $n$ such that $P(n+1) > P(n)$ has natural density equal to $\frac{1}{2}$. | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P9122",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P9122",
"citation": null,
"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_371 :
{ n | Nat.maxPrimeFac (n + 1) > Nat.maxPrimeFac n }.HasDensity (1/2) | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem erdos_371 :\n { n | Nat.maxPrimeFac (n + 1) > Nat.maxPrimeFac n }.HasDensity (1/2)",
"target": null,
"source_url": "https://theoremdb.org/statements/P9122",
"format": "declaration-or-fragment",
"sha256": "67b8409c9ccd5f1ba62cd9d61f3a9b3a4c25216637c... | [] | TheoremDB P9122 · Candidate awaiting source review | null | Source-reported assessment, retrieved 8 September 2026. Frontier Science has not independently reviewed this status.
Open. No signed Lean verification is attached to this problem or its selected resolution.
This record is in the source backlog; its novelty and unresolved status have not passed source review. | null | published | incremental | hard | [
"source-candidate",
"theoremdb"
] | null | null | null | null | [
{
"kind": "website",
"citation": "TheoremDB P9122",
"url": "https://theoremdb.org/statements/P9122",
"note": null,
"sort_order": 0
},
{
"kind": "formalization",
"citation": "Source formal statement (not locally verified)",
"url": "https://theoremdb.org/statements/P9122",
"not... | {
"source_record": "P9122",
"imported_at": "2026-09-08",
"status_as_of": "2026-09-08",
"evidence_label": "Source candidate · not reviewed",
"note": "Full public directory statement. Source lane: backlog; source resolution: open. Source verification claims are attributed only; no local Lean verification has be... | {"created_at":"2026-08-17T19:47:42+00:00","display_standard":{"editorial_review":"submitter_supplied","formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":9122,"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": "2026-09-08T03:32:53.085Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"bbc3eab9-d098-4111-95f9-a9ee17a42718","format_issues":null,"number":4000} |
1.0.0 | problem:theoremdb-p9124 | problem | theoremdb-p9124 | Minimum Triangle Area for Points in the Unit Disc | {
"main": "math",
"sub": "combinatorics",
"slug": "combinatorics",
"name": "Combinatorics",
"original_discipline": "mathematics"
} | [] | [
{
"code": "05",
"basis": "community-map",
"evidence": "Inferred from community field: combinatorics"
}
] | Let $\alpha(n)$ denote the minimum, over all configurations of $n$ points in the closed unit disc, of the maximum area of a triangle determined by three of the points. Determine whether there exists a function $o: \mathbb{N} \to \mathbb{R}$ with $\lim_{n \to \infty} o(n) = 0$ such that $\alpha(n) \ll n^{-2 + o(n)}$, wh... | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P9124",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P9124",
"citation": null,
"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_77 :
answer(sorry) ↔
∃ (o : ℕ → ℝ), Tendsto o atTop (𝓝 0) ∧
α ≪ fun n ↦ n ^ (-2 + o n) | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem green_77 :\n answer(sorry) ↔\n ∃ (o : ℕ → ℝ), Tendsto o atTop (𝓝 0) ∧\n α ≪ fun n ↦ n ^ (-2 + o n)",
"target": null,
"source_url": "https://theoremdb.org/statements/P9124",
"format": "declaration-or-fragment",
"sha256": "6bd490a650e871ed88... | [] | TheoremDB P9124 · Candidate awaiting source review | null | Source-reported assessment, retrieved 8 September 2026. Frontier Science has not independently reviewed this status.
Open. No signed Lean verification is attached to this problem or its selected resolution.
This record is in the source backlog; its novelty and unresolved status have not passed source review. | null | published | incremental | hard | [
"source-candidate",
"theoremdb"
] | null | null | null | null | [
{
"kind": "website",
"citation": "TheoremDB P9124",
"url": "https://theoremdb.org/statements/P9124",
"note": null,
"sort_order": 0
},
{
"kind": "formalization",
"citation": "Source formal statement (not locally verified)",
"url": "https://theoremdb.org/statements/P9124",
"not... | {
"source_record": "P9124",
"imported_at": "2026-09-08",
"status_as_of": "2026-09-08",
"evidence_label": "Source candidate · not reviewed",
"note": "Full public directory statement. Source lane: backlog; source resolution: open. Source verification claims are attributed only; no local Lean verification has be... | {"created_at":"2026-08-17T19:47:42+00:00","display_standard":{"editorial_review":"submitter_supplied","formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":9124,"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": "2026-09-08T03:33:04.049Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"3f665e5c-ffde-4eed-9b90-d34ccf90d3b8","format_issues":null,"number":6228} |
1.0.0 | problem:theoremdb-p9126 | problem | theoremdb-p9126 | Divisibility of Consecutive Integer Products in Erdős Problem 389 | {
"main": "math",
"sub": "number theory",
"slug": "number-theory",
"name": "Number theory",
"original_discipline": "mathematics"
} | [] | [
{
"code": "11",
"basis": "community-map",
"evidence": "Inferred from community field: number-theory"
}
] | For positive integers $n$ and $k$, consider the product of $k$ consecutive integers starting from $n$, denoted by
$$
P(n,k) = n(n+1)\cdots(n+k-1) = \prod_{i=0}^{k-1}(n+i),
$$
and the product of $k$ consecutive integers starting from $n+k$, denoted by
$$
Q(n,k) = (n+k)(n+k+1)\cdots(n+2k-1) = \prod_{i=0}^{k-1}(n+k+... | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P9126",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P9126",
"citation": null,
"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_389 : answer(sorry) ↔
∀ n ≥ 1, ∃ k ≥ 1, ∏ i ∈ Finset.range k, (n + i) ∣ ∏ i ∈ Finset.range k, (n + k + i) | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem erdos_389 : answer(sorry) ↔\n ∀ n ≥ 1, ∃ k ≥ 1, ∏ i ∈ Finset.range k, (n + i) ∣ ∏ i ∈ Finset.range k, (n + k + i)",
"target": null,
"source_url": "https://theoremdb.org/statements/P9126",
"format": "declaration-or-fragment",
"sha256": "d0197cb9d169... | [] | TheoremDB P9126 · Candidate awaiting source review | null | Source-reported assessment, retrieved 8 September 2026. Frontier Science has not independently reviewed this status.
Open. No signed Lean verification is attached to this problem or its selected resolution.
This record is in the source backlog; its novelty and unresolved status have not passed source review. | null | published | incremental | hard | [
"source-candidate",
"theoremdb"
] | null | null | null | null | [
{
"kind": "website",
"citation": "TheoremDB P9126",
"url": "https://theoremdb.org/statements/P9126",
"note": null,
"sort_order": 0
},
{
"kind": "formalization",
"citation": "Source formal statement (not locally verified)",
"url": "https://theoremdb.org/statements/P9126",
"not... | {
"source_record": "P9126",
"imported_at": "2026-09-08",
"status_as_of": "2026-09-08",
"evidence_label": "Source candidate · not reviewed",
"note": "Full public directory statement. Source lane: backlog; source resolution: open. Source verification claims are attributed only; no local Lean verification has be... | {"created_at":"2026-08-17T19:47:43+00:00","display_standard":{"editorial_review":"submitter_supplied","formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":9126,"public_status":{"evidence":{"hasFormalization":false,"hasInformalProof":false,"hasRecordedWork":false,"solutionRecord":null},"lean":{... | [
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"formalization-not-compiled-during-export",
"current-status-unassessed",
"source-full-text-not-present"
] | [] | {
"created_at": "2026-09-08T03:32:15.935Z",
"updated_at": "2026-09-08T03:32:15.935Z",
"published_at": "2026-09-08T03:32:53.285Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"79c9d624-c73a-4daf-9796-fcd0541ea0be","format_issues":null,"number":4043} |
1.0.0 | problem:theoremdb-p9128 | problem | theoremdb-p9128 | The Bunyakovsky conjecture on prime values of integer 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 $f$ be a polynomial with integer coefficients. Suppose that $f$ satisfies the following two conditions:
$i)$ (Bunyakovsky's condition) The leading coefficient of $f$ is positive, and there is no prime $p$ that divides $f(n)$ for every integer $n$.
$ii)$ (Schinzel's condition) The polynomial $f$ is irreducible ove... | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P9128",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P9128",
"citation": null,
"doi": null,
"arxiv_id": null,
"license": "Selected packets label their content CC0-1.0; original references retain their own terms",
"notes": "Selected problem... | theorem bunyakovsky_conjecture (f : ℤ[X]) :
BunyakovskyCondition f ∧ SchinzelCondition {f} →
Infinite {n : ℕ | (f.eval (n : ℤ)).natAbs.Prime} | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem bunyakovsky_conjecture (f : ℤ[X]) :\n BunyakovskyCondition f ∧ SchinzelCondition {f} →\n Infinite {n : ℕ | (f.eval (n : ℤ)).natAbs.Prime}",
"target": null,
"source_url": "https://theoremdb.org/statements/P9128",
"format": "declaration-or-fragment",
... | [] | TheoremDB P9128 · Candidate awaiting source review | null | Source-reported assessment, retrieved 8 September 2026. Frontier Science has not independently reviewed this status.
Open. No signed Lean verification is attached to this problem or its selected resolution.
This record is in the source backlog; its novelty and unresolved status have not passed source review. | null | published | incremental | hard | [
"source-candidate",
"theoremdb"
] | null | null | null | null | [
{
"kind": "website",
"citation": "TheoremDB P9128",
"url": "https://theoremdb.org/statements/P9128",
"note": null,
"sort_order": 0
},
{
"kind": "formalization",
"citation": "Source formal statement (not locally verified)",
"url": "https://theoremdb.org/statements/P9128",
"not... | {
"source_record": "P9128",
"imported_at": "2026-09-08",
"status_as_of": "2026-09-08",
"evidence_label": "Source candidate · not reviewed",
"note": "Full public directory statement. Source lane: backlog; source resolution: open. Source verification claims are attributed only; no local Lean verification has be... | {"created_at":"2026-08-17T19:47:43+00:00","display_standard":{"formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":9128,"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": "2026-09-08T03:33:06.403Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"be92c6e7-52ad-4a1b-9c46-5feafbbbaaa2","format_issues":null,"number":6735} |
1.0.0 | problem:theoremdb-p9130 | problem | theoremdb-p9130 | Density of integers with coprime harmonic numerator and least common multiple | {
"main": "math",
"sub": "number theory",
"slug": "number-theory",
"name": "Number theory",
"original_discipline": "mathematics"
} | [] | [
{
"code": "11",
"basis": "community-map",
"evidence": "Inferred from community field: number-theory"
}
] | For each positive integer $n$, let $L_n$ denote the least common multiple of $\{1, 2, \ldots, n\}$, and let $a_n$ be the unique positive integer defined by the identity
$$
\sum_{k=1}^{n} \frac{1}{k} = \frac{a_n}{L_n}.
$$
Let $S$ denote the set of positive integers $n$ for which $\gcd(a_n, L_n) = 1$. Define the natu... | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P9130",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P9130",
"citation": null,
"doi": null,
"arxiv_id": null,
"license": "Selected packets label their content CC0-1.0; original references retain their own terms",
"notes": "Selected problem... | theorem erdos_291.variants.shiu_heuristic_density_zero :
Filter.Tendsto (fun N : ℕ ↦
((((Finset.Icc 1 N).filter (fun n ↦ Nat.gcd (a n) (L n) = 1)).card : ℝ) / (N : ℝ)))
atTop (nhds 0) | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem erdos_291.variants.shiu_heuristic_density_zero :\n Filter.Tendsto (fun N : ℕ ↦\n ((((Finset.Icc 1 N).filter (fun n ↦ Nat.gcd (a n) (L n) = 1)).card : ℝ) / (N : ℝ)))\n atTop (nhds 0)",
"target": null,
"source_url": "https://theoremdb.org/statement... | [] | TheoremDB P9130 · Candidate awaiting source review | null | Source-reported assessment, retrieved 8 September 2026. Frontier Science has not independently reviewed this status.
Open. No signed Lean verification is attached to this problem or its selected resolution.
This record is in the source backlog; its novelty and unresolved status have not passed source review. | null | published | incremental | hard | [
"source-candidate",
"theoremdb"
] | null | null | null | null | [
{
"kind": "website",
"citation": "TheoremDB P9130",
"url": "https://theoremdb.org/statements/P9130",
"note": null,
"sort_order": 0
},
{
"kind": "formalization",
"citation": "Source formal statement (not locally verified)",
"url": "https://theoremdb.org/statements/P9130",
"not... | {
"source_record": "P9130",
"imported_at": "2026-09-08",
"status_as_of": "2026-09-08",
"evidence_label": "Source candidate · not reviewed",
"note": "Full public directory statement. Source lane: backlog; source resolution: open. Source verification claims are attributed only; no local Lean verification has be... | {"created_at":"2026-08-17T19:47:44+00:00","display_standard":{"editorial_review":"submitter_supplied","formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":9130,"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": "2026-09-08T03:32:53.094Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"de84e890-f14f-421f-b97e-abb207fed9bd","format_issues":null,"number":4002} |
1.0.0 | problem:theoremdb-p9132 | problem | theoremdb-p9132 | Lower bound for the van der Waerden number W(3,30) | {
"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 · P9132",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P9132",
"citation": null,
"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_30_lower : answer(sorry) ↔ W 3 30 ≥ 903 | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem W_3_30_lower : answer(sorry) ↔ W 3 30 ≥ 903",
"target": null,
"source_url": "https://theoremdb.org/statements/P9132",
"format": "declaration-or-fragment",
"sha256": "a86b2c6e0d71de167a551039396c78e25bfc75925d701f30f9b1592c742db72f",
"license": "Se... | [] | TheoremDB P9132 · Candidate awaiting source review | null | Source-reported assessment, retrieved 8 September 2026. Frontier Science has not independently reviewed this status.
Open. No signed Lean verification is attached to this problem or its selected resolution.
This record is in the source backlog; its novelty and unresolved status have not passed source review. | null | published | incremental | hard | [
"source-candidate",
"theoremdb"
] | null | null | null | null | [
{
"kind": "website",
"citation": "TheoremDB P9132",
"url": "https://theoremdb.org/statements/P9132",
"note": null,
"sort_order": 0
},
{
"kind": "formalization",
"citation": "Source formal statement (not locally verified)",
"url": "https://theoremdb.org/statements/P9132",
"not... | {
"source_record": "P9132",
"imported_at": "2026-09-08",
"status_as_of": "2026-09-08",
"evidence_label": "Source candidate · not reviewed",
"note": "Full public directory statement. Source lane: backlog; source resolution: open. Source verification claims are attributed only; no local Lean verification has be... | {"created_at":"2026-08-17T19:47:44+00:00","display_standard":{"formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":9132,"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": "2026-09-08T03:33:03.547Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"3ccf8df6-01ba-4af2-9741-6d8cb56db691","format_issues":null,"number":6118} |
1.0.0 | problem:theoremdb-p9134 | problem | theoremdb-p9134 | Non-existence of monochromatic quantum graphs with ternary integer weights | {
"main": "math",
"sub": "combinatorics",
"slug": "combinatorics",
"name": "Combinatorics",
"original_discipline": "mathematics"
} | [] | [
{
"code": "05",
"basis": "community-map",
"evidence": "Inferred from community field: combinatorics"
}
] | Let $N$ and $D$ be natural numbers with $N \geq 6$ even and $D \geq 3$. Consider a complete graph on $N$ vertices whose edges are coloured with $D$ colours and assigned integer weights in $\{-1, 0, 1\}$. For each vertex colouring $\iota$ (an assignment of one of the $D$ colours to each vertex), define the perfect-match... | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P9134",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P9134",
"citation": null,
"doi": null,
"arxiv_id": null,
"license": "Selected packets label their content CC0-1.0; original references retain their own terms",
"notes": "Selected problem... | theorem eqSystem_no_solution_ge6_ge3_trinary_int :
answer(sorry) ↔
∀ N D : Nat, N ≥ 6 → Even N → D ≥ 3 →
¬ ∃ W : WeightsN N D ℤ,
(∀ e, W e = (-1 : ℤ) ∨ W e = 0 ∨ W e = 1) ∧
EqSystemN N D W | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem eqSystem_no_solution_ge6_ge3_trinary_int :\n answer(sorry) ↔\n ∀ N D : Nat, N ≥ 6 → Even N → D ≥ 3 →\n ¬ ∃ W : WeightsN N D ℤ,\n (∀ e, W e = (-1 : ℤ) ∨ W e = 0 ∨ W e = 1) ∧\n EqSystemN N D W",
"target": null,
"source_url... | [] | TheoremDB P9134 · Candidate awaiting source review | null | Source-reported assessment, retrieved 8 September 2026. Frontier Science has not independently reviewed this status.
Open. No signed Lean verification is attached to this problem or its selected resolution.
This record is in the source backlog; its novelty and unresolved status have not passed source review. | null | published | incremental | hard | [
"source-candidate",
"theoremdb"
] | null | null | null | null | [
{
"kind": "website",
"citation": "TheoremDB P9134",
"url": "https://theoremdb.org/statements/P9134",
"note": null,
"sort_order": 0
},
{
"kind": "formalization",
"citation": "Source formal statement (not locally verified)",
"url": "https://theoremdb.org/statements/P9134",
"not... | {
"source_record": "P9134",
"imported_at": "2026-09-08",
"status_as_of": "2026-09-08",
"evidence_label": "Source candidate · not reviewed",
"note": "Full public directory statement. Source lane: backlog; source resolution: open. Source verification claims are attributed only; no local Lean verification has be... | {"created_at":"2026-08-17T19:47:45+00:00","display_standard":{"editorial_review":"submitter_supplied","formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":9134,"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": "2026-09-08T03:33:04.391Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"ed5f4c51-f04c-425d-a692-4e4363d9871b","format_issues":null,"number":6300} |
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