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1.0.0 | problem:theoremdb-p9736 | problem | theoremdb-p9736 | Brennan's Conjecture on the Universal Integral Means Spectrum | {
"main": "math",
"sub": "analysis",
"slug": "analysis",
"name": "Analysis",
"original_discipline": "mathematics"
} | [] | [] | Let $\mathbb{D} = \{z \in \mathbb{C} : |z| < 1\}$ denote the open unit disk in the complex plane. A function $f: \mathbb{C} \to \mathbb{C}$ is called normalized univalent on $\mathbb{D}$ if it is analytic on $\mathbb{D}$, injective on $\mathbb{D}$, and satisfies $f(0) = 0$ and $f'(0) = 1$. For such a function $f$ and a... | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P9736",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P9736",
"citation": null,
"doi": null,
"arxiv_id": null,
"license": "Selected packets label their content CC0-1.0; original references retain their own terms",
"notes": "Selected problem... | theorem brennan_universalSpectrum :
universalSpectrum (-2) = 1 | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem brennan_universalSpectrum :\n universalSpectrum (-2) = 1",
"target": null,
"source_url": "https://theoremdb.org/statements/P9736",
"format": "declaration-or-fragment",
"sha256": "a14d3df6c2921543503b330f3248ab1ece51a3fa131aa62a6919c64be7bbde02",
... | [] | TheoremDB P9736 · Candidate awaiting source review | null | Source-reported assessment, retrieved 8 September 2026. Frontier Science has not independently reviewed this status.
Open. No signed Lean verification is attached to this problem or its selected resolution.
This record is in the source backlog; its novelty and unresolved status have not passed source review. | null | published | incremental | hard | [
"source-candidate",
"theoremdb"
] | null | null | null | null | [
{
"kind": "website",
"citation": "TheoremDB P9736",
"url": "https://theoremdb.org/statements/P9736",
"note": null,
"sort_order": 0
},
{
"kind": "formalization",
"citation": "Source formal statement (not locally verified)",
"url": "https://theoremdb.org/statements/P9736",
"not... | {
"source_record": "P9736",
"imported_at": "2026-09-08",
"status_as_of": "2026-09-08",
"evidence_label": "Source candidate · not reviewed",
"note": "Full public directory statement. Source lane: backlog; source resolution: open. Source verification claims are attributed only; no local Lean verification has be... | {"created_at":"2026-08-17T20:02:10+00:00","display_standard":{"editorial_review":"submitter_supplied","formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":9736,"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": "2026-09-08T03:32:50.777Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"48023daf-5e17-4ad9-9f3f-4eda50091935","format_issues":null,"number":3490} |
1.0.0 | problem:theoremdb-p9738 | problem | theoremdb-p9738 | Existence of near-extremal additive bases in cyclic groups | {
"main": "math",
"sub": "combinatorics",
"slug": "combinatorics",
"name": "Combinatorics",
"original_discipline": "mathematics"
} | [] | [
{
"code": "05",
"basis": "community-map",
"evidence": "Inferred from community field: combinatorics"
}
] | Let $\mathbb{Z}/q\mathbb{Z}$ denote the cyclic group of order $q$, and let $A + A = \{a + b : a, b \in A\}$ denote the sumset of a subset $A \subseteq \mathbb{Z}/q\mathbb{Z}$. A subset $A$ is called an additive basis for $\mathbb{Z}/q\mathbb{Z}$ if $A + A = \mathbb{Z}/q\mathbb{Z}$. For a function $f(q)$, we write $|A| ... | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P9738",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P9738",
"citation": null,
"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_33 :
answer(sorry) ↔
∀ ε : ℝ, 0 < ε →
∃ᶠ q : ℕ+ in atTop,
∃ A : Finset (ZMod q),
A + A = Finset.univ ∧
|((A.card : ℝ) / Real.sqrt q - Real.sqrt 2)| < ε | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem green_33 :\n answer(sorry) ↔\n ∀ ε : ℝ, 0 < ε →\n ∃ᶠ q : ℕ+ in atTop,\n ∃ A : Finset (ZMod q),\n A + A = Finset.univ ∧\n |((A.card : ℝ) / Real.sqrt q - Real.sqrt 2)| < ε",
"target": null,
"source_url": "https://th... | [] | TheoremDB P9738 · Candidate awaiting source review | null | Source-reported assessment, retrieved 8 September 2026. Frontier Science has not independently reviewed this status.
Open. No signed Lean verification is attached to this problem or its selected resolution.
This record is in the source backlog; its novelty and unresolved status have not passed source review. | null | published | incremental | hard | [
"source-candidate",
"theoremdb"
] | null | null | null | null | [
{
"kind": "website",
"citation": "TheoremDB P9738",
"url": "https://theoremdb.org/statements/P9738",
"note": null,
"sort_order": 0
},
{
"kind": "formalization",
"citation": "Source formal statement (not locally verified)",
"url": "https://theoremdb.org/statements/P9738",
"not... | {
"source_record": "P9738",
"imported_at": "2026-09-08",
"status_as_of": "2026-09-08",
"evidence_label": "Source candidate · not reviewed",
"note": "Full public directory statement. Source lane: backlog; source resolution: open. Source verification claims are attributed only; no local Lean verification has be... | {"created_at":"2026-08-17T20:02:11+00:00","display_standard":{"editorial_review":"submitter_supplied","formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":9738,"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": "2026-09-08T03:32:58.788Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"0c89055f-30c1-4763-8e7a-629782b5c88c","format_issues":null,"number":5255} |
1.0.0 | problem:theoremdb-p9740 | problem | theoremdb-p9740 | Erdős Problem 595 on Infinite $K_4$-Free Graphs and Countable Decompositions | {
"main": "math",
"sub": "combinatorics",
"slug": "combinatorics",
"name": "Combinatorics",
"original_discipline": "mathematics"
} | [] | [
{
"code": "05",
"basis": "community-map",
"evidence": "Inferred from community field: combinatorics"
}
] | A simple graph $G$ with vertex set $V$ is called triangle-free if it contains no $K_3$, that is, no three distinct vertices are pairwise adjacent. A graph $G$ is called a countable union of triangle-free graphs if there exists a sequence $(H_i)_{i \in \mathbb{N}}$ of triangle-free graphs on the same vertex set $V$ such... | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P9740",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P9740",
"citation": null,
"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_595 : answer(sorry) ↔
∃ (V : Type*) (_ : Infinite V) (G : SimpleGraph V),
G.CliqueFree 4 ∧ ¬IsCountableUnionOfTriangleFree G | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem erdos_595 : answer(sorry) ↔\n ∃ (V : Type*) (_ : Infinite V) (G : SimpleGraph V),\n G.CliqueFree 4 ∧ ¬IsCountableUnionOfTriangleFree G",
"target": null,
"source_url": "https://theoremdb.org/statements/P9740",
"format": "declaration-or-fragment",
... | [] | TheoremDB P9740 · Candidate awaiting source review | null | Source-reported assessment, retrieved 8 September 2026. Frontier Science has not independently reviewed this status.
Open. No signed Lean verification is attached to this problem or its selected resolution.
This record is in the source backlog; its novelty and unresolved status have not passed source review. | null | published | incremental | hard | [
"source-candidate",
"theoremdb"
] | null | null | null | null | [
{
"kind": "website",
"citation": "TheoremDB P9740",
"url": "https://theoremdb.org/statements/P9740",
"note": null,
"sort_order": 0
},
{
"kind": "formalization",
"citation": "Source formal statement (not locally verified)",
"url": "https://theoremdb.org/statements/P9740",
"not... | {
"source_record": "P9740",
"imported_at": "2026-09-08",
"status_as_of": "2026-09-08",
"evidence_label": "Source candidate · not reviewed",
"note": "Full public directory statement. Source lane: backlog; source resolution: open. Source verification claims are attributed only; no local Lean verification has be... | {"created_at":"2026-08-17T20:02:12+00:00","display_standard":{"formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":9740,"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": "2026-09-08T03:32:55.699Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"ac9af50c-8029-4e87-9821-aa23d83c9844","format_issues":null,"number":4577} |
1.0.0 | problem:theoremdb-p9742 | problem | theoremdb-p9742 | Lower bound for the off-diagonal van der Waerden number W(3,37) | {
"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 · P9742",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P9742",
"citation": null,
"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_37_lower : answer(sorry) ↔ W 3 37 > 1338 | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem W_3_37_lower : answer(sorry) ↔ W 3 37 > 1338",
"target": null,
"source_url": "https://theoremdb.org/statements/P9742",
"format": "declaration-or-fragment",
"sha256": "848362aa5d0ae702da7b6c2e28e4c467f0b0d7b60416904b463ce262633ce30f",
"license": "S... | [] | TheoremDB P9742 · Candidate awaiting source review | null | Source-reported assessment, retrieved 8 September 2026. Frontier Science has not independently reviewed this status.
Open. No signed Lean verification is attached to this problem or its selected resolution.
This record is in the source backlog; its novelty and unresolved status have not passed source review. | null | published | incremental | hard | [
"source-candidate",
"theoremdb"
] | null | null | null | null | [
{
"kind": "website",
"citation": "TheoremDB P9742",
"url": "https://theoremdb.org/statements/P9742",
"note": null,
"sort_order": 0
},
{
"kind": "formalization",
"citation": "Source formal statement (not locally verified)",
"url": "https://theoremdb.org/statements/P9742",
"not... | {
"source_record": "P9742",
"imported_at": "2026-09-08",
"status_as_of": "2026-09-08",
"evidence_label": "Source candidate · not reviewed",
"note": "Full public directory statement. Source lane: backlog; source resolution: open. Source verification claims are attributed only; no local Lean verification has be... | {"created_at":"2026-08-17T20:02:14+00:00","display_standard":{"formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":9742,"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": "2026-09-08T03:33:03.489Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"6cac9992-48fd-4f6e-8328-863430637b9f","format_issues":null,"number":6106} |
1.0.0 | problem:theoremdb-p9744 | problem | theoremdb-p9744 | Termination of the iterated sum-of-divisors-minus-one map | {
"main": "math",
"sub": "number theory",
"slug": "number-theory",
"name": "Number theory",
"original_discipline": "mathematics"
} | [] | [
{
"code": "11",
"basis": "community-map",
"evidence": "Inferred from community field: number-theory"
}
] | For an integer $n > 1$, define the iteration $f(n) = \sigma(n) - 1$, where $\sigma(n)$ denotes the sum of all positive divisors of $n$. Let $f^{[i]}(n)$ denote the $i$-fold iterate of $f$ applied to $n$, with $f^{[0]}(n) = n$. Determine whether it is true that for every integer $n > 1$, there exists a non-negative inte... | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P9744",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P9744",
"citation": null,
"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_prime_termination :
answer(sorry) ↔ ∀ n > 1, ∃ i, (σ 1 · - 1)^[i] n |>.Prime | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem erdos_409.variants.sigma_prime_termination :\n answer(sorry) ↔ ∀ n > 1, ∃ i, (σ 1 · - 1)^[i] n |>.Prime",
"target": null,
"source_url": "https://theoremdb.org/statements/P9744",
"format": "declaration-or-fragment",
"sha256": "649ab02044ec4da55b8db0... | [] | TheoremDB P9744 · Candidate awaiting source review | null | Source-reported assessment, retrieved 8 September 2026. Frontier Science has not independently reviewed this status.
Open. No signed Lean verification is attached to this problem or its selected resolution.
This record is in the source backlog; its novelty and unresolved status have not passed source review. | null | published | incremental | hard | [
"source-candidate",
"theoremdb"
] | null | null | null | null | [
{
"kind": "website",
"citation": "TheoremDB P9744",
"url": "https://theoremdb.org/statements/P9744",
"note": null,
"sort_order": 0
},
{
"kind": "formalization",
"citation": "Source formal statement (not locally verified)",
"url": "https://theoremdb.org/statements/P9744",
"not... | {
"source_record": "P9744",
"imported_at": "2026-09-08",
"status_as_of": "2026-09-08",
"evidence_label": "Source candidate · not reviewed",
"note": "Full public directory statement. Source lane: backlog; source resolution: open. Source verification claims are attributed only; no local Lean verification has be... | {"created_at":"2026-08-17T20:02:15+00:00","display_standard":{"formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":9744,"public_status":{"evidence":{"hasFormalization":false,"hasInformalProof":false,"hasRecordedWork":false,"solutionRecord":null},"lean":{"kernelAccepted":false,"label":"","reaso... | [
"formalization-contains-placeholder",
"formalization-not-compiled-during-export",
"current-status-unassessed",
"source-full-text-not-present"
] | [] | {
"created_at": "2026-09-08T03:32:15.935Z",
"updated_at": "2026-09-08T03:32:15.935Z",
"published_at": "2026-09-08T03:33:06.297Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"f7a3830d-9f7e-45c0-a9ef-65f4c99e780f","format_issues":null,"number":6713} |
1.0.0 | problem:theoremdb-p9746 | problem | theoremdb-p9746 | Schinzel's generalization of the Erdős–Sierpiński Egyptian fraction 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 $a$ be a fixed positive integer. Determine whether, for all sufficiently large positive integers $n$, there exist distinct positive integers $x$, $y$, and $z$ with $1 \leq x < y < z$ such that $\frac{a}{n} = \frac{1}{x} + \frac{1}{y} + \frac{1}{z}$. | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P9746",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P9746",
"citation": null,
"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_242.variants.schinzel_generalization
(a : ℕ) (ha : 0 < a) :
∀ᶠ (n : ℕ) in Filter.atTop, ∃ x y z : ℕ, 1 ≤ x ∧ x < y ∧ y < z ∧
(a / n : ℚ) = 1 / x + 1 / y + 1 / z | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem erdos_242.variants.schinzel_generalization\n (a : ℕ) (ha : 0 < a) :\n ∀ᶠ (n : ℕ) in Filter.atTop, ∃ x y z : ℕ, 1 ≤ x ∧ x < y ∧ y < z ∧\n (a / n : ℚ) = 1 / x + 1 / y + 1 / z",
"target": null,
"source_url": "https://theoremdb.org/statements/P9746",
... | [] | TheoremDB P9746 · Candidate awaiting source review | null | Source-reported assessment, retrieved 8 September 2026. Frontier Science has not independently reviewed this status.
Open. No signed Lean verification is attached to this problem or its selected resolution.
This record is in the source backlog; its novelty and unresolved status have not passed source review. | null | published | incremental | hard | [
"source-candidate",
"theoremdb"
] | null | null | null | null | [
{
"kind": "website",
"citation": "TheoremDB P9746",
"url": "https://theoremdb.org/statements/P9746",
"note": null,
"sort_order": 0
},
{
"kind": "formalization",
"citation": "Source formal statement (not locally verified)",
"url": "https://theoremdb.org/statements/P9746",
"not... | {
"source_record": "P9746",
"imported_at": "2026-09-08",
"status_as_of": "2026-09-08",
"evidence_label": "Source candidate · not reviewed",
"note": "Full public directory statement. Source lane: backlog; source resolution: open. Source verification claims are attributed only; no local Lean verification has be... | {"created_at":"2026-08-17T20:02:15+00:00","display_standard":{"editorial_review":"submitter_supplied","formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":9746,"public_status":{"evidence":{"hasFormalization":false,"hasInformalProof":false,"hasRecordedWork":false,"solutionRecord":null},"lean":{... | [
"formalization-not-compiled-during-export",
"current-status-unassessed",
"source-full-text-not-present"
] | [] | {
"created_at": "2026-09-08T03:32:15.935Z",
"updated_at": "2026-09-08T03:32:15.935Z",
"published_at": "2026-09-08T03:33:05.728Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"da8616e1-1194-4e82-b74c-03b957757420","format_issues":null,"number":6590} |
1.0.0 | problem:theoremdb-p9748 | problem | theoremdb-p9748 | Infinitely Many Missing Values of Hofstadter's Q-Sequence Variant | {
"main": "math",
"sub": "number theory",
"slug": "number-theory",
"name": "Number theory",
"original_discipline": "mathematics"
} | [] | [
{
"code": "11",
"basis": "community-map",
"evidence": "Inferred from community field: number-theory"
}
] | Define a function $f : \mathbb{Z}^+ \to \mathbb{Z}^+$ by $f(1) = f(2) = 1$ and for all integers $n > 2$,
$$
f(n) = f(n - f(n-1)) + f(n - f(n-2)).
$$
(It is not known whether $f(n)$ is well-defined for all $n$.)
Determine whether the set $\{n \in \mathbb{Z}^+ : f(x) \neq n \text{ for all } x \in \mathbb{Z}^+\}$ is ... | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P9748",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P9748",
"citation": null,
"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 : answer(sorry) ↔ Set.Infinite {n | ∀ x, f x ≠ n} | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem erdos_422 : answer(sorry) ↔ Set.Infinite {n | ∀ x, f x ≠ n}",
"target": null,
"source_url": "https://theoremdb.org/statements/P9748",
"format": "declaration-or-fragment",
"sha256": "a06b0380238bbc05b7f8b6058f3cd83d35d1b79aa705a9bd23d6f8123b82989d",
... | [] | TheoremDB P9748 · Candidate awaiting source review | null | Source-reported assessment, retrieved 8 September 2026. Frontier Science has not independently reviewed this status.
Open. No signed Lean verification is attached to this problem or its selected resolution.
This record is in the source backlog; its novelty and unresolved status have not passed source review. | null | published | incremental | hard | [
"source-candidate",
"theoremdb"
] | null | null | null | null | [
{
"kind": "website",
"citation": "TheoremDB P9748",
"url": "https://theoremdb.org/statements/P9748",
"note": null,
"sort_order": 0
},
{
"kind": "formalization",
"citation": "Source formal statement (not locally verified)",
"url": "https://theoremdb.org/statements/P9748",
"not... | {
"source_record": "P9748",
"imported_at": "2026-09-08",
"status_as_of": "2026-09-08",
"evidence_label": "Source candidate · not reviewed",
"note": "Full public directory statement. Source lane: backlog; source resolution: open. Source verification claims are attributed only; no local Lean verification has be... | {"created_at":"2026-08-17T20:02:16+00:00","display_standard":{"editorial_review":"submitter_supplied","formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":9748,"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": "2026-09-08T03:33:00.943Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"c66a63f9-0882-4080-b1d3-b1f835132bc4","format_issues":null,"number":5624} |
1.0.0 | problem:theoremdb-p9750 | problem | theoremdb-p9750 | Existence of a covering system with all moduli odd and greater than 1 | {
"main": "math",
"sub": "number theory",
"slug": "number-theory",
"name": "Number theory",
"original_discipline": "mathematics"
} | [] | [
{
"code": "11",
"basis": "community-map",
"evidence": "Inferred from community field: number-theory"
}
] | A strict covering system of the integers is a collection of congruence classes such that every integer belongs to exactly one class in the collection, where the moduli are required to be distinct. The modulus of a congruence class is the positive integer that defines the residue classes modulo that integer. For an idea... | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P9750",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P9750",
"citation": null,
"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_7 : answer(sorry) ↔
∃ (C : StrictCoveringSystem ℤ), ∀ i,
¬ C.moduli i ≤ Ideal.span {2} ∧ C.moduli i ≠ ⊤ | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem erdos_7 : answer(sorry) ↔\n ∃ (C : StrictCoveringSystem ℤ), ∀ i,\n ¬ C.moduli i ≤ Ideal.span {2} ∧ C.moduli i ≠ ⊤",
"target": null,
"source_url": "https://theoremdb.org/statements/P9750",
"format": "declaration-or-fragment",
"sha256": "f81736c... | [] | TheoremDB P9750 · Candidate awaiting source review | null | Source-reported assessment, retrieved 8 September 2026. Frontier Science has not independently reviewed this status.
Open. No signed Lean verification is attached to this problem or its selected resolution.
This record is in the source backlog; its novelty and unresolved status have not passed source review. | null | published | incremental | hard | [
"source-candidate",
"theoremdb"
] | null | null | null | null | [
{
"kind": "website",
"citation": "TheoremDB P9750",
"url": "https://theoremdb.org/statements/P9750",
"note": null,
"sort_order": 0
},
{
"kind": "formalization",
"citation": "Source formal statement (not locally verified)",
"url": "https://theoremdb.org/statements/P9750",
"not... | {
"source_record": "P9750",
"imported_at": "2026-09-08",
"status_as_of": "2026-09-08",
"evidence_label": "Source candidate · not reviewed",
"note": "Full public directory statement. Source lane: backlog; source resolution: open. Source verification claims are attributed only; no local Lean verification has be... | {"created_at":"2026-08-17T20:02:18+00:00","display_standard":{"editorial_review":"submitter_supplied","formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":9750,"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": "2026-09-08T03:32:58.421Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"43a2f9a8-1a03-4df0-b0af-68ffca98ef96","format_issues":null,"number":5179} |
1.0.0 | problem:theoremdb-p9752 | problem | theoremdb-p9752 | Connectedness of inverses for connected bijections of Euclidean spaces | {
"main": "math",
"sub": "analysis",
"slug": "analysis",
"name": "Analysis",
"original_discipline": "mathematics"
} | [] | [] | For an integer $n > 1$, let $\mathbb{R}^n$ denote the $n$-dimensional Euclidean space with its standard topology. A function $f : X \to Y$ between topological spaces is called *connected* if for every connected subset $s \subseteq X$, the image $f(s)$ is connected in $Y$. Suppose $f : \mathbb{R}^n \to \mathbb{R}^n$ is ... | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P9752",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P9752",
"citation": null,
"doi": null,
"arxiv_id": null,
"license": "Selected packets label their content CC0-1.0; original references retain their own terms",
"notes": "Selected problem... | theorem mathoverflow_235893 :
answer(sorry) ↔ ∀ n > 1, ∀ (f : ℝ^n ≃ ℝ^n), IsConnectedMap f → IsConnectedMap f.symm | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem mathoverflow_235893 :\n answer(sorry) ↔ ∀ n > 1, ∀ (f : ℝ^n ≃ ℝ^n), IsConnectedMap f → IsConnectedMap f.symm",
"target": null,
"source_url": "https://theoremdb.org/statements/P9752",
"format": "declaration-or-fragment",
"sha256": "c65ba0e3d83fe0ead... | [] | TheoremDB P9752 · Candidate awaiting source review | null | Source-reported assessment, retrieved 8 September 2026. Frontier Science has not independently reviewed this status.
Open. No signed Lean verification is attached to this problem or its selected resolution.
This record is in the source backlog; its novelty and unresolved status have not passed source review. | null | published | incremental | hard | [
"source-candidate",
"theoremdb"
] | null | null | null | null | [
{
"kind": "website",
"citation": "TheoremDB P9752",
"url": "https://theoremdb.org/statements/P9752",
"note": null,
"sort_order": 0
},
{
"kind": "formalization",
"citation": "Source formal statement (not locally verified)",
"url": "https://theoremdb.org/statements/P9752",
"not... | {
"source_record": "P9752",
"imported_at": "2026-09-08",
"status_as_of": "2026-09-08",
"evidence_label": "Source candidate · not reviewed",
"note": "Full public directory statement. Source lane: backlog; source resolution: open. Source verification claims are attributed only; no local Lean verification has be... | {"created_at":"2026-08-17T20:02:20+00:00","display_standard":{"editorial_review":"submitter_supplied","formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":9752,"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": "2026-09-08T03:32:52.587Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"b2da1f2d-7a9e-4e64-8d48-3744faa12891","format_issues":null,"number":3892} |
1.0.0 | problem:theoremdb-p9754 | problem | theoremdb-p9754 | Asymptotic growth of the largest $k$-term progression-free subset of $\{1,\dots,N\}$ | {
"main": "math",
"sub": "number theory",
"slug": "number-theory",
"name": "Number theory",
"original_discipline": "mathematics"
} | [] | [
{
"code": "11",
"basis": "community-map",
"evidence": "Inferred from community field: number-theory"
}
] | For a positive integer $k$, let $r_k(N)$ denote the maximum cardinality of a subset $A \subseteq \{1, 2, \dots, N\}$ that contains no non-trivial $k$-term arithmetic progression, meaning no $k$ distinct elements of the form $a, a+d, a+2d, \dots, a+(k-1)d$ with common difference $d > 0$. Determine a function $f_k: \math... | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P9754",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P9754",
"citation": null,
"doi": null,
"arxiv_id": null,
"license": "Selected packets label their content CC0-1.0; original references retain their own terms",
"notes": "Selected problem... | theorem erdos_142 (k : ℕ) : (fun N => (r k N : ℝ)) =Θ[atTop] (answer(sorry) : ℕ → ℝ) | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem erdos_142 (k : ℕ) : (fun N => (r k N : ℝ)) =Θ[atTop] (answer(sorry) : ℕ → ℝ)",
"target": null,
"source_url": "https://theoremdb.org/statements/P9754",
"format": "declaration-or-fragment",
"sha256": "9d93d70e1a8688c5b279ef9f8fe93d554abc0848a9426d254a9d... | [] | TheoremDB P9754 · Candidate awaiting source review | null | Source-reported assessment, retrieved 8 September 2026. Frontier Science has not independently reviewed this status.
Open. No signed Lean verification is attached to this problem or its selected resolution.
This record is in the source backlog; its novelty and unresolved status have not passed source review. | null | published | incremental | hard | [
"source-candidate",
"theoremdb"
] | null | null | null | null | [
{
"kind": "website",
"citation": "TheoremDB P9754",
"url": "https://theoremdb.org/statements/P9754",
"note": null,
"sort_order": 0
},
{
"kind": "formalization",
"citation": "Source formal statement (not locally verified)",
"url": "https://theoremdb.org/statements/P9754",
"not... | {
"source_record": "P9754",
"imported_at": "2026-09-08",
"status_as_of": "2026-09-08",
"evidence_label": "Source candidate · not reviewed",
"note": "Full public directory statement. Source lane: backlog; source resolution: open. Source verification claims are attributed only; no local Lean verification has be... | {"created_at":"2026-08-17T20:02:23+00:00","display_standard":{"editorial_review":"submitter_supplied","formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":9754,"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": "2026-09-08T03:32:50.183Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"304b5ffa-7a50-4d97-8f0d-8884d60a92ec","format_issues":null,"number":3368} |
1.0.0 | problem:theoremdb-p9756 | problem | theoremdb-p9756 | Upper bound for the cofactor threshold | {
"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 $h(n)$ denote the cofactor threshold, defined as the largest integer $m$ such that for every set $A$ of $n$ non-negative integers, the set $$\left\{ \frac{a}{(a,b)} : a,b \in A \right\}$$ has cardinality at least $m$, where $(a,b)$ denotes the greatest common divisor of $a$ and $b$. Prov... | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P9756",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P9756",
"citation": null,
"doi": null,
"arxiv_id": null,
"license": "Selected packets label their content CC0-1.0; original references retain their own terms",
"notes": "Selected problem... | theorem erdos_539.variants.isBigO_sq :
(fun n ↦ (cofactorThreshold n : ℝ)) =O[atTop] fun n ↦ √n | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem erdos_539.variants.isBigO_sq :\n (fun n ↦ (cofactorThreshold n : ℝ)) =O[atTop] fun n ↦ √n",
"target": null,
"source_url": "https://theoremdb.org/statements/P9756",
"format": "declaration-or-fragment",
"sha256": "b3f8da0dc2686f568e63045ea629e7cfdb80... | [] | TheoremDB P9756 · Candidate awaiting source review | null | Source-reported assessment, retrieved 8 September 2026. Frontier Science has not independently reviewed this status.
Open. No signed Lean verification is attached to this problem or its selected resolution.
This record is in the source backlog; its novelty and unresolved status have not passed source review. | null | published | incremental | hard | [
"source-candidate",
"theoremdb"
] | null | null | null | null | [
{
"kind": "website",
"citation": "TheoremDB P9756",
"url": "https://theoremdb.org/statements/P9756",
"note": null,
"sort_order": 0
},
{
"kind": "formalization",
"citation": "Source formal statement (not locally verified)",
"url": "https://theoremdb.org/statements/P9756",
"not... | {
"source_record": "P9756",
"imported_at": "2026-09-08",
"status_as_of": "2026-09-08",
"evidence_label": "Source candidate · not reviewed",
"note": "Full public directory statement. Source lane: backlog; source resolution: open. Source verification claims are attributed only; no local Lean verification has be... | {"created_at":"2026-08-17T20:02:23+00:00","display_standard":{"editorial_review":"submitter_supplied","formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":9756,"public_status":{"evidence":{"hasFormalization":false,"hasInformalProof":false,"hasRecordedWork":false,"solutionRecord":null},"lean":{... | [
"formalization-not-compiled-during-export",
"current-status-unassessed",
"source-full-text-not-present"
] | [] | {
"created_at": "2026-09-08T03:32:15.935Z",
"updated_at": "2026-09-08T03:32:15.935Z",
"published_at": "2026-09-08T03:33:07.328Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"bedd3ec2-98f2-4b13-83b7-56ec4a052496","format_issues":null,"number":6936} |
1.0.0 | problem:theoremdb-p9758 | problem | theoremdb-p9758 | Green's Sieve Problem for 1000 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 $N$ be a positive integer. Let $p_1 < p_2 < \cdots < p_{1000}$ be primes such that $p_{1000}^{10} < N^9$. For each $i \in \{1, 2, \ldots, 1000\}$, let $A_i$ be a subset of $\mathbb{Z}/p_i\mathbb{Z}$ with $|A_i| = \lfloor p_i/2 \rfloor$. Define the remaining set $\mathcal{R}$ to be those integers $x$ with $1 \leq x ... | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P9758",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P9758",
"citation": null,
"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_44 :
answer(sorry) ↔ ∀ (N : ℕ) (p : Fin 1000 → ℕ) (A : (i : Fin 1000) → Finset (ZMod (p i))),
let remaining := (Finset.Icc 1 N).filter (fun x => ∀ i, (x : ZMod (p i)) ∉ A i)
(∀ i, (p i).Prime) →
StrictMono p →
(p 999) ^ 10 < N ^ 9 →
(∀ i, (A i).card = (p i) / 2) →
1... | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem green_44 :\n answer(sorry) ↔ ∀ (N : ℕ) (p : Fin 1000 → ℕ) (A : (i : Fin 1000) → Finset (ZMod (p i))),\n let remaining := (Finset.Icc 1 N).filter (fun x => ∀ i, (x : ZMod (p i)) ∉ A i)\n (∀ i, (p i).Prime) →\n StrictMono p →\n (p 999) ^ 10 < N ^... | [] | TheoremDB P9758 · Candidate awaiting source review | null | Source-reported assessment, retrieved 8 September 2026. Frontier Science has not independently reviewed this status.
Open. No signed Lean verification is attached to this problem or its selected resolution.
This record is in the source backlog; its novelty and unresolved status have not passed source review. | null | published | incremental | hard | [
"source-candidate",
"theoremdb"
] | null | null | null | null | [
{
"kind": "website",
"citation": "TheoremDB P9758",
"url": "https://theoremdb.org/statements/P9758",
"note": null,
"sort_order": 0
},
{
"kind": "formalization",
"citation": "Source formal statement (not locally verified)",
"url": "https://theoremdb.org/statements/P9758",
"not... | {
"source_record": "P9758",
"imported_at": "2026-09-08",
"status_as_of": "2026-09-08",
"evidence_label": "Source candidate · not reviewed",
"note": "Full public directory statement. Source lane: backlog; source resolution: open. Source verification claims are attributed only; no local Lean verification has be... | {"created_at":"2026-08-17T20:02:23+00:00","display_standard":{"editorial_review":"submitter_supplied","formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":9758,"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": "2026-09-08T03:32:59.475Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"12305cb4-bd4a-4828-98f4-6f65d42c776f","format_issues":null,"number":5403} |
1.0.0 | problem:theoremdb-p9760 | problem | theoremdb-p9760 | Weak Grimm Conjecture | {
"main": "math",
"sub": "number theory",
"slug": "number-theory",
"name": "Number theory",
"original_discipline": "mathematics"
} | [] | [
{
"code": "11",
"basis": "community-map",
"evidence": "Inferred from community field: number-theory"
}
] | Let $n$ and $k$ be positive integers such that each of the $k$ consecutive integers $n, n+1, \dots, n+k-1$ is composite. Must the product $n(n+1)\cdots(n+k-1)$ have at least $k$ distinct prime divisors? Equivalently, do there exist $k$ distinct primes $p_0, p_1, \dots, p_{k-1}$ such that each $p_i$ divides at least one... | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P9760",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P9760",
"citation": null,
"doi": null,
"arxiv_id": null,
"license": "Selected packets label their content CC0-1.0; original references retain their own terms",
"notes": "Selected problem... | theorem grimm_conjecture_weak (n k : ℕ) (hn : 1 ≤ n) (hk : 1 ≤ k)
(h : ∀ i : Fin k, (n + i).Composite) :
∃ ps : Fin k ↪ ℕ, ∀ i : Fin k, (ps i).Prime ∧ ∃ j : Fin k, ps i ∣ (n + j) | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem grimm_conjecture_weak (n k : ℕ) (hn : 1 ≤ n) (hk : 1 ≤ k)\n (h : ∀ i : Fin k, (n + i).Composite) :\n ∃ ps : Fin k ↪ ℕ, ∀ i : Fin k, (ps i).Prime ∧ ∃ j : Fin k, ps i ∣ (n + j)",
"target": null,
"source_url": "https://theoremdb.org/statements/P9760",
... | [] | TheoremDB P9760 · Candidate awaiting source review | null | Source-reported assessment, retrieved 8 September 2026. Frontier Science has not independently reviewed this status.
Open. No signed Lean verification is attached to this problem or its selected resolution.
This record is in the source backlog; its novelty and unresolved status have not passed source review. | null | published | incremental | hard | [
"source-candidate",
"theoremdb"
] | null | null | null | null | [
{
"kind": "website",
"citation": "TheoremDB P9760",
"url": "https://theoremdb.org/statements/P9760",
"note": null,
"sort_order": 0
},
{
"kind": "formalization",
"citation": "Source formal statement (not locally verified)",
"url": "https://theoremdb.org/statements/P9760",
"not... | {
"source_record": "P9760",
"imported_at": "2026-09-08",
"status_as_of": "2026-09-08",
"evidence_label": "Source candidate · not reviewed",
"note": "Full public directory statement. Source lane: backlog; source resolution: open. Source verification claims are attributed only; no local Lean verification has be... | {"created_at":"2026-08-17T20:02:24+00:00","display_standard":{"editorial_review":"submitter_supplied","formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":9760,"public_status":{"evidence":{"hasFormalization":false,"hasInformalProof":false,"hasRecordedWork":false,"solutionRecord":null},"lean":{... | [
"formalization-not-compiled-during-export",
"current-status-unassessed",
"source-full-text-not-present"
] | [] | {
"created_at": "2026-09-08T03:32:15.935Z",
"updated_at": "2026-09-08T03:32:15.935Z",
"published_at": "2026-09-08T03:33:07.455Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"2d8d44e9-fd06-42e1-aa94-2db28d40b2a8","format_issues":null,"number":6963} |
1.0.0 | problem:theoremdb-p9762 | problem | theoremdb-p9762 | Linear upper bound for unit distances in convex polygons | {
"main": "math",
"sub": "geometry and topology",
"slug": "geometry-and-topology",
"name": "Geometry and topology",
"original_discipline": "mathematics"
} | [] | [] | For each integer $n \geq 3$, let $U_c(n)$ denote the maximum number of unit-distance pairs among all sets of $n$ points in $\mathbb{R}^2$ that form the vertices of a convex polygon. Here, a unit-distance pair means a pair of points at Euclidean distance exactly $1$. Does there exist a constant $C > 0$ such that $U_c(n)... | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P9762",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P9762",
"citation": null,
"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_96 :
answer(sorry) ↔ (fun n => (maxConvexUnitDistances n : ℝ)) =O[atTop] fun n => (n : ℝ) | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem erdos_96 :\n answer(sorry) ↔ (fun n => (maxConvexUnitDistances n : ℝ)) =O[atTop] fun n => (n : ℝ)",
"target": null,
"source_url": "https://theoremdb.org/statements/P9762",
"format": "declaration-or-fragment",
"sha256": "dfb373ff622967ea3af0b2f1a186... | [] | TheoremDB P9762 · Candidate awaiting source review | null | Source-reported assessment, retrieved 8 September 2026. Frontier Science has not independently reviewed this status.
Open. No signed Lean verification is attached to this problem or its selected resolution.
This record is in the source backlog; its novelty and unresolved status have not passed source review. | null | published | incremental | hard | [
"source-candidate",
"theoremdb"
] | null | null | null | null | [
{
"kind": "website",
"citation": "TheoremDB P9762",
"url": "https://theoremdb.org/statements/P9762",
"note": null,
"sort_order": 0
},
{
"kind": "formalization",
"citation": "Source formal statement (not locally verified)",
"url": "https://theoremdb.org/statements/P9762",
"not... | {
"source_record": "P9762",
"imported_at": "2026-09-08",
"status_as_of": "2026-09-08",
"evidence_label": "Source candidate · not reviewed",
"note": "Full public directory statement. Source lane: backlog; source resolution: open. Source verification claims are attributed only; no local Lean verification has be... | {"created_at":"2026-08-17T20:02:26+00:00","display_standard":{"editorial_review":"submitter_supplied","formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":9762,"public_status":{"evidence":{"hasFormalization":false,"hasInformalProof":false,"hasRecordedWork":false,"solutionRecord":null},"lean":{... | [
"formalization-contains-placeholder",
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"created_at": "2026-09-08T03:32:15.935Z",
"updated_at": "2026-09-08T03:32:15.935Z",
"published_at": "2026-09-08T03:33:03.255Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"264b8ec1-591e-4ec5-bc85-2b636428d708","format_issues":null,"number":6056} |
1.0.0 | problem:theoremdb-p9764 | problem | theoremdb-p9764 | Non-existence of real monochromatic quantum graphs with six vertices and five colours | {
"main": "math",
"sub": "combinatorics",
"slug": "combinatorics",
"name": "Combinatorics",
"original_discipline": "mathematics"
} | [] | [
{
"code": "05",
"basis": "community-map",
"evidence": "Inferred from community field: combinatorics"
}
] | Let $N$ be a positive even integer and let $D$ be a positive integer. An edge of the complete graph $K_N$ is an unordered pair of distinct vertices, identified with the set $\{0, 1, \ldots, N-1\}$. A $D$-colouring of an edge is an unordered pair of (not necessarily distinct) colours from $\{0, 1, \ldots, D-1\}$. A weig... | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P9764",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P9764",
"citation": null,
"doi": null,
"arxiv_id": null,
"license": "Selected packets label their content CC0-1.0; original references retain their own terms",
"notes": "Selected problem... | theorem eqSystem6_no_solution_d5_real :
answer(sorry) ↔
¬ ∃ W : WeightsN 6 5 ℝ, EqSystemN 6 5 W | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem eqSystem6_no_solution_d5_real :\n answer(sorry) ↔\n ¬ ∃ W : WeightsN 6 5 ℝ, EqSystemN 6 5 W",
"target": null,
"source_url": "https://theoremdb.org/statements/P9764",
"format": "declaration-or-fragment",
"sha256": "328e344628d02546a1142a9f70e2f... | [] | TheoremDB P9764 · Candidate awaiting source review | null | Source-reported assessment, retrieved 8 September 2026. Frontier Science has not independently reviewed this status.
Open. No signed Lean verification is attached to this problem or its selected resolution.
This record is in the source backlog; its novelty and unresolved status have not passed source review. | null | published | incremental | hard | [
"source-candidate",
"theoremdb"
] | null | null | null | null | [
{
"kind": "website",
"citation": "TheoremDB P9764",
"url": "https://theoremdb.org/statements/P9764",
"note": null,
"sort_order": 0
},
{
"kind": "formalization",
"citation": "Source formal statement (not locally verified)",
"url": "https://theoremdb.org/statements/P9764",
"not... | {
"source_record": "P9764",
"imported_at": "2026-09-08",
"status_as_of": "2026-09-08",
"evidence_label": "Source candidate · not reviewed",
"note": "Full public directory statement. Source lane: backlog; source resolution: open. Source verification claims are attributed only; no local Lean verification has be... | {"created_at":"2026-08-17T20:02:27+00:00","display_standard":{"editorial_review":"submitter_supplied","formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":9764,"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": "2026-09-08T03:33:04.420Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"c1a9b616-3a28-4ba6-9ff0-dbf699182890","format_issues":null,"number":6306} |
1.0.0 | problem:theoremdb-p9766 | problem | theoremdb-p9766 | Infinitude of odd floor-powers of 3/2 | {
"main": "math",
"sub": "number theory",
"slug": "number-theory",
"name": "Number theory",
"original_discipline": "mathematics"
} | [] | [
{
"code": "11",
"basis": "community-map",
"evidence": "Inferred from community field: number-theory"
}
] | For each positive integer $n$, consider the integer $\lfloor (3/2)^n \rfloor$, the greatest integer not exceeding $(3/2)^n$. Determine whether the set of positive integers $n$ for which $\lfloor (3/2)^n \rfloor$ is odd is infinite. | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P9766",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P9766",
"citation": null,
"doi": null,
"arxiv_id": null,
"license": "Selected packets label their content CC0-1.0; original references retain their own terms",
"notes": "Selected problem... | theorem erdos_349.variants.floor_3_halves_odd :
answer(sorry) ↔ {n | Odd ⌊(3/2 : ℝ) ^ n⌋}.Infinite | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem erdos_349.variants.floor_3_halves_odd :\n answer(sorry) ↔ {n | Odd ⌊(3/2 : ℝ) ^ n⌋}.Infinite",
"target": null,
"source_url": "https://theoremdb.org/statements/P9766",
"format": "declaration-or-fragment",
"sha256": "e189663720ecd92460f267bf3cf41cf6d... | [] | TheoremDB P9766 · Candidate awaiting source review | null | Source-reported assessment, retrieved 8 September 2026. Frontier Science has not independently reviewed this status.
Open. No signed Lean verification is attached to this problem or its selected resolution.
This record is in the source backlog; its novelty and unresolved status have not passed source review. | null | published | incremental | hard | [
"source-candidate",
"theoremdb"
] | null | null | null | null | [
{
"kind": "website",
"citation": "TheoremDB P9766",
"url": "https://theoremdb.org/statements/P9766",
"note": null,
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},
{
"kind": "formalization",
"citation": "Source formal statement (not locally verified)",
"url": "https://theoremdb.org/statements/P9766",
"not... | {
"source_record": "P9766",
"imported_at": "2026-09-08",
"status_as_of": "2026-09-08",
"evidence_label": "Source candidate · not reviewed",
"note": "Full public directory statement. Source lane: backlog; source resolution: open. Source verification claims are attributed only; no local Lean verification has be... | {"created_at":"2026-08-17T20:02:27+00:00","display_standard":{"editorial_review":"submitter_supplied","formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":9766,"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": "2026-09-08T03:33:01.181Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"fb639cbd-2ba4-476c-a8ec-ef7b875c0453","format_issues":null,"number":5675} |
1.0.0 | problem:theoremdb-p9768 | problem | theoremdb-p9768 | Lower bound for the two-color van der Waerden number W(3, 29) | {
"main": "math",
"sub": "combinatorics",
"slug": "combinatorics",
"name": "Combinatorics",
"original_discipline": "mathematics"
} | [] | [
{
"code": "05",
"basis": "community-map",
"evidence": "Inferred from community field: combinatorics"
}
] | For positive integers $k$ and $r$, the two-color van der Waerden number $W(k, r)$ is defined as the least positive integer $N$ such that every coloring of $\{1, 2, \ldots, N\}$ with two colors (say red and blue) contains either a monochromatic arithmetic progression of length $k$ in the first color or a monochromatic a... | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P9768",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P9768",
"citation": null,
"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_29_lower : answer(sorry) ↔ W 3 29 ≥ 868 | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem W_3_29_lower : answer(sorry) ↔ W 3 29 ≥ 868",
"target": null,
"source_url": "https://theoremdb.org/statements/P9768",
"format": "declaration-or-fragment",
"sha256": "caa765849eb5981b7eb3f4c616845648e30c6962f4017336c5883f1b98e205af",
"license": "Se... | [] | TheoremDB P9768 · Candidate awaiting source review | null | Source-reported assessment, retrieved 8 September 2026. Frontier Science has not independently reviewed this status.
Open. No signed Lean verification is attached to this problem or its selected resolution.
This record is in the source backlog; its novelty and unresolved status have not passed source review. | null | published | incremental | hard | [
"source-candidate",
"theoremdb"
] | null | null | null | null | [
{
"kind": "website",
"citation": "TheoremDB P9768",
"url": "https://theoremdb.org/statements/P9768",
"note": null,
"sort_order": 0
},
{
"kind": "formalization",
"citation": "Source formal statement (not locally verified)",
"url": "https://theoremdb.org/statements/P9768",
"not... | {
"source_record": "P9768",
"imported_at": "2026-09-08",
"status_as_of": "2026-09-08",
"evidence_label": "Source candidate · not reviewed",
"note": "Full public directory statement. Source lane: backlog; source resolution: open. Source verification claims are attributed only; no local Lean verification has be... | {"created_at":"2026-08-17T20:02:28+00:00","display_standard":{"formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":9768,"public_status":{"evidence":{"hasFormalization":false,"hasInformalProof":false,"hasRecordedWork":false,"solutionRecord":null},"lean":{"kernelAccepted":false,"label":"","reaso... | [
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] | [] | {
"created_at": "2026-09-08T03:32:15.935Z",
"updated_at": "2026-09-08T03:32:15.935Z",
"published_at": "2026-09-08T03:33:03.523Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"6164224e-64c6-454e-b7b1-07b35738e3c3","format_issues":null,"number":6113} |
1.0.0 | problem:theoremdb-p9770 | problem | theoremdb-p9770 | Asymptotic upper bound for gaps between consecutive squarefree numbers | {
"main": "math",
"sub": "number theory",
"slug": "number-theory",
"name": "Number theory",
"original_discipline": "mathematics"
} | [] | [
{
"code": "11",
"basis": "community-map",
"evidence": "Inferred from community field: number-theory"
}
] | Let $s_1 < s_2 < \dots$ denote the sequence of squarefree numbers, where a positive integer is called squarefree if it is not divisible by the square of any prime. Does there exist a function $c : \mathbb{N} \to \mathbb{R}$ such that $c(n) = o(1)$ as $n \to \infty$ and, for all sufficiently large $n$, the inequality
$... | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P9770",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P9770",
"citation": null,
"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_208.parts.ii : answer(sorry) ↔ ∃ (c : ℕ → ℝ), (c =o[atTop] (1 : ℕ → ℝ)) ∧ ∀ᶠ n in atTop,
s (n + 1) - s n ≤ (1 + (c n)) * (π^2 / 6) * log (s n) / log (log (s n)) | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem erdos_208.parts.ii : answer(sorry) ↔ ∃ (c : ℕ → ℝ), (c =o[atTop] (1 : ℕ → ℝ)) ∧ ∀ᶠ n in atTop,\n s (n + 1) - s n ≤ (1 + (c n)) * (π^2 / 6) * log (s n) / log (log (s n))",
"target": null,
"source_url": "https://theoremdb.org/statements/P9770",
"format... | [] | TheoremDB P9770 · Candidate awaiting source review | null | Source-reported assessment, retrieved 8 September 2026. Frontier Science has not independently reviewed this status.
Open. No signed Lean verification is attached to this problem or its selected resolution.
This record is in the source backlog; its novelty and unresolved status have not passed source review. | null | published | incremental | hard | [
"source-candidate",
"theoremdb"
] | null | null | null | null | [
{
"kind": "website",
"citation": "TheoremDB P9770",
"url": "https://theoremdb.org/statements/P9770",
"note": null,
"sort_order": 0
},
{
"kind": "formalization",
"citation": "Source formal statement (not locally verified)",
"url": "https://theoremdb.org/statements/P9770",
"not... | {
"source_record": "P9770",
"imported_at": "2026-09-08",
"status_as_of": "2026-09-08",
"evidence_label": "Source candidate · not reviewed",
"note": "Full public directory statement. Source lane: backlog; source resolution: open. Source verification claims are attributed only; no local Lean verification has be... | {"created_at":"2026-08-17T20:02:28+00:00","display_standard":{"editorial_review":"submitter_supplied","formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":9770,"public_status":{"evidence":{"hasFormalization":false,"hasInformalProof":false,"hasRecordedWork":false,"solutionRecord":null},"lean":{... | [
"formalization-contains-placeholder",
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"current-status-unassessed",
"source-full-text-not-present"
] | [] | {
"created_at": "2026-09-08T03:32:15.935Z",
"updated_at": "2026-09-08T03:32:15.935Z",
"published_at": "2026-09-08T03:32:50.414Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"a2f512e3-1240-4309-a4bc-d77945d18acb","format_issues":null,"number":3415} |
1.0.0 | problem:theoremdb-p9772 | problem | theoremdb-p9772 | Pyber's conjecture on abelian subgroups of finite $p$-groups | {
"main": "math",
"sub": "algebra",
"slug": "algebra",
"name": "Algebra",
"original_discipline": "mathematics"
} | [] | [] | Let $p$ be a prime number, let $G$ be a finite $p$-group, and let $k$ be a natural number. Suppose that every abelian normal subgroup of $G$ has order at most $p^k$. Does every abelian subgroup of $G$ have order at most $p^{2k}$? | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P9772",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P9772",
"citation": null,
"doi": null,
"arxiv_id": null,
"license": "Selected packets label their content CC0-1.0; original references retain their own terms",
"notes": "Selected problem... | theorem kourovka.«20.76» : answer(sorry) ↔
∀ᵉ (p : ℕ) (hp : p.Prime) (G : Type) (_ : Group G) (hg : IsPGroup p G) (_ : Finite G) (k : ℕ)
(h : ∀ H: Subgroup G, H.Normal ∧ IsMulCommutative H → Nat.card H ≤ p ^ k),
(∀ H : Subgroup G, IsMulCommutative H → Nat.card H ≤ p ^ (2 * k)) | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem kourovka.«20.76» : answer(sorry) ↔\n ∀ᵉ (p : ℕ) (hp : p.Prime) (G : Type) (_ : Group G) (hg : IsPGroup p G) (_ : Finite G) (k : ℕ)\n (h : ∀ H: Subgroup G, H.Normal ∧ IsMulCommutative H → Nat.card H ≤ p ^ k),\n (∀ H : Subgroup G, IsMulCommutative H → Nat.card... | [] | TheoremDB P9772 · Candidate awaiting source review | null | Source-reported assessment, retrieved 8 September 2026. Frontier Science has not independently reviewed this status.
Open. No signed Lean verification is attached to this problem or its selected resolution.
This record is in the source backlog; its novelty and unresolved status have not passed source review. | null | published | incremental | hard | [
"source-candidate",
"theoremdb"
] | null | null | null | null | [
{
"kind": "website",
"citation": "TheoremDB P9772",
"url": "https://theoremdb.org/statements/P9772",
"note": null,
"sort_order": 0
},
{
"kind": "formalization",
"citation": "Source formal statement (not locally verified)",
"url": "https://theoremdb.org/statements/P9772",
"not... | {
"source_record": "P9772",
"imported_at": "2026-09-08",
"status_as_of": "2026-09-08",
"evidence_label": "Source candidate · not reviewed",
"note": "Full public directory statement. Source lane: backlog; source resolution: open. Source verification claims are attributed only; no local Lean verification has be... | {"created_at":"2026-08-17T20:02:31+00:00","display_standard":{"editorial_review":"submitter_supplied","formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":9772,"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": "2026-09-08T03:33:05.338Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"37edbf47-be76-499b-86d5-bd0ce5920e87","format_issues":null,"number":6506} |
1.0.0 | problem:theoremdb-p9774 | problem | theoremdb-p9774 | Non-existence of monochromatic quantum graphs with eight vertices, three colors, and 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 $N$ be a positive integer. A perfect matching on $N$ vertices is a partition of the vertex set into $N/2$ disjoint edges. Let $D$ be a positive integer. An edge coloring with $D$ colors assigns to each unordered pair of distinct vertices a color from $\{1, \dots, D\}$. A vertex coloring $\iota$ assigns to each vert... | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P9774",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P9774",
"citation": null,
"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_int :
answer(sorry) ↔
¬ ∃ W : WeightsN 8 3 ℤ, EqSystemN 8 3 W | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem eqSystem8_no_solution_d3_int :\n answer(sorry) ↔\n ¬ ∃ W : WeightsN 8 3 ℤ, EqSystemN 8 3 W",
"target": null,
"source_url": "https://theoremdb.org/statements/P9774",
"format": "declaration-or-fragment",
"sha256": "a77249bf4269fc38dd059d20640bec... | [] | TheoremDB P9774 · Candidate awaiting source review | null | Source-reported assessment, retrieved 8 September 2026. Frontier Science has not independently reviewed this status.
Open. No signed Lean verification is attached to this problem or its selected resolution.
This record is in the source backlog; its novelty and unresolved status have not passed source review. | null | published | incremental | hard | [
"source-candidate",
"theoremdb"
] | null | null | null | null | [
{
"kind": "website",
"citation": "TheoremDB P9774",
"url": "https://theoremdb.org/statements/P9774",
"note": null,
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},
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"citation": "Source formal statement (not locally verified)",
"url": "https://theoremdb.org/statements/P9774",
"not... | {
"source_record": "P9774",
"imported_at": "2026-09-08",
"status_as_of": "2026-09-08",
"evidence_label": "Source candidate · not reviewed",
"note": "Full public directory statement. Source lane: backlog; source resolution: open. Source verification claims are attributed only; no local Lean verification has be... | {"created_at":"2026-08-17T20:02:32+00:00","display_standard":{"editorial_review":"submitter_supplied","formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":9774,"public_status":{"evidence":{"hasFormalization":false,"hasInformalProof":false,"hasRecordedWork":false,"solutionRecord":null},"lean":{... | [
"formalization-contains-placeholder",
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"current-status-unassessed",
"source-full-text-not-present"
] | [] | {
"created_at": "2026-09-08T03:32:15.935Z",
"updated_at": "2026-09-08T03:32:15.935Z",
"published_at": "2026-09-08T03:33:04.310Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"a4541e1f-836c-420d-8dc1-5fe407c3c08d","format_issues":null,"number":6284} |
1.0.0 | problem:theoremdb-p9776 | problem | theoremdb-p9776 | Deaconescu's Prime Between Consecutive Primorial Gaps | {
"main": "math",
"sub": "number theory",
"slug": "number-theory",
"name": "Number theory",
"original_discipline": "mathematics"
} | [] | [
{
"code": "11",
"basis": "community-map",
"evidence": "Inferred from community field: number-theory"
}
] | For each integer $n \geq 1$, let $P_n$ denote the product of the first $n+1$ prime numbers, that is, $P_n = p_1 p_2 \cdots p_{n+1}$ where $p_i$ denotes the $i$-th prime number. Determine whether there always exists a prime number $p$ such that $p_{n+1} < p < P_n$ and $P_n + p$ is also prime. | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P9776",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P9776",
"citation": null,
"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_779 (n : ℕ) (hn : n ≥ 1): let P := ∏ i ∈ range (n + 1), nth Nat.Prime i
∃ p, p.Prime ∧ (P + p).Prime ∧ nth Nat.Prime n < p ∧ p < P | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem erdos_779 (n : ℕ) (hn : n ≥ 1): let P := ∏ i ∈ range (n + 1), nth Nat.Prime i\n ∃ p, p.Prime ∧ (P + p).Prime ∧ nth Nat.Prime n < p ∧ p < P",
"target": null,
"source_url": "https://theoremdb.org/statements/P9776",
"format": "declaration-or-fragment",
... | [] | TheoremDB P9776 · Candidate awaiting source review | null | Source-reported assessment, retrieved 8 September 2026. Frontier Science has not independently reviewed this status.
Open. No signed Lean verification is attached to this problem or its selected resolution.
This record is in the source backlog; its novelty and unresolved status have not passed source review. | null | published | incremental | hard | [
"source-candidate",
"theoremdb"
] | null | null | null | null | [
{
"kind": "website",
"citation": "TheoremDB P9776",
"url": "https://theoremdb.org/statements/P9776",
"note": null,
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},
{
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"citation": "Source formal statement (not locally verified)",
"url": "https://theoremdb.org/statements/P9776",
"not... | {
"source_record": "P9776",
"imported_at": "2026-09-08",
"status_as_of": "2026-09-08",
"evidence_label": "Source candidate · not reviewed",
"note": "Full public directory statement. Source lane: backlog; source resolution: open. Source verification claims are attributed only; no local Lean verification has be... | {"created_at":"2026-08-17T20:02:33+00:00","display_standard":{"editorial_review":"submitter_supplied","formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":9776,"public_status":{"evidence":{"hasFormalization":false,"hasInformalProof":false,"hasRecordedWork":false,"solutionRecord":null},"lean":{... | [
"formalization-not-compiled-during-export",
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"created_at": "2026-09-08T03:32:15.935Z",
"updated_at": "2026-09-08T03:32:15.935Z",
"published_at": "2026-09-08T03:32:52.920Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"83c4f900-fb87-4203-aed2-e1ca2b68e154","format_issues":null,"number":3964} |
1.0.0 | problem:theoremdb-p9778 | problem | theoremdb-p9778 | Infinitely many primes with prime floor multiples for irrational multipliers | {
"main": "math",
"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 $\alpha$, define the prime set of $\alpha$ to be the set of primes $p$ such that $\lfloor \alpha p \rfloor$ is also prime, where $\lfloor x \rfloor$ denotes the greatest integer less than or equal to $x$. Determine whether the following holds: for every real number $\alpha > 1$ that is irrational, the... | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P9778",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P9778",
"citation": null,
"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_972 : answer(sorry) ↔ ∀ α > 1, Irrational α → (primeSet α).Infinite | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem erdos_972 : answer(sorry) ↔ ∀ α > 1, Irrational α → (primeSet α).Infinite",
"target": null,
"source_url": "https://theoremdb.org/statements/P9778",
"format": "declaration-or-fragment",
"sha256": "2862d06eeb933b7fd3952ebc9bdc1c672c1965f26a22abf711a9829... | [] | TheoremDB P9778 · Candidate awaiting source review | null | Source-reported assessment, retrieved 8 September 2026. Frontier Science has not independently reviewed this status.
Open. No signed Lean verification is attached to this problem or its selected resolution.
This record is in the source backlog; its novelty and unresolved status have not passed source review. | null | published | incremental | hard | [
"source-candidate",
"theoremdb"
] | null | null | null | null | [
{
"kind": "website",
"citation": "TheoremDB P9778",
"url": "https://theoremdb.org/statements/P9778",
"note": null,
"sort_order": 0
},
{
"kind": "formalization",
"citation": "Source formal statement (not locally verified)",
"url": "https://theoremdb.org/statements/P9778",
"not... | {
"source_record": "P9778",
"imported_at": "2026-09-08",
"status_as_of": "2026-09-08",
"evidence_label": "Source candidate · not reviewed",
"note": "Full public directory statement. Source lane: backlog; source resolution: open. Source verification claims are attributed only; no local Lean verification has be... | {"created_at":"2026-08-17T20:02:34+00:00","display_standard":{"editorial_review":"submitter_supplied","formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":9778,"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": "2026-09-08T03:33:01.015Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"96ddbfde-a6a7-42ed-b85b-9428fca6a50e","format_issues":null,"number":5638} |
1.0.0 | problem:theoremdb-p9780 | problem | theoremdb-p9780 | Erdős Distinct Distances Lower Bound Conjecture | {
"main": "math",
"sub": "geometry and topology",
"slug": "geometry-and-topology",
"name": "Geometry and topology",
"original_discipline": "mathematics"
} | [] | [] | For a positive integer $n$, let $f(n)$ denote the minimum, over all sets of $n$ distinct points in the Euclidean plane $\mathbb{R}^2$, of the number of distinct pairwise distances determined by those points. Determine whether there exists a positive constant $C$ such that for all sufficiently large $n$, every set of $n... | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P9780",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P9780",
"citation": null,
"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_89 :
(fun (n : ℕ) => n/(n : ℝ).log.sqrt) =O[atTop] (fun n => (minimalDistinctDistances n : ℝ)) | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem erdos_89 :\n (fun (n : ℕ) => n/(n : ℝ).log.sqrt) =O[atTop] (fun n => (minimalDistinctDistances n : ℝ))",
"target": null,
"source_url": "https://theoremdb.org/statements/P9780",
"format": "declaration-or-fragment",
"sha256": "0e2e07466a332ddfce64314... | [] | TheoremDB P9780 · Candidate awaiting source review | null | Source-reported assessment, retrieved 8 September 2026. Frontier Science has not independently reviewed this status.
Open. No signed Lean verification is attached to this problem or its selected resolution.
This record is in the source backlog; its novelty and unresolved status have not passed source review. | null | published | incremental | hard | [
"source-candidate",
"theoremdb"
] | null | null | null | null | [
{
"kind": "website",
"citation": "TheoremDB P9780",
"url": "https://theoremdb.org/statements/P9780",
"note": null,
"sort_order": 0
},
{
"kind": "formalization",
"citation": "Source formal statement (not locally verified)",
"url": "https://theoremdb.org/statements/P9780",
"not... | {
"source_record": "P9780",
"imported_at": "2026-09-08",
"status_as_of": "2026-09-08",
"evidence_label": "Source candidate · not reviewed",
"note": "Full public directory statement. Source lane: backlog; source resolution: open. Source verification claims are attributed only; no local Lean verification has be... | {"created_at":"2026-08-17T20:02:36+00:00","display_standard":{"formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":9780,"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": "2026-09-08T03:32:54.812Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"d2c9e8fd-fafa-4425-a206-6e77e39e8a79","format_issues":null,"number":4383} |
1.0.0 | problem:theoremdb-p9782 | problem | theoremdb-p9782 | The Casas-Alvero Conjecture | {
"main": "math",
"sub": "algebra",
"slug": "algebra",
"name": "Algebra",
"original_discipline": "mathematics"
} | [] | [] | Let $K$ be a field of characteristic zero and let $P \in K[X]$ be a monic polynomial. For a non-negative integer $i$, the $i$th Hasse derivative of $P$, denoted $P^{[i]}$, is the polynomial defined by the identity $P(X+Y) = \sum_{i=0}^{\deg P} P^{[i]}(X) Y^i$ in $K[X,Y]$. We say that $P$ has the Casas-Alvero property i... | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P9782",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P9782",
"citation": null,
"doi": null,
"arxiv_id": null,
"license": "Selected packets label their content CC0-1.0; original references retain their own terms",
"notes": "Selected problem... | theorem casas_alvero_conjecture (hP' : HasCasasAlveroProp P) :
∃ α : K, P = (X - C α) ^ P.natDegree | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem casas_alvero_conjecture (hP' : HasCasasAlveroProp P) :\n ∃ α : K, P = (X - C α) ^ P.natDegree",
"target": null,
"source_url": "https://theoremdb.org/statements/P9782",
"format": "declaration-or-fragment",
"sha256": "23fdc10442df48d1a4eefc500f824bd1... | [] | TheoremDB P9782 · Candidate awaiting source review | null | Source-reported assessment, retrieved 8 September 2026. Frontier Science has not independently reviewed this status.
Open. No signed Lean verification is attached to this problem or its selected resolution.
This record is in the source backlog; its novelty and unresolved status have not passed source review. | null | published | incremental | hard | [
"source-candidate",
"theoremdb"
] | null | null | null | null | [
{
"kind": "website",
"citation": "TheoremDB P9782",
"url": "https://theoremdb.org/statements/P9782",
"note": null,
"sort_order": 0
},
{
"kind": "formalization",
"citation": "Source formal statement (not locally verified)",
"url": "https://theoremdb.org/statements/P9782",
"not... | {
"source_record": "P9782",
"imported_at": "2026-09-08",
"status_as_of": "2026-09-08",
"evidence_label": "Source candidate · not reviewed",
"note": "Full public directory statement. Source lane: backlog; source resolution: open. Source verification claims are attributed only; no local Lean verification has be... | {"created_at":"2026-08-17T20:02:36+00:00","display_standard":{"formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":9782,"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": "2026-09-08T03:33:06.407Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"e65e3bbc-df6c-4a84-a7d7-cb81961c5954","format_issues":null,"number":6736} |
1.0.0 | problem:theoremdb-p9784 | problem | theoremdb-p9784 | Erdős's problem on integers with many prime factors in short intervals | {
"main": "math",
"sub": "number theory",
"slug": "number-theory",
"name": "Number theory",
"original_discipline": "mathematics"
} | [] | [
{
"code": "11",
"basis": "community-map",
"evidence": "Inferred from community field: number-theory"
}
] | Let $p_1 = 2 < p_2 < p_3 < \cdots$ denote the sequence of prime numbers in increasing order. For a positive integer $m$, let $\omega(m)$ denote the number of distinct prime factors of $m$. For integers $k \geq 2$ and $n$, let $[n, n + p_1 p_2 \cdots p_k)$ denote the half-open interval of integers $m$ satisfying $n \leq... | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P9784",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P9784",
"citation": null,
"doi": null,
"arxiv_id": null,
"license": "Selected packets label their content CC0-1.0; original references retain their own terms",
"notes": "Selected problem... | theorem erdos_891 :
answer(sorry) ↔
∀ k ≥ 2, ∀ᶠ n in atTop,
∃ m ∈ Ico n (n + ∏ i ∈ range k, i.nth Nat.Prime), k < ω m | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem erdos_891 :\n answer(sorry) ↔\n ∀ k ≥ 2, ∀ᶠ n in atTop,\n ∃ m ∈ Ico n (n + ∏ i ∈ range k, i.nth Nat.Prime), k < ω m",
"target": null,
"source_url": "https://theoremdb.org/statements/P9784",
"format": "declaration-or-fragment",
"sha256": "... | [] | TheoremDB P9784 · Candidate awaiting source review | null | Source-reported assessment, retrieved 8 September 2026. Frontier Science has not independently reviewed this status.
Open. No signed Lean verification is attached to this problem or its selected resolution.
This record is in the source backlog; its novelty and unresolved status have not passed source review. | null | published | incremental | hard | [
"source-candidate",
"theoremdb"
] | null | null | null | null | [
{
"kind": "website",
"citation": "TheoremDB P9784",
"url": "https://theoremdb.org/statements/P9784",
"note": null,
"sort_order": 0
},
{
"kind": "formalization",
"citation": "Source formal statement (not locally verified)",
"url": "https://theoremdb.org/statements/P9784",
"not... | {
"source_record": "P9784",
"imported_at": "2026-09-08",
"status_as_of": "2026-09-08",
"evidence_label": "Source candidate · not reviewed",
"note": "Full public directory statement. Source lane: backlog; source resolution: open. Source verification claims are attributed only; no local Lean verification has be... | {"created_at":"2026-08-17T20:02:36+00:00","display_standard":{"editorial_review":"submitter_supplied","formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":9784,"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": "2026-09-08T03:32:56.487Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"aa908bd4-eb56-47f0-bf2e-d92644fc0d23","format_issues":null,"number":4751} |
1.0.0 | problem:theoremdb-p9786 | problem | theoremdb-p9786 | Completeness of an interleaved floor-multiples sequence with ratio 2 | {
"main": "math",
"sub": "number theory",
"slug": "number-theory",
"name": "Number theory",
"original_discipline": "mathematics"
} | [] | [
{
"code": "11",
"basis": "community-map",
"evidence": "Inferred from community field: number-theory"
}
] | For positive real numbers $\alpha$ and $\beta$ such that $\alpha/\beta$ is irrational, consider the sequence obtained by interleaving the sequences $\lfloor 2^n \alpha \rfloor$ and $\lfloor 2^n \beta \rfloor$ for $n = 0, 1, 2, \ldots$. That is, the sequence is
$$
\lfloor \alpha \rfloor, \lfloor \beta \rfloor, \lfloor ... | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P9786",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P9786",
"citation": null,
"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_354.parts.i : answer(sorry) ↔ ∀ᵉ (α > 0) (β > 0), Irrational (α / β) →
IsAddCompleteNatSeq' (FloorMultiples.interleave α β 2) | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem erdos_354.parts.i : answer(sorry) ↔ ∀ᵉ (α > 0) (β > 0), Irrational (α / β) →\n IsAddCompleteNatSeq' (FloorMultiples.interleave α β 2)",
"target": null,
"source_url": "https://theoremdb.org/statements/P9786",
"format": "declaration-or-fragment",
"sh... | [] | TheoremDB P9786 · Candidate awaiting source review | null | Source-reported assessment, retrieved 8 September 2026. Frontier Science has not independently reviewed this status.
Open. No signed Lean verification is attached to this problem or its selected resolution.
This record is in the source backlog; its novelty and unresolved status have not passed source review. | null | published | incremental | hard | [
"source-candidate",
"theoremdb"
] | null | null | null | null | [
{
"kind": "website",
"citation": "TheoremDB P9786",
"url": "https://theoremdb.org/statements/P9786",
"note": null,
"sort_order": 0
},
{
"kind": "formalization",
"citation": "Source formal statement (not locally verified)",
"url": "https://theoremdb.org/statements/P9786",
"not... | {
"source_record": "P9786",
"imported_at": "2026-09-08",
"status_as_of": "2026-09-08",
"evidence_label": "Source candidate · not reviewed",
"note": "Full public directory statement. Source lane: backlog; source resolution: open. Source verification claims are attributed only; no local Lean verification has be... | {"created_at":"2026-08-17T20:02:39+00:00","display_standard":{"editorial_review":"submitter_supplied","formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":9786,"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": "2026-09-08T03:32:52.489Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"5c656061-e8d7-43ab-a1ed-a92ea814fd93","format_issues":null,"number":3871} |
1.0.0 | problem:theoremdb-p9788 | problem | theoremdb-p9788 | The Riemann Hypothesis | {
"main": "math",
"sub": "number theory",
"slug": "number-theory",
"name": "Number theory",
"original_discipline": "mathematics"
} | [] | [
{
"code": "11",
"basis": "community-map",
"evidence": "Inferred from community field: number-theory"
}
] | The Riemann zeta function $\zeta(s)$ is initially defined for complex numbers $s$ with real part greater than $1$ by the absolutely convergent series $\zeta(s) = \sum_{n=1}^{\infty} n^{-s}$, and admits a unique meromorphic continuation to the entire complex plane with a single simple pole at $s = 1$. The trivial zeros ... | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P9788",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P9788",
"citation": null,
"doi": null,
"arxiv_id": null,
"license": "Selected packets label their content CC0-1.0; original references retain their own terms",
"notes": "Selected problem... | theorem riemannHypothesis : RiemannHypothesis | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem riemannHypothesis : RiemannHypothesis",
"target": null,
"source_url": "https://theoremdb.org/statements/P9788",
"format": "declaration-or-fragment",
"sha256": "1527a6aa9b0567bdecadef77760aaf4a4a082d867de76b800865a7261ea32c33",
"license": "Selected... | [] | TheoremDB P9788 · Candidate awaiting source review | null | Source-reported assessment, retrieved 8 September 2026. Frontier Science has not independently reviewed this status.
Open. No signed Lean verification is attached to this problem or its selected resolution.
This record is in the source backlog; its novelty and unresolved status have not passed source review. | null | published | incremental | hard | [
"source-candidate",
"theoremdb"
] | null | null | null | null | [
{
"kind": "website",
"citation": "TheoremDB P9788",
"url": "https://theoremdb.org/statements/P9788",
"note": null,
"sort_order": 0
},
{
"kind": "formalization",
"citation": "Source formal statement (not locally verified)",
"url": "https://theoremdb.org/statements/P9788",
"not... | {
"source_record": "P9788",
"imported_at": "2026-09-08",
"status_as_of": "2026-09-08",
"evidence_label": "Source candidate · not reviewed",
"note": "Full public directory statement. Source lane: backlog; source resolution: open. Source verification claims are attributed only; no local Lean verification has be... | {"created_at":"2026-08-17T20:02:40+00:00","display_standard":{"editorial_review":"submitter_supplied","formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":9788,"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": "2026-09-08T03:33:06.745Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"38f04e74-9449-43f9-a79d-cad38914d157","format_issues":null,"number":6808} |
1.0.0 | problem:theoremdb-p9790 | problem | theoremdb-p9790 | Generalized Riemann Hypothesis for Dirichlet L-functions | {
"main": "math",
"sub": "number theory",
"slug": "number-theory",
"name": "Number theory",
"original_discipline": "mathematics"
} | [] | [
{
"code": "11",
"basis": "community-map",
"evidence": "Inferred from community field: number-theory"
}
] | Let $q$ be a positive integer and let $\chi$ be a primitive Dirichlet character modulo $q$. The Dirichlet $L$-function associated to $\chi$ is denoted $L(\chi, s)$. The trivial zeros of $L(\chi, s)$ are defined as follows: if $q = 1$ (so $\chi = 1$ and $L(\chi, s)$ is the Riemann zeta function $\zeta(s)$), the trivial ... | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P9790",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P9790",
"citation": null,
"doi": null,
"arxiv_id": null,
"license": "Selected packets label their content CC0-1.0; original references retain their own terms",
"notes": "Selected problem... | theorem generalized_riemann_hypothesis (q : ℕ) [NeZero q] (χ : DirichletCharacter ℂ q)
(hχ : χ.IsPrimitive) (s : ℂ) (hs : χ.LFunction s = 0)
(hs_nontrivial : s ∉ Int.cast '' trivialZeros χ) :
s.re = 1 / 2 | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem generalized_riemann_hypothesis (q : ℕ) [NeZero q] (χ : DirichletCharacter ℂ q)\n (hχ : χ.IsPrimitive) (s : ℂ) (hs : χ.LFunction s = 0)\n (hs_nontrivial : s ∉ Int.cast '' trivialZeros χ) :\n s.re = 1 / 2",
"target": null,
"source_url": "https://theore... | [] | TheoremDB P9790 · Candidate awaiting source review | null | Source-reported assessment, retrieved 8 September 2026. Frontier Science has not independently reviewed this status.
Open. No signed Lean verification is attached to this problem or its selected resolution.
This record is in the source backlog; its novelty and unresolved status have not passed source review. | null | published | incremental | hard | [
"source-candidate",
"theoremdb"
] | null | null | null | null | [
{
"kind": "website",
"citation": "TheoremDB P9790",
"url": "https://theoremdb.org/statements/P9790",
"note": null,
"sort_order": 0
},
{
"kind": "formalization",
"citation": "Source formal statement (not locally verified)",
"url": "https://theoremdb.org/statements/P9790",
"not... | {
"source_record": "P9790",
"imported_at": "2026-09-08",
"status_as_of": "2026-09-08",
"evidence_label": "Source candidate · not reviewed",
"note": "Full public directory statement. Source lane: backlog; source resolution: open. Source verification claims are attributed only; no local Lean verification has be... | {"created_at":"2026-08-17T20:02:40+00:00","display_standard":{"editorial_review":"submitter_supplied","formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":9790,"public_status":{"evidence":{"hasFormalization":false,"hasInformalProof":false,"hasRecordedWork":false,"solutionRecord":null},"lean":{... | [
"formalization-not-compiled-during-export",
"current-status-unassessed",
"source-full-text-not-present"
] | [] | {
"created_at": "2026-09-08T03:32:15.935Z",
"updated_at": "2026-09-08T03:32:15.935Z",
"published_at": "2026-09-08T03:32:59.295Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"94b33a7a-62eb-456a-9a67-ae15d33040ff","format_issues":null,"number":5364} |
1.0.0 | problem:theoremdb-p9792 | problem | theoremdb-p9792 | The Collatz 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"
}
] | Define the Collatz step as the function on natural numbers that sends an even number $n$ to $n/2$ and an odd number $n$ to $3n+1$. For any positive integer $n$, define the Collatz sequence starting from $n$ to be the sequence whose first term is $n$ and whose each subsequent term is obtained by applying the Collatz ste... | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P9792",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P9792",
"citation": null,
"doi": null,
"arxiv_id": null,
"license": "Selected packets label their content CC0-1.0; original references retain their own terms",
"notes": "Selected problem... | theorem collatz_conjecture (n : ℕ) (hn : n > 0) : ∃ m, collatzStep^[m] n = 1 | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem collatz_conjecture (n : ℕ) (hn : n > 0) : ∃ m, collatzStep^[m] n = 1",
"target": null,
"source_url": "https://theoremdb.org/statements/P9792",
"format": "declaration-or-fragment",
"sha256": "7d2b60c6ac95f1ba7feeef456162fddf07fbcf7f5a156c321995fba78f86... | [] | TheoremDB P9792 · Candidate awaiting source review | null | Source-reported assessment, retrieved 8 September 2026. Frontier Science has not independently reviewed this status.
Open. No signed Lean verification is attached to this problem or its selected resolution.
This record is in the source backlog; its novelty and unresolved status have not passed source review. | null | published | incremental | hard | [
"source-candidate",
"theoremdb"
] | null | null | null | null | [
{
"kind": "website",
"citation": "TheoremDB P9792",
"url": "https://theoremdb.org/statements/P9792",
"note": null,
"sort_order": 0
},
{
"kind": "formalization",
"citation": "Source formal statement (not locally verified)",
"url": "https://theoremdb.org/statements/P9792",
"not... | {
"source_record": "P9792",
"imported_at": "2026-09-08",
"status_as_of": "2026-09-08",
"evidence_label": "Source candidate · not reviewed",
"note": "Full public directory statement. Source lane: backlog; source resolution: open. Source verification claims are attributed only; no local Lean verification has be... | {"created_at":"2026-08-17T20:02:41+00:00","display_standard":{"formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":9792,"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": "2026-09-08T03:33:06.412Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"ee8fbacb-78e4-4433-92f5-c2f5c69e21ae","format_issues":null,"number":6737} |
1.0.0 | problem:theoremdb-p9794 | problem | theoremdb-p9794 | The Pierce–Birkhoff conjecture on piecewise-polynomial functions | {
"main": "math",
"sub": "algebra",
"slug": "algebra",
"name": "Algebra",
"original_discipline": "mathematics"
} | [] | [] | Let $n$ be a natural number and let $f : \mathbb{R}^n \to \mathbb{R}$ be a function. A subset $S \subseteq \mathbb{R}^n$ is called semi-algebraic if it can be expressed as a finite union of sets defined by multivariate polynomial equations and strict inequalities; that is, there exist finite index sets and families of ... | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P9794",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P9794",
"citation": null,
"doi": null,
"arxiv_id": null,
"license": "Selected packets label their content CC0-1.0; original references retain their own terms",
"notes": "Selected problem... | theorem pierce_birkhoff_conjecture {n : ℕ} (f : (Fin n → ℝ) → ℝ)
(hf : IsPiecewiseMvPolynomial f) :
∃ (ι κ : Type) (g : ι → κ → MvPolynomial (Fin n) ℝ), Finite ι ∧ Finite κ ∧
∀ x, f x = ⨆ i, ⨅ j, MvPolynomial.eval x (g i j) | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem pierce_birkhoff_conjecture {n : ℕ} (f : (Fin n → ℝ) → ℝ)\n (hf : IsPiecewiseMvPolynomial f) :\n ∃ (ι κ : Type) (g : ι → κ → MvPolynomial (Fin n) ℝ), Finite ι ∧ Finite κ ∧\n ∀ x, f x = ⨆ i, ⨅ j, MvPolynomial.eval x (g i j)",
"target": null,
"source_... | [] | TheoremDB P9794 · Candidate awaiting source review | null | Source-reported assessment, retrieved 8 September 2026. Frontier Science has not independently reviewed this status.
Open. No signed Lean verification is attached to this problem or its selected resolution.
This record is in the source backlog; its novelty and unresolved status have not passed source review. | null | published | incremental | hard | [
"source-candidate",
"theoremdb"
] | null | null | null | null | [
{
"kind": "website",
"citation": "TheoremDB P9794",
"url": "https://theoremdb.org/statements/P9794",
"note": null,
"sort_order": 0
},
{
"kind": "formalization",
"citation": "Source formal statement (not locally verified)",
"url": "https://theoremdb.org/statements/P9794",
"not... | {
"source_record": "P9794",
"imported_at": "2026-09-08",
"status_as_of": "2026-09-08",
"evidence_label": "Source candidate · not reviewed",
"note": "Full public directory statement. Source lane: backlog; source resolution: open. Source verification claims are attributed only; no local Lean verification has be... | {"created_at":"2026-08-17T20:02:43+00:00","display_standard":{"formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":9794,"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": "2026-09-08T03:33:06.703Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"01dd6524-3865-4639-bca2-7604d5a0eb88","format_issues":null,"number":6800} |
1.0.0 | problem:theoremdb-p9796 | problem | theoremdb-p9796 | Infinitude of real quadratic fields with class number one | {
"main": "math",
"sub": "number theory",
"slug": "number-theory",
"name": "Number theory",
"original_discipline": "mathematics"
} | [] | [
{
"code": "11",
"basis": "community-map",
"evidence": "Inferred from community field: number-theory"
}
] | For a squarefree integer $d > 1$, let $\mathbb{Q}(\sqrt{d})$ denote the real quadratic field obtained by adjoining a root of the polynomial $X^2 - d$ to $\mathbb{Q}$. The class number of a number field is the order of its ideal class group, which measures the failure of unique factorization in the ring of integers. We ... | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P9796",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P9796",
"citation": null,
"doi": null,
"arxiv_id": null,
"license": "Selected packets label their content CC0-1.0; original references retain their own terms",
"notes": "Selected problem... | theorem class_number_problem :
{ d : ℤ | Squarefree d ∧ d > 1 ∧ IsClassNumberOne d }.Infinite | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem class_number_problem :\n { d : ℤ | Squarefree d ∧ d > 1 ∧ IsClassNumberOne d }.Infinite",
"target": null,
"source_url": "https://theoremdb.org/statements/P9796",
"format": "declaration-or-fragment",
"sha256": "e23f653f2c96f2e4d00d1c9ae0dc2e49f6a1c8... | [] | TheoremDB P9796 · Candidate awaiting source review | null | Source-reported assessment, retrieved 8 September 2026. Frontier Science has not independently reviewed this status.
Open. No signed Lean verification is attached to this problem or its selected resolution.
This record is in the source backlog; its novelty and unresolved status have not passed source review. | null | published | incremental | hard | [
"source-candidate",
"theoremdb"
] | null | null | null | null | [
{
"kind": "website",
"citation": "TheoremDB P9796",
"url": "https://theoremdb.org/statements/P9796",
"note": null,
"sort_order": 0
},
{
"kind": "formalization",
"citation": "Source formal statement (not locally verified)",
"url": "https://theoremdb.org/statements/P9796",
"not... | {
"source_record": "P9796",
"imported_at": "2026-09-08",
"status_as_of": "2026-09-08",
"evidence_label": "Source candidate · not reviewed",
"note": "Full public directory statement. Source lane: backlog; source resolution: open. Source verification claims are attributed only; no local Lean verification has be... | {"created_at":"2026-08-17T20:02:44+00:00","display_standard":{"editorial_review":"submitter_supplied","formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":9796,"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": "2026-09-08T03:33:01.248Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"f741602c-e0ff-4cb4-8fd4-de21deddf066","format_issues":null,"number":5689} |
1.0.0 | problem:theoremdb-p9798 | problem | theoremdb-p9798 | Asymptotic growth of minimum distinct distances for point sets in general position | {
"main": "math",
"sub": "geometry and topology",
"slug": "geometry-and-topology",
"name": "Geometry and topology",
"original_discipline": "mathematics"
} | [] | [] | For each natural number $n$, let $h(n)$ denote the minimum number of distinct distances determined by any set of $n$ points in the Euclidean plane $\mathbb{R}^2$ that is in general position, meaning no three points are collinear and no four points are concyclic (lie on a common circle). Does the ratio $h(n)/n$ tend to ... | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P9798",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P9798",
"citation": null,
"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_98 :
answer(sorry) ↔ Tendsto (fun n : ℕ ↦ ((h n : ℝ) / (n : ℝ))) atTop atTop | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem erdos_98 :\n answer(sorry) ↔ Tendsto (fun n : ℕ ↦ ((h n : ℝ) / (n : ℝ))) atTop atTop",
"target": null,
"source_url": "https://theoremdb.org/statements/P9798",
"format": "declaration-or-fragment",
"sha256": "4fc5f2a3986878f8b6bd2f9fe199643860f06034c... | [] | TheoremDB P9798 · Candidate awaiting source review | null | Source-reported assessment, retrieved 8 September 2026. Frontier Science has not independently reviewed this status.
Open. No signed Lean verification is attached to this problem or its selected resolution.
This record is in the source backlog; its novelty and unresolved status have not passed source review. | null | published | incremental | hard | [
"source-candidate",
"theoremdb"
] | null | null | null | null | [
{
"kind": "website",
"citation": "TheoremDB P9798",
"url": "https://theoremdb.org/statements/P9798",
"note": null,
"sort_order": 0
},
{
"kind": "formalization",
"citation": "Source formal statement (not locally verified)",
"url": "https://theoremdb.org/statements/P9798",
"not... | {
"source_record": "P9798",
"imported_at": "2026-09-08",
"status_as_of": "2026-09-08",
"evidence_label": "Source candidate · not reviewed",
"note": "Full public directory statement. Source lane: backlog; source resolution: open. Source verification claims are attributed only; no local Lean verification has be... | {"created_at":"2026-08-17T20:02:45+00:00","display_standard":{"editorial_review":"submitter_supplied","formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":9798,"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": "2026-09-08T03:32:50.149Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"621922cc-7cd9-483a-a61e-562f783d4179","format_issues":null,"number":3361} |
1.0.0 | problem:theoremdb-p9800 | problem | theoremdb-p9800 | Breakdown of periodic Navier–Stokes solutions in three dimensions | {
"main": "math",
"sub": "partial differential equations",
"slug": "partial-differential-equations",
"name": "Partial differential equations",
"original_discipline": "mathematics"
} | [] | [
{
"code": "35",
"basis": "community-map",
"evidence": "Inferred from community field: partial-differential-equations"
}
] | Let $\nu > 0$ be a positive real number. Does there exist a smooth, divergence-free, 1-periodic initial velocity field $u_0 : \mathbb{R}^3 \to \mathbb{R}^3$ and a smooth external force $f : \mathbb{R}^3 \times [0,\infty) \to \mathbb{R}^3$ that is 1-periodic in space and whose derivatives decay faster than any polynomia... | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P9800",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P9800",
"citation": null,
"doi": null,
"arxiv_id": null,
"license": "Selected packets label their content CC0-1.0; original references retain their own terms",
"notes": "Selected problem... | theorem navier_stokes_breakdown_periodic (nu : ℝ) (hnu : nu > 0) :
∃ (u₀ : ℝ³ → ℝ³) (f : ℝ³ → ℝ → ℝ³),
InitialVelocityConditionPeriodic u₀ ∧ ForceConditionPeriodic f ∧
¬ (∃ v p, NavierStokesExistenceAndSmoothnessPeriodic nu u₀ f v p) | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem navier_stokes_breakdown_periodic (nu : ℝ) (hnu : nu > 0) :\n ∃ (u₀ : ℝ³ → ℝ³) (f : ℝ³ → ℝ → ℝ³),\n InitialVelocityConditionPeriodic u₀ ∧ ForceConditionPeriodic f ∧\n ¬ (∃ v p, NavierStokesExistenceAndSmoothnessPeriodic nu u₀ f v p)",
"target": null,
... | [] | TheoremDB P9800 · Candidate awaiting source review | null | Source-reported assessment, retrieved 8 September 2026. Frontier Science has not independently reviewed this status.
Open. No signed Lean verification is attached to this problem or its selected resolution.
This record is in the source backlog; its novelty and unresolved status have not passed source review. | null | published | incremental | hard | [
"source-candidate",
"theoremdb"
] | null | null | null | null | [
{
"kind": "website",
"citation": "TheoremDB P9800",
"url": "https://theoremdb.org/statements/P9800",
"note": null,
"sort_order": 0
},
{
"kind": "formalization",
"citation": "Source formal statement (not locally verified)",
"url": "https://theoremdb.org/statements/P9800",
"not... | {
"source_record": "P9800",
"imported_at": "2026-09-08",
"status_as_of": "2026-09-08",
"evidence_label": "Source candidate · not reviewed",
"note": "Full public directory statement. Source lane: backlog; source resolution: open. Source verification claims are attributed only; no local Lean verification has be... | {"created_at":"2026-08-17T20:02:46+00:00","display_standard":{"editorial_review":"submitter_supplied","formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":9800,"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": "2026-09-08T03:32:50.762Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"2f188ea6-5a89-49aa-9f67-547102e76009","format_issues":null,"number":3487} |
1.0.0 | problem:theoremdb-p9802 | problem | theoremdb-p9802 | Surányi's Conjecture on Factorial Equations | {
"main": "math",
"sub": "number theory",
"slug": "number-theory",
"name": "Number theory",
"original_discipline": "mathematics"
} | [] | [
{
"code": "11",
"basis": "community-map",
"evidence": "Inferred from community field: number-theory"
}
] | For integers $n$, $a$, and $b$ all greater than $1$, consider the equation $n! = a! \cdot b!$ subject to the conditions $b \leq a$ and $a + 1 \neq n$. Determine whether the only solution to this equation is $(n, a, b) = (10, 7, 6)$. | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P9802",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P9802",
"citation": null,
"doi": null,
"arxiv_id": null,
"license": "Selected packets label their content CC0-1.0; original references retain their own terms",
"notes": "Selected problem... | theorem erdos_373.variants.suranyi :
{(n, a, b) : ℕ × ℕ × ℕ | n ! = a ! * b ! ∧ 1 < n ∧ 1 < a ∧ 1 < b ∧ b ≤ a ∧ a + 1 ≠ n}
= {(10, 7, 6)} | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem erdos_373.variants.suranyi :\n {(n, a, b) : ℕ × ℕ × ℕ | n ! = a ! * b ! ∧ 1 < n ∧ 1 < a ∧ 1 < b ∧ b ≤ a ∧ a + 1 ≠ n}\n = {(10, 7, 6)}",
"target": null,
"source_url": "https://theoremdb.org/statements/P9802",
"format": "declaration-or-fragment",
... | [] | TheoremDB P9802 · Candidate awaiting source review | null | Source-reported assessment, retrieved 8 September 2026. Frontier Science has not independently reviewed this status.
Open. No signed Lean verification is attached to this problem or its selected resolution.
This record is in the source backlog; its novelty and unresolved status have not passed source review. | null | published | incremental | hard | [
"source-candidate",
"theoremdb"
] | null | null | null | null | [
{
"kind": "website",
"citation": "TheoremDB P9802",
"url": "https://theoremdb.org/statements/P9802",
"note": null,
"sort_order": 0
},
{
"kind": "formalization",
"citation": "Source formal statement (not locally verified)",
"url": "https://theoremdb.org/statements/P9802",
"not... | {
"source_record": "P9802",
"imported_at": "2026-09-08",
"status_as_of": "2026-09-08",
"evidence_label": "Source candidate · not reviewed",
"note": "Full public directory statement. Source lane: backlog; source resolution: open. Source verification claims are attributed only; no local Lean verification has be... | {"created_at":"2026-08-17T20:02:47+00:00","display_standard":{"formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":9802,"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": "2026-09-08T03:33:06.242Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"b0a16f48-9d85-4833-8ecc-81b09b3dc3d1","format_issues":null,"number":6701} |
1.0.0 | problem:theoremdb-p9804 | problem | theoremdb-p9804 | Non-existence of monochromatic quantum graphs with ten vertices and six colours | {
"main": "math",
"sub": "combinatorics",
"slug": "combinatorics",
"name": "Combinatorics",
"original_discipline": "mathematics"
} | [] | [
{
"code": "05",
"basis": "community-map",
"evidence": "Inferred from community field: combinatorics"
}
] | Let $N \geq 2$ be an even integer and let $D \geq 1$ be an integer. An edge of the complete graph $K_N$ on vertices $\{1, \dots, N\}$ is a 2-element subset of $\{1, \dots, N\}$. A vertex colouring is a function $\iota : \{1, \dots, N\} \to \{1, \dots, D\}$. A perfect matching of $K_N$ is a set of $N/2$ pairwise disjoin... | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P9804",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P9804",
"citation": null,
"doi": null,
"arxiv_id": null,
"license": "Selected packets label their content CC0-1.0; original references retain their own terms",
"notes": "Selected problem... | theorem eqSystem10_no_solution_d6 :
answer(sorry) ↔
¬ ∃ W : WeightsN 10 6 ℂ, EqSystemN 10 6 W | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem eqSystem10_no_solution_d6 :\n answer(sorry) ↔\n ¬ ∃ W : WeightsN 10 6 ℂ, EqSystemN 10 6 W",
"target": null,
"source_url": "https://theoremdb.org/statements/P9804",
"format": "declaration-or-fragment",
"sha256": "edf4fe7c9d226d4c99af32e8c0eb546... | [] | TheoremDB P9804 · Candidate awaiting source review | null | Source-reported assessment, retrieved 8 September 2026. Frontier Science has not independently reviewed this status.
Open. No signed Lean verification is attached to this problem or its selected resolution.
This record is in 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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"note": null,
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},
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"url": "https://theoremdb.org/statements/P9804",
"not... | {
"source_record": "P9804",
"imported_at": "2026-09-08",
"status_as_of": "2026-09-08",
"evidence_label": "Source candidate · not reviewed",
"note": "Full public directory statement. Source lane: backlog; source resolution: open. Source verification claims are attributed only; no local Lean verification has be... | {"created_at":"2026-08-17T20:02:47+00:00","display_standard":{"editorial_review":"submitter_supplied","formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":9804,"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": "2026-09-08T03:33:04.386Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"38c779ac-a9f1-4df7-9730-73cf0ed6c375","format_issues":null,"number":6299} |
1.0.0 | problem:theoremdb-p9806 | problem | theoremdb-p9806 | Total Domination Number Bound via Havel–Hakimi Zero Step and Minimum Triangle Frequency | {
"main": "math",
"sub": "combinatorics",
"slug": "combinatorics",
"name": "Combinatorics",
"original_discipline": "mathematics"
} | [] | [
{
"code": "05",
"basis": "community-map",
"evidence": "Inferred from community field: combinatorics"
}
] | Let $G$ be a simple connected graph with vertex set $V$ and $n = |V| > 2$. For a vertex $v \in V$, let $T(v)$ denote the number of triangles (3-cliques) containing $v$. Define the minimum triangle count $t_{\min} = \min_{v \in V} T(v)$, and let $\mathrm{freq}(t_{\min})$ be the number of vertices $v \in V$ such that $T(... | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P9806",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P9806",
"citation": null,
"doi": null,
"arxiv_id": null,
"license": "Selected packets label their content CC0-1.0; original references retain their own terms",
"notes": "Selected problem... | theorem conjecture291 (G : SimpleGraph α) [DecidableRel G.Adj] (h : G.Connected)
(hn : 2 < Fintype.card α) :
G.totalDominationNumber ≤
havelHakimiZeroStep G + freqMinTriangles G | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem conjecture291 (G : SimpleGraph α) [DecidableRel G.Adj] (h : G.Connected)\n (hn : 2 < Fintype.card α) :\n G.totalDominationNumber ≤\n havelHakimiZeroStep G + freqMinTriangles G",
"target": null,
"source_url": "https://theoremdb.org/statements/P9806",
... | [] | TheoremDB P9806 · Candidate awaiting source review | null | Source-reported assessment, retrieved 8 September 2026. Frontier Science has not independently reviewed this status.
Open. No signed Lean verification is attached to this problem or its selected resolution.
This record is in the source backlog; its novelty and unresolved status have not passed source review. | null | published | incremental | hard | [
"source-candidate",
"theoremdb"
] | null | null | null | null | [
{
"kind": "website",
"citation": "TheoremDB P9806",
"url": "https://theoremdb.org/statements/P9806",
"note": null,
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},
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"kind": "formalization",
"citation": "Source formal statement (not locally verified)",
"url": "https://theoremdb.org/statements/P9806",
"not... | {
"source_record": "P9806",
"imported_at": "2026-09-08",
"status_as_of": "2026-09-08",
"evidence_label": "Source candidate · not reviewed",
"note": "Full public directory statement. Source lane: backlog; source resolution: open. Source verification claims are attributed only; no local Lean verification has be... | {"created_at":"2026-08-17T20:02:49+00:00","display_standard":{"editorial_review":"submitter_supplied","formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":9806,"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": "2026-09-08T03:33:06.928Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"bf06a3e9-1372-47d6-bd2d-794dcfbf06a5","format_issues":null,"number":6848} |
1.0.0 | problem:theoremdb-p9808 | problem | theoremdb-p9808 | The Maximum Number of Mutually Unbiased Bases in Complex Hilbert Spaces | {
"main": "math",
"sub": "combinatorics",
"slug": "combinatorics",
"name": "Combinatorics",
"original_discipline": "mathematics"
} | [] | [
{
"code": "05",
"basis": "community-map",
"evidence": "Inferred from community field: combinatorics"
}
] | For each integer $d \ge 2$, determine the maximum number of mutually unbiased orthonormal bases that can exist in the complex Hilbert space $\mathbb{C}^d$. Two orthonormal bases $\mathcal{B}_r = \{e_0^{(r)}, \dots, e_{d-1}^{(r)}\}$ and $\mathcal{B}_s = \{e_0^{(s)}, \dots, e_{d-1}^{(s)}\}$ of $\mathbb{C}^d$ are said to ... | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P9808",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P9808",
"citation": null,
"doi": null,
"arxiv_id": null,
"license": "Selected packets label their content CC0-1.0; original references retain their own terms",
"notes": "Selected problem... | theorem mutuallyUnbiasedBases (d : ℕ) (hd : 2 ≤ d) :
IsMaxMUBCount d ((answer(sorry) : ℕ → ℕ) d) | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem mutuallyUnbiasedBases (d : ℕ) (hd : 2 ≤ d) :\n IsMaxMUBCount d ((answer(sorry) : ℕ → ℕ) d)",
"target": null,
"source_url": "https://theoremdb.org/statements/P9808",
"format": "declaration-or-fragment",
"sha256": "c7e14261f4e8ada70f0d2040704ecac608a... | [] | TheoremDB P9808 · Candidate awaiting source review | null | Source-reported assessment, retrieved 8 September 2026. Frontier Science has not independently reviewed this status.
Open. No signed Lean verification is attached to this problem or its selected resolution.
This record is in the source backlog; its novelty and unresolved status have not passed source review. | null | published | incremental | hard | [
"source-candidate",
"theoremdb"
] | null | null | null | null | [
{
"kind": "website",
"citation": "TheoremDB P9808",
"url": "https://theoremdb.org/statements/P9808",
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},
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"kind": "formalization",
"citation": "Source formal statement (not locally verified)",
"url": "https://theoremdb.org/statements/P9808",
"not... | {
"source_record": "P9808",
"imported_at": "2026-09-08",
"status_as_of": "2026-09-08",
"evidence_label": "Source candidate · not reviewed",
"note": "Full public directory statement. Source lane: backlog; source resolution: open. Source verification claims are attributed only; no local Lean verification has be... | {"created_at":"2026-08-17T20:02:51+00:00","display_standard":{"formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":9808,"public_status":{"evidence":{"hasFormalization":false,"hasInformalProof":false,"hasRecordedWork":false,"solutionRecord":null},"lean":{"kernelAccepted":false,"label":"","reaso... | [
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"updated_at": "2026-09-08T03:32:15.935Z",
"published_at": "2026-09-08T03:33:06.644Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"a5b6cb66-067e-4795-b20d-245e100ca991","format_issues":null,"number":6787} |
1.0.0 | problem:theoremdb-p9810 | problem | theoremdb-p9810 | Explicit lower bound for the monotone distinct-sums 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$, let $h(n)$ denote the largest integer $k$ for which there exist integers $1 \le a_1 \le a_2 \le \dots \le a_k \le n$ such that all sums of the form $\sum_{u \le i \le v} a_i$ with $1 \le u \le v \le k$ are distinct. Determine an explicit function $g : \mathbb{N} \to \mathbb{R}$ such that ... | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P9810",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P9810",
"citation": null,
"doi": null,
"arxiv_id": null,
"license": "Selected packets label their content CC0-1.0; original references retain their own terms",
"notes": "Selected problem... | theorem erdos_357.variants.monotone.parts.ii.bigO_version :
(answer(sorry) : ℕ → ℝ) =O[atTop] (fun n ↦ (h n : ℝ)) | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem erdos_357.variants.monotone.parts.ii.bigO_version :\n (answer(sorry) : ℕ → ℝ) =O[atTop] (fun n ↦ (h n : ℝ))",
"target": null,
"source_url": "https://theoremdb.org/statements/P9810",
"format": "declaration-or-fragment",
"sha256": "93b3a93c37c07f6115... | [] | TheoremDB P9810 · Candidate awaiting source review | null | Source-reported assessment, retrieved 8 September 2026. Frontier Science has not independently reviewed this status.
Open. No signed Lean verification is attached to this problem or its selected resolution.
This record is in the source backlog; its novelty and unresolved status have not passed source review. | null | published | incremental | hard | [
"source-candidate",
"theoremdb"
] | null | null | null | null | [
{
"kind": "website",
"citation": "TheoremDB P9810",
"url": "https://theoremdb.org/statements/P9810",
"note": null,
"sort_order": 0
},
{
"kind": "formalization",
"citation": "Source formal statement (not locally verified)",
"url": "https://theoremdb.org/statements/P9810",
"not... | {
"source_record": "P9810",
"imported_at": "2026-09-08",
"status_as_of": "2026-09-08",
"evidence_label": "Source candidate · not reviewed",
"note": "Full public directory statement. Source lane: backlog; source resolution: open. Source verification claims are attributed only; no local Lean verification has be... | {"created_at":"2026-08-17T20:02:52+00:00","display_standard":{"editorial_review":"submitter_supplied","formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":9810,"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": "2026-09-08T03:32:58.902Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"abb6e37c-b78b-4de8-b33a-ddd86cdb6bb4","format_issues":null,"number":5279} |
1.0.0 | problem:theoremdb-p9812 | problem | theoremdb-p9812 | Asymptotic upper bound for the monotone distinct-sums function | {
"main": "math",
"sub": "number theory",
"slug": "number-theory",
"name": "Number theory",
"original_discipline": "mathematics"
} | [] | [
{
"code": "11",
"basis": "community-map",
"evidence": "Inferred from community field: number-theory"
}
] | Let $h(n)$ be the maximum integer $k$ such that there exist integers $1 \le a_1 \le a_2 \le \dots \le a_k \le n$ with the property that all sums of consecutive terms are distinct; that is, for any two intervals of indices $[u,v]$ and $[u',v']$, if $\sum_{i=u}^{v} a_i = \sum_{i=u'}^{v'} a_i$ then the intervals are ident... | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P9812",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P9812",
"citation": null,
"doi": null,
"arxiv_id": null,
"license": "Selected packets label their content CC0-1.0; original references retain their own terms",
"notes": "Selected problem... | theorem erdos_357.variants.monotone.parts.ii.littleO_version_symm :
(fun n ↦ (h n : ℝ)) =o[atTop] (answer(sorry) : ℕ → ℝ) | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem erdos_357.variants.monotone.parts.ii.littleO_version_symm :\n (fun n ↦ (h n : ℝ)) =o[atTop] (answer(sorry) : ℕ → ℝ)",
"target": null,
"source_url": "https://theoremdb.org/statements/P9812",
"format": "declaration-or-fragment",
"sha256": "e06ba0a254... | [] | TheoremDB P9812 · Candidate awaiting source review | null | Source-reported assessment, retrieved 8 September 2026. Frontier Science has not independently reviewed this status.
Open. No signed Lean verification is attached to this problem or its selected resolution.
This record is in the source backlog; its novelty and unresolved status have not passed source review. | null | published | incremental | hard | [
"source-candidate",
"theoremdb"
] | null | null | null | null | [
{
"kind": "website",
"citation": "TheoremDB P9812",
"url": "https://theoremdb.org/statements/P9812",
"note": null,
"sort_order": 0
},
{
"kind": "formalization",
"citation": "Source formal statement (not locally verified)",
"url": "https://theoremdb.org/statements/P9812",
"not... | {
"source_record": "P9812",
"imported_at": "2026-09-08",
"status_as_of": "2026-09-08",
"evidence_label": "Source candidate · not reviewed",
"note": "Full public directory statement. Source lane: backlog; source resolution: open. Source verification claims are attributed only; no local Lean verification has be... | {"created_at":"2026-08-17T20:02:53+00:00","display_standard":{"editorial_review":"submitter_supplied","formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":9812,"public_status":{"evidence":{"hasFormalization":false,"hasInformalProof":false,"hasRecordedWork":false,"solutionRecord":null},"lean":{... | [
"formalization-contains-placeholder",
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"current-status-unassessed",
"source-full-text-not-present"
] | [] | {
"created_at": "2026-09-08T03:32:15.935Z",
"updated_at": "2026-09-08T03:32:15.935Z",
"published_at": "2026-09-08T03:32:50.486Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"45cb4849-cf48-4fa8-9ae0-ffbf6c802675","format_issues":null,"number":3427} |
1.0.0 | problem:theoremdb-p9814 | problem | theoremdb-p9814 | Transcendence of at least one of $2^t$ and $3^t$ for irrational $t$ | {
"main": "math",
"sub": "number theory",
"slug": "number-theory",
"name": "Number theory",
"original_discipline": "mathematics"
} | [] | [
{
"code": "11",
"basis": "community-map",
"evidence": "Inferred from community field: number-theory"
}
] | Let $t$ be an irrational real number. Determine whether at least one of the real numbers $2^t$ and $3^t$ is transcendental over $\mathbb{Q}$. | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P9814",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P9814",
"citation": null,
"doi": null,
"arxiv_id": null,
"license": "Selected packets label their content CC0-1.0; original references retain their own terms",
"notes": "Selected problem... | theorem two_pow_three_pow_transcendental (t : ℝ) (h : Irrational t) :
Transcendental ℚ (2 ^ t : ℝ) ∨ Transcendental ℚ (3 ^ t : ℝ) | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem two_pow_three_pow_transcendental (t : ℝ) (h : Irrational t) :\n Transcendental ℚ (2 ^ t : ℝ) ∨ Transcendental ℚ (3 ^ t : ℝ)",
"target": null,
"source_url": "https://theoremdb.org/statements/P9814",
"format": "declaration-or-fragment",
"sha256": "ed... | [] | TheoremDB P9814 · Candidate awaiting source review | null | Source-reported assessment, retrieved 8 September 2026. Frontier Science has not independently reviewed this status.
Open. No signed Lean verification is attached to this problem or its selected resolution.
This record is in the source backlog; its novelty and unresolved status have not passed source review. | null | published | incremental | hard | [
"source-candidate",
"theoremdb"
] | null | null | null | null | [
{
"kind": "website",
"citation": "TheoremDB P9814",
"url": "https://theoremdb.org/statements/P9814",
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"sort_order": 0
},
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"kind": "formalization",
"citation": "Source formal statement (not locally verified)",
"url": "https://theoremdb.org/statements/P9814",
"not... | {
"source_record": "P9814",
"imported_at": "2026-09-08",
"status_as_of": "2026-09-08",
"evidence_label": "Source candidate · not reviewed",
"note": "Full public directory statement. Source lane: backlog; source resolution: open. Source verification claims are attributed only; no local Lean verification has be... | {"created_at":"2026-08-17T20:02:53+00:00","display_standard":{"editorial_review":"submitter_supplied","formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":9814,"public_status":{"evidence":{"hasFormalization":false,"hasInformalProof":false,"hasRecordedWork":false,"solutionRecord":null},"lean":{... | [
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"current-status-unassessed",
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"created_at": "2026-09-08T03:32:15.935Z",
"updated_at": "2026-09-08T03:32:15.935Z",
"published_at": "2026-09-08T03:33:06.984Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"93f4c2ad-f56c-497b-a25b-94b6255b4e1c","format_issues":null,"number":6860} |
1.0.0 | problem:theoremdb-p9816 | problem | theoremdb-p9816 | Asymptotic bound for the count of elliptic curves over ℚ with rank at least 21 | {
"main": "math",
"sub": "number theory",
"slug": "number-theory",
"name": "Number theory",
"original_discipline": "mathematics"
} | [] | [
{
"code": "11",
"basis": "community-map",
"evidence": "Inferred from community field: number-theory"
}
] | For an elliptic curve $E$ over $\mathbb{Q}$, the rank of $E$ is the free rank of the finitely generated abelian group $E(\mathbb{Q})$ of rational points, which is finite by the Mordell–Weil theorem. Every elliptic curve over $\mathbb{Q}$ is isomorphic to a unique curve in reduced form with Weierstrass equation $y^2 = x... | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P9816",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P9816",
"citation": null,
"doi": null,
"arxiv_id": null,
"license": "Selected packets label their content CC0-1.0; original references retain their own terms",
"notes": "Selected problem... | theorem twentyone_le_rank_height_count_asymptotic :
∃ f : ℕ → ℝ, atTop.Tendsto f (𝓝 0) ∧
∀ H : ℕ, 1 < H → {E ∈ heightLE H | 21 ≤ E.rank}.ncard ≤ (H : ℝ) ^ f H | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem twentyone_le_rank_height_count_asymptotic :\n ∃ f : ℕ → ℝ, atTop.Tendsto f (𝓝 0) ∧\n ∀ H : ℕ, 1 < H → {E ∈ heightLE H | 21 ≤ E.rank}.ncard ≤ (H : ℝ) ^ f H",
"target": null,
"source_url": "https://theoremdb.org/statements/P9816",
"format": "declar... | [] | TheoremDB P9816 · Candidate awaiting source review | null | Source-reported assessment, retrieved 8 September 2026. Frontier Science has not independently reviewed this status.
Open. No signed Lean verification is attached to this problem or its selected resolution.
This record is in the source backlog; its novelty and unresolved status have not passed source review. | null | published | incremental | hard | [
"source-candidate",
"theoremdb"
] | null | null | null | null | [
{
"kind": "website",
"citation": "TheoremDB P9816",
"url": "https://theoremdb.org/statements/P9816",
"note": null,
"sort_order": 0
},
{
"kind": "formalization",
"citation": "Source formal statement (not locally verified)",
"url": "https://theoremdb.org/statements/P9816",
"not... | {
"source_record": "P9816",
"imported_at": "2026-09-08",
"status_as_of": "2026-09-08",
"evidence_label": "Source candidate · not reviewed",
"note": "Full public directory statement. Source lane: backlog; source resolution: open. Source verification claims are attributed only; no local Lean verification has be... | {"created_at":"2026-08-17T20:02:56+00:00","display_standard":{"editorial_review":"submitter_supplied","formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":9816,"public_status":{"evidence":{"hasFormalization":false,"hasInformalProof":false,"hasRecordedWork":false,"solutionRecord":null},"lean":{... | [
"formalization-not-compiled-during-export",
"current-status-unassessed",
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] | [] | {
"created_at": "2026-09-08T03:32:15.935Z",
"updated_at": "2026-09-08T03:32:15.935Z",
"published_at": "2026-09-08T03:32:49.927Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"f0ed20cb-c93e-4145-ae40-f26d68e917d0","format_issues":null,"number":3315} |
1.0.0 | problem:theoremdb-p9818 | problem | theoremdb-p9818 | Non-existence of monochromatic quantum graphs with six vertices and 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$ be a positive even integer and let $D$ be a positive integer. A quantum graph with $N$ vertices and $D$ colours consists of a complete graph on $N$ vertices together with a weight function $W$ assigning to each unordered pair of distinct vertices $\{i,j\}$ and each colour $c \in \{1, \dots, D\}$ an integer weig... | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P9818",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P9818",
"citation": null,
"doi": null,
"arxiv_id": null,
"license": "Selected packets label their content CC0-1.0; original references retain their own terms",
"notes": "Selected problem... | theorem eqSystem6_no_solution_ge3_int :
answer(sorry) ↔
∀ D : Nat, D ≥ 3 →
¬ ∃ W : WeightsN 6 D ℤ, EqSystemN 6 D W | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem eqSystem6_no_solution_ge3_int :\n answer(sorry) ↔\n ∀ D : Nat, D ≥ 3 →\n ¬ ∃ W : WeightsN 6 D ℤ, EqSystemN 6 D W",
"target": null,
"source_url": "https://theoremdb.org/statements/P9818",
"format": "declaration-or-fragment",
"sha256": "e... | [] | TheoremDB P9818 · Candidate awaiting source review | null | Source-reported assessment, retrieved 8 September 2026. Frontier Science has not independently reviewed this status.
Open. No signed Lean verification is attached to this problem or its selected resolution.
This record is in the source backlog; its novelty and unresolved status have not passed source review. | null | published | incremental | hard | [
"source-candidate",
"theoremdb"
] | null | null | null | null | [
{
"kind": "website",
"citation": "TheoremDB P9818",
"url": "https://theoremdb.org/statements/P9818",
"note": null,
"sort_order": 0
},
{
"kind": "formalization",
"citation": "Source formal statement (not locally verified)",
"url": "https://theoremdb.org/statements/P9818",
"not... | {
"source_record": "P9818",
"imported_at": "2026-09-08",
"status_as_of": "2026-09-08",
"evidence_label": "Source candidate · not reviewed",
"note": "Full public directory statement. Source lane: backlog; source resolution: open. Source verification claims are attributed only; no local Lean verification has be... | {"created_at":"2026-08-17T20:02:56+00:00","display_standard":{"editorial_review":"submitter_supplied","formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":9818,"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": "2026-09-08T03:33:04.342Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"fd39f2e8-3cba-4d30-8ff4-fbd3549ddb18","format_issues":null,"number":6290} |
1.0.0 | problem:theoremdb-p9820 | problem | theoremdb-p9820 | Convex decomposability of weak tiling measures into proper tilings | {
"main": "math",
"sub": "analysis",
"slug": "analysis",
"name": "Analysis",
"original_discipline": "mathematics"
} | [] | [] | Let $\Omega \subseteq \mathbb{R}$ be a finite union of intervals and let $\nu$ be a weak tiling measure for $\Omega$. Must $\nu$ be expressible as a convex combination of proper tiling measures? That is, do there exist sets $T_i \subseteq \mathbb{R}$ and coefficients $c_i \geq 0$ for $i \in \mathbb{N}$ with $\sum_{i=0}... | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P9820",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P9820",
"citation": null,
"doi": null,
"arxiv_id": null,
"license": "Selected packets label their content CC0-1.0; original references retain their own terms",
"notes": "Selected problem... | theorem problem_4_3 :
answer(sorry) ↔ ∀ (Ω : Set ℝ) (_ : IsFiniteUnionOfIntervals Ω)
(ν : Measure ℝ) (_ : IsWeakTilingMeasure Ω ν),
∃ (T : ℕ → Set ℝ) (c : ℕ → ℝ≥0), (∀ i, IsProperTiling Ω (T i)) ∧ ∑' i : ℕ, c i = 1 ∧
ν = Measure.sum
(fun i => (c i : ℝ≥0∞) • Measure.sum (fun t : T i => Meas... | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem problem_4_3 :\n answer(sorry) ↔ ∀ (Ω : Set ℝ) (_ : IsFiniteUnionOfIntervals Ω)\n (ν : Measure ℝ) (_ : IsWeakTilingMeasure Ω ν),\n ∃ (T : ℕ → Set ℝ) (c : ℕ → ℝ≥0), (∀ i, IsProperTiling Ω (T i)) ∧ ∑' i : ℕ, c i = 1 ∧\n ν = Measure.sum\n (fun i ... | [] | TheoremDB P9820 · Candidate awaiting source review | null | Source-reported assessment, retrieved 8 September 2026. Frontier Science has not independently reviewed this status.
Open. No signed Lean verification is attached to this problem or its selected resolution.
This record is in the source backlog; its novelty and unresolved status have not passed source review. | null | published | incremental | hard | [
"source-candidate",
"theoremdb"
] | null | null | null | null | [
{
"kind": "website",
"citation": "TheoremDB P9820",
"url": "https://theoremdb.org/statements/P9820",
"note": null,
"sort_order": 0
},
{
"kind": "formalization",
"citation": "Source formal statement (not locally verified)",
"url": "https://theoremdb.org/statements/P9820",
"not... | {
"source_record": "P9820",
"imported_at": "2026-09-08",
"status_as_of": "2026-09-08",
"evidence_label": "Source candidate · not reviewed",
"note": "Full public directory statement. Source lane: backlog; source resolution: open. Source verification claims are attributed only; no local Lean verification has be... | {"created_at":"2026-08-17T20:02:57+00:00","display_standard":{"editorial_review":"submitter_supplied","formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":9820,"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": "2026-09-08T03:32:52.707Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"e2b8c436-06be-4d0c-92e7-ab91e2732837","format_issues":null,"number":3918} |
1.0.0 | problem:theoremdb-p9822 | problem | theoremdb-p9822 | Fusibility of numbers generated by successive fusion with shifted successors | {
"main": "math",
"sub": "combinatorics",
"slug": "combinatorics",
"name": "Combinatorics",
"original_discipline": "mathematics"
} | [] | [
{
"code": "05",
"basis": "community-map",
"evidence": "Inferred from community field: combinatorics"
}
] | A rational number is called fusible if it belongs to the smallest set containing $0$ and closed under the binary operation
$$
a \sim b = \frac{a + b + 1}{2}
$$
whenever $|a - b| < 1$.
Let $x$ and $y$ be fusible numbers with $x < y$, and suppose there is no fusible number strictly between $x$ and $y$. Define $m = y... | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P9822",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P9822",
"citation": null,
"doi": null,
"arxiv_id": null,
"license": "Selected packets label their content CC0-1.0; original references retain their own terms",
"notes": "Selected problem... | theorem conj_7_1 (x y q : ℚ) (n : ℕ) (fus_x : IsFusible x) (fus_y : IsFusible y) (lt : x < y)
(nmem_Ioo : ∀ z, IsFusible z → z ∉ Set.Ioo x y) :
let m := y - x
let ℓ (n : ℕ) := y + 1 - m / 2 ^ n
IsFusible q → q ∈ Set.Ico (ℓ n) (ℓ (n + 1)) → IsFusible (2 * q - 1 - x - (2 - 1 / 2 ^ n) * m) | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem conj_7_1 (x y q : ℚ) (n : ℕ) (fus_x : IsFusible x) (fus_y : IsFusible y) (lt : x < y)\n (nmem_Ioo : ∀ z, IsFusible z → z ∉ Set.Ioo x y) :\n let m := y - x\n let ℓ (n : ℕ) := y + 1 - m / 2 ^ n\n IsFusible q → q ∈ Set.Ico (ℓ n) (ℓ (n + 1)) → IsFusible (2 * ... | [] | TheoremDB P9822 · Candidate awaiting source review | null | Source-reported assessment, retrieved 8 September 2026. Frontier Science has not independently reviewed this status.
Open. No signed Lean verification is attached to this problem or its selected resolution.
This record is in the source backlog; its novelty and unresolved status have not passed source review. | null | published | incremental | hard | [
"source-candidate",
"theoremdb"
] | null | null | null | null | [
{
"kind": "website",
"citation": "TheoremDB P9822",
"url": "https://theoremdb.org/statements/P9822",
"note": null,
"sort_order": 0
},
{
"kind": "formalization",
"citation": "Source formal statement (not locally verified)",
"url": "https://theoremdb.org/statements/P9822",
"not... | {
"source_record": "P9822",
"imported_at": "2026-09-08",
"status_as_of": "2026-09-08",
"evidence_label": "Source candidate · not reviewed",
"note": "Full public directory statement. Source lane: backlog; source resolution: open. Source verification claims are attributed only; no local Lean verification has be... | {"created_at":"2026-08-17T20:02:57+00:00","display_standard":{"editorial_review":"submitter_supplied","formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":9822,"public_status":{"evidence":{"hasFormalization":false,"hasInformalProof":false,"hasRecordedWork":false,"solutionRecord":null},"lean":{... | [
"formalization-not-compiled-during-export",
"current-status-unassessed",
"source-full-text-not-present"
] | [] | {
"created_at": "2026-09-08T03:32:15.935Z",
"updated_at": "2026-09-08T03:32:15.935Z",
"published_at": "2026-09-08T03:32:59.260Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"a6614e88-6458-45bf-bb50-3d1b13151b5c","format_issues":null,"number":5358} |
1.0.0 | problem:theoremdb-p9824 | problem | theoremdb-p9824 | Subpolynomial bound for four-term progression-free sets in vector spaces over the field of five elements | {
"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}_5$ denote the finite field with five elements. For each positive integer $n$, let $\mathbb{F}_5^n$ denote the $n$-dimensional vector space over $\mathbb{F}_5$, and let $N = 5^n$. For a finite abelian group $G$ and an integer $k \geq 3$, the quantity $r_k(G)$ denotes the maximum cardinality of a subset o... | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P9824",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P9824",
"citation": null,
"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_9_iii : answer(sorry) ↔
∃ c > (0 : ℝ), (fun (n : ℕ) ↦ ((Finset.univ : Finset (𝔽₅ n)).maxAPFreeCard 4 : ℝ))
≪ fun (n : ℕ) ↦ ((5 : ℝ) ^ n) ^ (1 - c) | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem green_9_iii : answer(sorry) ↔\n ∃ c > (0 : ℝ), (fun (n : ℕ) ↦ ((Finset.univ : Finset (𝔽₅ n)).maxAPFreeCard 4 : ℝ))\n ≪ fun (n : ℕ) ↦ ((5 : ℝ) ^ n) ^ (1 - c)",
"target": null,
"source_url": "https://theoremdb.org/statements/P9824",
"format": "decl... | [] | TheoremDB P9824 · Candidate awaiting source review | null | Source-reported assessment, retrieved 8 September 2026. Frontier Science has not independently reviewed this status.
Open. No signed Lean verification is attached to this problem or its selected resolution.
This record is in the source backlog; its novelty and unresolved status have not passed source review. | null | published | incremental | hard | [
"source-candidate",
"theoremdb"
] | null | null | null | null | [
{
"kind": "website",
"citation": "TheoremDB P9824",
"url": "https://theoremdb.org/statements/P9824",
"note": null,
"sort_order": 0
},
{
"kind": "formalization",
"citation": "Source formal statement (not locally verified)",
"url": "https://theoremdb.org/statements/P9824",
"not... | {
"source_record": "P9824",
"imported_at": "2026-09-08",
"status_as_of": "2026-09-08",
"evidence_label": "Source candidate · not reviewed",
"note": "Full public directory statement. Source lane: backlog; source resolution: open. Source verification claims are attributed only; no local Lean verification has be... | {"created_at":"2026-08-17T20:02:59+00:00","display_standard":{"editorial_review":"submitter_supplied","formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":9824,"public_status":{"evidence":{"hasFormalization":false,"hasInformalProof":false,"hasRecordedWork":false,"solutionRecord":null},"lean":{... | [
"formalization-contains-placeholder",
"formalization-not-compiled-during-export",
"current-status-unassessed",
"source-full-text-not-present"
] | [] | {
"created_at": "2026-09-08T03:32:15.935Z",
"updated_at": "2026-09-08T03:32:15.935Z",
"published_at": "2026-09-08T03:33:06.153Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"e67d813e-9992-4aef-b5a4-e18c0e9877f8","format_issues":null,"number":6682} |
1.0.0 | problem:theoremdb-p9826 | problem | theoremdb-p9826 | Polynomial logarithmic bound for the practical number representation function of factorials | {
"main": "math",
"sub": "number theory",
"slug": "number-theory",
"name": "Number theory",
"original_discipline": "mathematics"
} | [] | [
{
"code": "11",
"basis": "community-map",
"evidence": "Inferred from community field: number-theory"
}
] | For a positive integer $n$, let $h(n)$ denote the maximum, over all integers $m$ with $1 \le m \le n$, of the minimum number of distinct divisors of $n$ required to represent $m$ as their sum. A positive integer $n$ is called practical if every integer $m$ with $1 \le m \le n$ can be expressed as a sum of distinct divi... | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P9826",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P9826",
"citation": null,
"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_18c : answer(sorry) ↔
∃ C : ℝ, 0 < C ∧ ∀ᶠ n : ℕ in atTop, (practicalH n.factorial : ℝ) < (log n) ^ C | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem erdos_18c : answer(sorry) ↔\n ∃ C : ℝ, 0 < C ∧ ∀ᶠ n : ℕ in atTop, (practicalH n.factorial : ℝ) < (log n) ^ C",
"target": null,
"source_url": "https://theoremdb.org/statements/P9826",
"format": "declaration-or-fragment",
"sha256": "c129f2aaf8723a78b... | [] | TheoremDB P9826 · Candidate awaiting source review | null | Source-reported assessment, retrieved 8 September 2026. Frontier Science has not independently reviewed this status.
Open. No signed Lean verification is attached to this problem or its selected resolution.
This record is in the source backlog; its novelty and unresolved status have not passed source review. | null | published | incremental | hard | [
"source-candidate",
"theoremdb"
] | null | null | null | null | [
{
"kind": "website",
"citation": "TheoremDB P9826",
"url": "https://theoremdb.org/statements/P9826",
"note": null,
"sort_order": 0
},
{
"kind": "formalization",
"citation": "Source formal statement (not locally verified)",
"url": "https://theoremdb.org/statements/P9826",
"not... | {
"source_record": "P9826",
"imported_at": "2026-09-08",
"status_as_of": "2026-09-08",
"evidence_label": "Source candidate · not reviewed",
"note": "Full public directory statement. Source lane: backlog; source resolution: open. Source verification claims are attributed only; no local Lean verification has be... | {"created_at":"2026-08-17T20:03:01+00:00","display_standard":{"formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":9826,"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": "2026-09-08T03:33:04.889Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"198e8972-c69f-488c-8ab2-6b1cda908b55","format_issues":null,"number":6408} |
1.0.0 | problem:theoremdb-p9828 | problem | theoremdb-p9828 | Lower bound for the corner density exponent in finite vector spaces | {
"main": "math",
"sub": "combinatorics",
"slug": "combinatorics",
"name": "Combinatorics",
"original_discipline": "mathematics"
} | [] | [
{
"code": "05",
"basis": "community-map",
"evidence": "Inferred from community field: combinatorics"
}
] | For each positive integer $n$, 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 \times \mathbb{F}_2^n$ and an element $d \in \mathbb{F}_2^n$, a corner in $A$ with common difference $d$ is a triple of points $(x,y)$, $(x+d,y)$, $(x,y+... | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P9828",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P9828",
"citation": null,
"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_19.lower : C >= 3.13 | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem green_19.lower : C >= 3.13",
"target": null,
"source_url": "https://theoremdb.org/statements/P9828",
"format": "declaration-or-fragment",
"sha256": "02e7b5d1409661d30a452b6cf2ee5dd72bd1fb123eaf0d18c209bf714f89507b",
"license": "Selected packets la... | [] | TheoremDB P9828 · Candidate awaiting source review | null | Source-reported assessment, retrieved 8 September 2026. Frontier Science has not independently reviewed this status.
Open. No signed Lean verification is attached to this problem or its selected resolution.
This record is in the source backlog; its novelty and unresolved status have not passed source review. | null | published | incremental | hard | [
"source-candidate",
"theoremdb"
] | null | null | null | null | [
{
"kind": "website",
"citation": "TheoremDB P9828",
"url": "https://theoremdb.org/statements/P9828",
"note": null,
"sort_order": 0
},
{
"kind": "formalization",
"citation": "Source formal statement (not locally verified)",
"url": "https://theoremdb.org/statements/P9828",
"not... | {
"source_record": "P9828",
"imported_at": "2026-09-08",
"status_as_of": "2026-09-08",
"evidence_label": "Source candidate · not reviewed",
"note": "Full public directory statement. Source lane: backlog; source resolution: open. Source verification claims are attributed only; no local Lean verification has be... | {"created_at":"2026-08-17T20:03:01+00:00","display_standard":{"editorial_review":"submitter_supplied","formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":9828,"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": "2026-09-08T03:33:03.450Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"762a4eaf-6609-4625-84ec-36b3d8f5b32f","format_issues":null,"number":6098} |
1.0.0 | problem:theoremdb-p9830 | problem | theoremdb-p9830 | Continuous surjection from a homogeneous compact Hausdorff space | {
"main": "math",
"sub": "geometry and topology",
"slug": "geometry-and-topology",
"name": "Geometry and topology",
"original_discipline": "mathematics"
} | [] | [] | A topological space is called *homogeneous* if for every pair of points $x, y$ in the space, there exists a homeomorphism of the space onto itself that maps $x$ to $y$. Determine whether every compact Hausdorff space is the continuous image of some homogeneous compact Hausdorff space. That is, for every compact Hausdor... | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P9830",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P9830",
"citation": null,
"doi": null,
"arxiv_id": null,
"license": "Selected packets label their content CC0-1.0; original references retain their own terms",
"notes": "Selected problem... | theorem homogeneousSpace_exists_surjective :
answer(sorry) ↔ ∀ (X : Type) (_ : TopologicalSpace X), T2Space X → CompactSpace X →
∃ (Y : Type) (_ : TopologicalSpace Y), T2Space Y ∧ CompactSpace Y ∧ HomogeneousSpace Y ∧
∃ f : Y → X, Continuous f ∧ f.Surjective | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem homogeneousSpace_exists_surjective :\n answer(sorry) ↔ ∀ (X : Type) (_ : TopologicalSpace X), T2Space X → CompactSpace X →\n ∃ (Y : Type) (_ : TopologicalSpace Y), T2Space Y ∧ CompactSpace Y ∧ HomogeneousSpace Y ∧\n ∃ f : Y → X, Continuous f ∧ f.Surjec... | [] | TheoremDB P9830 · Candidate awaiting source review | null | Source-reported assessment, retrieved 8 September 2026. Frontier Science has not independently reviewed this status.
Open. No signed Lean verification is attached to this problem or its selected resolution.
This record is in the source backlog; its novelty and unresolved status have not passed source review. | null | published | incremental | hard | [
"source-candidate",
"theoremdb"
] | null | null | null | null | [
{
"kind": "website",
"citation": "TheoremDB P9830",
"url": "https://theoremdb.org/statements/P9830",
"note": null,
"sort_order": 0
},
{
"kind": "formalization",
"citation": "Source formal statement (not locally verified)",
"url": "https://theoremdb.org/statements/P9830",
"not... | {
"source_record": "P9830",
"imported_at": "2026-09-08",
"status_as_of": "2026-09-08",
"evidence_label": "Source candidate · not reviewed",
"note": "Full public directory statement. Source lane: backlog; source resolution: open. Source verification claims are attributed only; no local Lean verification has be... | {"created_at":"2026-08-17T20:03:01+00:00","display_standard":{"editorial_review":"submitter_supplied","formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":9830,"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": "2026-09-08T03:32:52.645Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"e4edaf05-8947-41e7-847e-eea8b25b0821","format_issues":null,"number":3905} |
1.0.0 | problem:theoremdb-p9832 | problem | theoremdb-p9832 | Convergence of the Reciprocal Sum for Good Sets | {
"main": "math",
"sub": "number theory",
"slug": "number-theory",
"name": "Number theory",
"original_discipline": "mathematics"
} | [] | [
{
"code": "11",
"basis": "community-map",
"evidence": "Inferred from community field: number-theory"
}
] | A set $A$ of natural numbers is called good if it is infinite and there do not exist distinct elements $a, b, c \in A$ such that $a$ divides $b + c$ with $b > a$ and $c > a$. Determine whether the following statement is true: For every good set $A \subseteq \mathbb{N}$, the series $\sum_{n \in A} \frac{1}{n}$ converges... | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P9832",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P9832",
"citation": null,
"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_12.parts.iii :
answer(sorry) ↔ ∀ (A : Set ℕ), IsGood A → Summable (fun (n : A) ↦ (1 / n : ℝ)) | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem erdos_12.parts.iii :\n answer(sorry) ↔ ∀ (A : Set ℕ), IsGood A → Summable (fun (n : A) ↦ (1 / n : ℝ))",
"target": null,
"source_url": "https://theoremdb.org/statements/P9832",
"format": "declaration-or-fragment",
"sha256": "ec6813b626b6d10de7adf2e7... | [] | TheoremDB P9832 · Candidate awaiting source review | null | Source-reported assessment, retrieved 8 September 2026. Frontier Science has not independently reviewed this status.
Open. No signed Lean verification is attached to this problem or its selected resolution.
This record is in the source backlog; its novelty and unresolved status have not passed source review. | null | published | incremental | hard | [
"source-candidate",
"theoremdb"
] | null | null | null | null | [
{
"kind": "website",
"citation": "TheoremDB P9832",
"url": "https://theoremdb.org/statements/P9832",
"note": null,
"sort_order": 0
},
{
"kind": "formalization",
"citation": "Source formal statement (not locally verified)",
"url": "https://theoremdb.org/statements/P9832",
"not... | {
"source_record": "P9832",
"imported_at": "2026-09-08",
"status_as_of": "2026-09-08",
"evidence_label": "Source candidate · not reviewed",
"note": "Full public directory statement. Source lane: backlog; source resolution: open. Source verification claims are attributed only; no local Lean verification has be... | {"created_at":"2026-08-17T20:03:01+00:00","display_standard":{"editorial_review":"submitter_supplied","formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":9832,"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": "2026-09-08T03:32:52.681Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"0cdb94ab-b23d-419a-b62d-123fe1e3046f","format_issues":null,"number":3913} |
1.0.0 | problem:theoremdb-p9834 | problem | theoremdb-p9834 | Least side length for packing 17 unit squares into a square | {
"main": "math",
"sub": "geometry and topology",
"slug": "geometry-and-topology",
"name": "Geometry and topology",
"original_discipline": "mathematics"
} | [] | [] | For a positive real number $x$, let the open square of side length $x$ be the set
$$
\{(a,b)\in\mathbb{R}^2 : 0 < a < x \text{ and } 0 < b < x\}.
$$
A packing of 17 unit squares into a square of side length $x$ consists of 17 isometric embeddings of the open unit square $\{(a,b)\in\mathbb{R}^2 : 0 < a < 1 \text{ and ... | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P9834",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P9834",
"citation": null,
"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_seventeen_square_packing_in_square :
IsLeast {x : ℝ | Nonempty (Packing 17 UnitSquare (Square x))} answer(sorry) | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem least_seventeen_square_packing_in_square :\n IsLeast {x : ℝ | Nonempty (Packing 17 UnitSquare (Square x))} answer(sorry)",
"target": null,
"source_url": "https://theoremdb.org/statements/P9834",
"format": "declaration-or-fragment",
"sha256": "49aea... | [] | TheoremDB P9834 · Candidate awaiting source review | null | Source-reported assessment, retrieved 8 September 2026. Frontier Science has not independently reviewed this status.
Open. No signed Lean verification is attached to this problem or its selected resolution.
This record is in the source backlog; its novelty and unresolved status have not passed source review. | null | published | incremental | hard | [
"source-candidate",
"theoremdb"
] | null | null | null | null | [
{
"kind": "website",
"citation": "TheoremDB P9834",
"url": "https://theoremdb.org/statements/P9834",
"note": null,
"sort_order": 0
},
{
"kind": "formalization",
"citation": "Source formal statement (not locally verified)",
"url": "https://theoremdb.org/statements/P9834",
"not... | {
"source_record": "P9834",
"imported_at": "2026-09-08",
"status_as_of": "2026-09-08",
"evidence_label": "Source candidate · not reviewed",
"note": "Full public directory statement. Source lane: backlog; source resolution: open. Source verification claims are attributed only; no local Lean verification has be... | {"created_at":"2026-08-17T20:03:01+00:00","display_standard":{"editorial_review":"submitter_supplied","formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":9834,"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": "2026-09-08T03:33:03.116Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"a9e877da-d3fb-43b5-9680-2d7f00476b87","format_issues":null,"number":6026} |
1.0.0 | problem:theoremdb-p9836 | problem | theoremdb-p9836 | Asymptotic upper bound for the size of minimal vanishing signed unit fraction sets | {
"main": "math",
"sub": "combinatorics",
"slug": "combinatorics",
"name": "Combinatorics",
"original_discipline": "mathematics"
} | [] | [
{
"code": "05",
"basis": "community-map",
"evidence": "Inferred from community field: combinatorics"
}
] | For each positive integer $N$, let $c(N)$ denote the maximum cardinality of a subset $A \subseteq \{1, 2, \ldots, N\}$ for which there exists a function $\delta: A \to \{-1, 1\}$ such that
$$
\sum_{n \in A} \frac{\delta(n)}{n} = 0
$$
and such that for every nonempty proper subset $A' \subsetneq A$,
$$
\sum_{n \in... | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P9836",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P9836",
"citation": null,
"doi": null,
"arxiv_id": null,
"license": "Selected packets label their content CC0-1.0; original references retain their own terms",
"notes": "Selected problem... | theorem erdos_319.variants.isBigO (N : ℕ) (c : ℕ → ℝ)
(h : ∀ N, IsGreatest
{ (#A : ℝ) | (A) (_ : A ⊆ Finset.Icc 1 N)
(_ : ∃ δ : ℕ → ℤˣ, ∑ n ∈ A, (δ n : ℚ) / n = 0 ∧
∀ A' ⊂ A, A'.Nonempty → ∑ n ∈ A', (δ n : ℚ) / n ≠ 0) } (c N)) :
c =O[atTop] (answer(sorry) : ℕ → ℝ) | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem erdos_319.variants.isBigO (N : ℕ) (c : ℕ → ℝ)\n (h : ∀ N, IsGreatest\n { (#A : ℝ) | (A) (_ : A ⊆ Finset.Icc 1 N)\n (_ : ∃ δ : ℕ → ℤˣ, ∑ n ∈ A, (δ n : ℚ) / n = 0 ∧\n ∀ A' ⊂ A, A'.Nonempty → ∑ n ∈ A', (δ n : ℚ) / n ≠ 0) } (c N)) :\n c =O[atTop] (an... | [] | TheoremDB P9836 · Candidate awaiting source review | null | Source-reported assessment, retrieved 8 September 2026. Frontier Science has not independently reviewed this status.
Open. No signed Lean verification is attached to this problem or its selected resolution.
This record is in the source backlog; its novelty and unresolved status have not passed source review. | null | published | incremental | hard | [
"source-candidate",
"theoremdb"
] | null | null | null | null | [
{
"kind": "website",
"citation": "TheoremDB P9836",
"url": "https://theoremdb.org/statements/P9836",
"note": null,
"sort_order": 0
},
{
"kind": "formalization",
"citation": "Source formal statement (not locally verified)",
"url": "https://theoremdb.org/statements/P9836",
"not... | {
"source_record": "P9836",
"imported_at": "2026-09-08",
"status_as_of": "2026-09-08",
"evidence_label": "Source candidate · not reviewed",
"note": "Full public directory statement. Source lane: backlog; source resolution: open. Source verification claims are attributed only; no local Lean verification has be... | {"created_at":"2026-08-17T20:03:02+00:00","display_standard":{"editorial_review":"submitter_supplied","formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":9836,"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": "2026-09-08T03:32:50.493Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"e76317ac-cee6-4bf7-87cb-780338a7fa55","format_issues":null,"number":3428} |
1.0.0 | problem:theoremdb-p9838 | problem | theoremdb-p9838 | Non-existence of ternary-integer monochromatic quantum graphs with six vertices and arbitrarily many colours | {
"main": "math",
"sub": "combinatorics",
"slug": "combinatorics",
"name": "Combinatorics",
"original_discipline": "mathematics"
} | [] | [
{
"code": "05",
"basis": "community-map",
"evidence": "Inferred from community field: combinatorics"
}
] | Let $N$ be a positive even integer and let $D$ be a positive integer. A coloured edge of the complete graph $K_N$ on vertices $\{1, \dots, N\}$ is a pair $(e, c)$ where $e$ is an edge of $K_N$ and $c \in \{1, \dots, D\}$ is a colour. A weight function on coloured edges with values in a ring $\alpha$ assigns to each col... | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P9838",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P9838",
"citation": null,
"doi": null,
"arxiv_id": null,
"license": "Selected packets label their content CC0-1.0; original references retain their own terms",
"notes": "Selected problem... | theorem eqSystem6_no_solution_ge3_trinary_int :
answer(sorry) ↔
∀ D : Nat, D ≥ 3 →
¬ ∃ W : WeightsN 6 D ℤ,
(∀ e, W e = (-1 : ℤ) ∨ W e = 0 ∨ W e = 1) ∧
EqSystemN 6 D W | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem eqSystem6_no_solution_ge3_trinary_int :\n answer(sorry) ↔\n ∀ D : Nat, D ≥ 3 →\n ¬ ∃ W : WeightsN 6 D ℤ,\n (∀ e, W e = (-1 : ℤ) ∨ W e = 0 ∨ W e = 1) ∧\n EqSystemN 6 D W",
"target": null,
"source_url": "https://theoremdb.... | [] | TheoremDB P9838 · Candidate awaiting source review | null | Source-reported assessment, retrieved 8 September 2026. Frontier Science has not independently reviewed this status.
Open. No signed Lean verification is attached to this problem or its selected resolution.
This record is in the source backlog; its novelty and unresolved status have not passed source review. | null | published | incremental | hard | [
"source-candidate",
"theoremdb"
] | null | null | null | null | [
{
"kind": "website",
"citation": "TheoremDB P9838",
"url": "https://theoremdb.org/statements/P9838",
"note": null,
"sort_order": 0
},
{
"kind": "formalization",
"citation": "Source formal statement (not locally verified)",
"url": "https://theoremdb.org/statements/P9838",
"not... | {
"source_record": "P9838",
"imported_at": "2026-09-08",
"status_as_of": "2026-09-08",
"evidence_label": "Source candidate · not reviewed",
"note": "Full public directory statement. Source lane: backlog; source resolution: open. Source verification claims are attributed only; no local Lean verification has be... | {"created_at":"2026-08-17T20:03:05+00:00","display_standard":{"editorial_review":"submitter_supplied","formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":9838,"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": "2026-09-08T03:33:04.429Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"1d5ce68a-0b98-4ec2-a837-259272665d28","format_issues":null,"number":6308} |
1.0.0 | problem:theoremdb-p9840 | problem | theoremdb-p9840 | Irrationality of a weighted exponential sum for rapidly growing integer sequences | {
"main": "math",
"sub": "number theory",
"slug": "number-theory",
"name": "Number theory",
"original_discipline": "mathematics"
} | [] | [
{
"code": "11",
"basis": "community-map",
"evidence": "Inferred from community field: number-theory"
}
] | Let $\mathbb{N}$ denote the set of positive integers and let $\mathbb{Z}$ denote the set of integers. Let $a : \mathbb{N} \to \mathbb{Z}$ be a strictly increasing sequence of integers, and suppose that $\frac{a_n}{n} \to \infty$ as $n \to \infty$. Let $s \in \mathbb{R}$ be the sum of the series $\sum_{n=1}^{\infty} \fr... | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P9840",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P9840",
"citation": null,
"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_260 : answer(sorry) ↔
∀ a : ℕ → ℤ, ∀ s : ℝ,
StrictMono a →
Tendsto (fun n => (a n : ℝ ) / n ) atTop atTop →
HasSum (fun n => (a n : ℝ ) / 2 ^ a n) s → Irrational s | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem erdos_260 : answer(sorry) ↔\n ∀ a : ℕ → ℤ, ∀ s : ℝ,\n StrictMono a →\n Tendsto (fun n => (a n : ℝ ) / n ) atTop atTop →\n HasSum (fun n => (a n : ℝ ) / 2 ^ a n) s → Irrational s",
"target": null,... | [] | TheoremDB P9840 · Candidate awaiting source review | null | Source-reported assessment, retrieved 8 September 2026. Frontier Science has not independently reviewed this status.
Open. No signed Lean verification is attached to this problem or its selected resolution.
This record is in the source backlog; its novelty and unresolved status have not passed source review. | null | published | incremental | hard | [
"source-candidate",
"theoremdb"
] | null | null | null | null | [
{
"kind": "website",
"citation": "TheoremDB P9840",
"url": "https://theoremdb.org/statements/P9840",
"note": null,
"sort_order": 0
},
{
"kind": "formalization",
"citation": "Source formal statement (not locally verified)",
"url": "https://theoremdb.org/statements/P9840",
"not... | {
"source_record": "P9840",
"imported_at": "2026-09-08",
"status_as_of": "2026-09-08",
"evidence_label": "Source candidate · not reviewed",
"note": "Full public directory statement. Source lane: backlog; source resolution: open. Source verification claims are attributed only; no local Lean verification has be... | {"created_at":"2026-08-17T20:03:07+00:00","display_standard":{"editorial_review":"submitter_supplied","formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":9840,"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": "2026-09-08T03:33:01.390Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"1ed18297-15e6-48ff-938e-f1d84925b1c7","format_issues":null,"number":5720} |
1.0.0 | problem:theoremdb-p9842 | problem | theoremdb-p9842 | Sidorenko's Conjecture on Homomorphism Densities | {
"main": "math",
"sub": "combinatorics",
"slug": "combinatorics",
"name": "Combinatorics",
"original_discipline": "mathematics"
} | [] | [
{
"code": "05",
"basis": "community-map",
"evidence": "Inferred from community field: combinatorics"
}
] | Let $H$ be a finite bipartite simple graph and let $G$ be a finite simple graph. For a simple graph $F$, the homomorphism density $t(F, G)$ is the probability that a uniformly random function from the vertex set of $F$ to the vertex set of $G$ preserves adjacency. Let $K_2$ denote the complete graph on 2 vertices, whic... | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P9842",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P9842",
"citation": null,
"doi": null,
"arxiv_id": null,
"license": "Selected packets label their content CC0-1.0; original references retain their own terms",
"notes": "Selected problem... | theorem sidorenko_conjecture : answer(sorry) ↔
∀ {V W : Type} [Fintype V] [Fintype W] [DecidableEq V] [DecidableEq W] [Nonempty W]
(H : SimpleGraph V) (G : SimpleGraph W)
[DecidableRel H.Adj] [DecidableRel G.Adj],
H.IsBipartite →
homDensity (completeGraph (Fin 2)) G ^ H.edgeFinset.card ≤ hom... | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem sidorenko_conjecture : answer(sorry) ↔\n ∀ {V W : Type} [Fintype V] [Fintype W] [DecidableEq V] [DecidableEq W] [Nonempty W]\n (H : SimpleGraph V) (G : SimpleGraph W)\n [DecidableRel H.Adj] [DecidableRel G.Adj],\n H.IsBipartite →\n homDensity (... | [] | TheoremDB P9842 · Candidate awaiting source review | null | Source-reported assessment, retrieved 8 September 2026. Frontier Science has not independently reviewed this status.
Open. No signed Lean verification is attached to this problem or its selected resolution.
This record is in the source backlog; its novelty and unresolved status have not passed source review. | null | published | incremental | hard | [
"source-candidate",
"theoremdb"
] | null | null | null | null | [
{
"kind": "website",
"citation": "TheoremDB P9842",
"url": "https://theoremdb.org/statements/P9842",
"note": null,
"sort_order": 0
},
{
"kind": "formalization",
"citation": "Source formal statement (not locally verified)",
"url": "https://theoremdb.org/statements/P9842",
"not... | {
"source_record": "P9842",
"imported_at": "2026-09-08",
"status_as_of": "2026-09-08",
"evidence_label": "Source candidate · not reviewed",
"note": "Full public directory statement. Source lane: backlog; source resolution: open. Source verification claims are attributed only; no local Lean verification has be... | {"created_at":"2026-08-17T20:03:08+00:00","display_standard":{"editorial_review":"submitter_supplied","formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":9842,"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": "2026-09-08T03:33:05.839Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"7cb2b719-2b46-4607-a72f-3166a8ab2117","format_issues":null,"number":6614} |
1.0.0 | problem:theoremdb-p9844 | problem | theoremdb-p9844 | Covering Integers by Prime Congruence Classes | {
"main": "math",
"sub": "number theory",
"slug": "number-theory",
"name": "Number theory",
"original_discipline": "mathematics"
} | [] | [
{
"code": "11",
"basis": "community-map",
"evidence": "Inferred from community field: number-theory"
}
] | For a positive integer $n$, consider the set of primes $p$ with $2 \leq p \leq n$. For each such prime $p$, choose a residue class $a_p \pmod{p}$. An integer $m$ with $1 \leq m \leq n$ is said to be covered by this choice if it satisfies the congruence $m \equiv a_p \pmod{p}$ for at least two distinct primes $p$ in the... | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P9844",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P9844",
"citation": null,
"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_689 :
answer(sorry) ↔ ∀ᶠ n in .atTop, ∃ a : ℕ → ℕ, ∀ m ∈ Finset.Icc 1 n,
2 ≤ (Finset.Icc 1 n |>.filter fun p => p.Prime ∧ a p ≡ m [MOD p]).card | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem erdos_689 :\n answer(sorry) ↔ ∀ᶠ n in .atTop, ∃ a : ℕ → ℕ, ∀ m ∈ Finset.Icc 1 n,\n 2 ≤ (Finset.Icc 1 n |>.filter fun p => p.Prime ∧ a p ≡ m [MOD p]).card",
"target": null,
"source_url": "https://theoremdb.org/statements/P9844",
"format": "declarat... | [] | TheoremDB P9844 · Candidate awaiting source review | null | Source-reported assessment, retrieved 8 September 2026. Frontier Science has not independently reviewed this status.
Open. No signed Lean verification is attached to this problem or its selected resolution.
This record is in the source backlog; its novelty and unresolved status have not passed source review. | null | published | incremental | hard | [
"source-candidate",
"theoremdb"
] | null | null | null | null | [
{
"kind": "website",
"citation": "TheoremDB P9844",
"url": "https://theoremdb.org/statements/P9844",
"note": null,
"sort_order": 0
},
{
"kind": "formalization",
"citation": "Source formal statement (not locally verified)",
"url": "https://theoremdb.org/statements/P9844",
"not... | {
"source_record": "P9844",
"imported_at": "2026-09-08",
"status_as_of": "2026-09-08",
"evidence_label": "Source candidate · not reviewed",
"note": "Full public directory statement. Source lane: backlog; source resolution: open. Source verification claims are attributed only; no local Lean verification has be... | {"created_at":"2026-08-17T20:03:44+00:00","display_standard":{"editorial_review":"submitter_supplied","formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":9844,"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": "2026-09-08T03:32:52.836Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"1ce1481d-19f4-4bc0-ac9d-2b6d6fd6f3e4","format_issues":null,"number":3946} |
1.0.0 | problem:theoremdb-p9846 | problem | theoremdb-p9846 | Brennan's Conjecture for Bounded Univalent Functions | {
"main": "math",
"sub": "analysis",
"slug": "analysis",
"name": "Analysis",
"original_discipline": "mathematics"
} | [] | [] | Let $\mathbb{D} = \{z \in \mathbb{C} : |z| < 1\}$ denote the open unit disk in the complex plane. A function $f: \mathbb{C} \to \mathbb{C}$ is called normalized univalent on $\mathbb{D}$ if $f$ is analytic on $\mathbb{D}$, injective on $\mathbb{D}$, and satisfies $f(0) = 0$ and $f'(0) = 1$. For such a function $f$ and ... | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P9846",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P9846",
"citation": null,
"doi": null,
"arxiv_id": null,
"license": "Selected packets label their content CC0-1.0; original references retain their own terms",
"notes": "Selected problem... | theorem brennan_universalSpectrumBounded :
universalSpectrumBounded (-2) = 1 | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem brennan_universalSpectrumBounded :\n universalSpectrumBounded (-2) = 1",
"target": null,
"source_url": "https://theoremdb.org/statements/P9846",
"format": "declaration-or-fragment",
"sha256": "4681957ef86a7b7e617e81c6102f3a8bbbb152839a6846692242731... | [] | TheoremDB P9846 · Candidate awaiting source review | null | Source-reported assessment, retrieved 8 September 2026. Frontier Science has not independently reviewed this status.
Open. No signed Lean verification is attached to this problem or its selected resolution.
This record is in the source backlog; its novelty and unresolved status have not passed source review. | null | published | incremental | hard | [
"source-candidate",
"theoremdb"
] | null | null | null | null | [
{
"kind": "website",
"citation": "TheoremDB P9846",
"url": "https://theoremdb.org/statements/P9846",
"note": null,
"sort_order": 0
},
{
"kind": "formalization",
"citation": "Source formal statement (not locally verified)",
"url": "https://theoremdb.org/statements/P9846",
"not... | {
"source_record": "P9846",
"imported_at": "2026-09-08",
"status_as_of": "2026-09-08",
"evidence_label": "Source candidate · not reviewed",
"note": "Full public directory statement. Source lane: backlog; source resolution: open. Source verification claims are attributed only; no local Lean verification has be... | {"created_at":"2026-08-17T20:03:54+00:00","display_standard":{"editorial_review":"submitter_supplied","formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":9846,"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": "2026-09-08T03:32:50.772Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"d889cd57-e0f1-4fbe-bcab-a3e86668a4fe","format_issues":null,"number":3489} |
1.0.0 | problem:theoremdb-p9848 | problem | theoremdb-p9848 | Membership of 33 in the difference set of the greedy Sidon sequence | {
"main": "math",
"sub": "combinatorics",
"slug": "combinatorics",
"name": "Combinatorics",
"original_discipline": "mathematics"
} | [] | [
{
"code": "05",
"basis": "community-map",
"evidence": "Inferred from community field: combinatorics"
}
] | Let $A = \{1, 2, 4, 8, 13, 21, 31, 45, 66, 81, 97, \ldots\}$ be the greedy Sidon sequence, defined recursively by beginning with $1$ and iteratively including the smallest positive integer that preserves the Sidon property (that is, all sums $a + b$ with $a \leq b$ in the set are distinct). Does the integer $33$ belong... | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P9848",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P9848",
"citation": null,
"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._33_mem_sub : answer(sorry) ↔
33 ∈ Set.range greedySidon - Set.range greedySidon | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem erdos_340.variants._33_mem_sub : answer(sorry) ↔\n 33 ∈ Set.range greedySidon - Set.range greedySidon",
"target": null,
"source_url": "https://theoremdb.org/statements/P9848",
"format": "declaration-or-fragment",
"sha256": "e6ee7faf10efa2f96776046a... | [] | TheoremDB P9848 · Candidate awaiting source review | null | Source-reported assessment, retrieved 8 September 2026. Frontier Science has not independently reviewed this status.
Open. No signed Lean verification is attached to this problem or its selected resolution.
This record is in the source backlog; its novelty and unresolved status have not passed source review. | null | published | incremental | hard | [
"source-candidate",
"theoremdb"
] | null | null | null | null | [
{
"kind": "website",
"citation": "TheoremDB P9848",
"url": "https://theoremdb.org/statements/P9848",
"note": null,
"sort_order": 0
},
{
"kind": "formalization",
"citation": "Source formal statement (not locally verified)",
"url": "https://theoremdb.org/statements/P9848",
"not... | {
"source_record": "P9848",
"imported_at": "2026-09-08",
"status_as_of": "2026-09-08",
"evidence_label": "Source candidate · not reviewed",
"note": "Full public directory statement. Source lane: backlog; source resolution: open. Source verification claims are attributed only; no local Lean verification has be... | {"created_at":"2026-08-17T20:03:55+00:00","display_standard":{"formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":9848,"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": "2026-09-08T03:33:03.878Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"45910888-bda4-4c13-b586-d649bf11e521","format_issues":null,"number":6190} |
1.0.0 | problem:theoremdb-p9850 | problem | theoremdb-p9850 | Better lower bound for the four-term arithmetic progression colouring number | {
"main": "math",
"sub": "combinatorics",
"slug": "combinatorics",
"name": "Combinatorics",
"original_discipline": "mathematics"
} | [] | [
{
"code": "05",
"basis": "community-map",
"evidence": "Inferred from community field: combinatorics"
}
] | For each positive integer $n$, let $h(n)$ denote the smallest integer $k$ such that the set $\{1, \ldots, n\}$ can be coloured with $k$ colours in such a way that every four-term arithmetic progression contained in $\{1, \ldots, n\}$ contains at least three distinct colours. Determine whether there exists a function $L... | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P9850",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P9850",
"citation": null,
"doi": null,
"arxiv_id": null,
"license": "Selected packets label their content CC0-1.0; original references retain their own terms",
"notes": "Selected problem... | theorem erdos_160.better_lower:
let lower_bound : ℕ → ℝ := answer(sorry)
(lower_bound =O[atTop] fun n => (erdos_160.h n : ℝ)) ∧
∀ c > 0,
(fun (n : ℕ) => exp (c * log n ^ ((1 : ℝ) / 12))) =O[atTop] (fun n => (erdos_160.h n : ℝ)) →
∀ c > 0, (fun (n : ℕ) => exp (c * log n ^ ((1 : ℝ) / 12))) =o[atTop] l... | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem erdos_160.better_lower:\n let lower_bound : ℕ → ℝ := answer(sorry)\n (lower_bound =O[atTop] fun n => (erdos_160.h n : ℝ)) ∧\n ∀ c > 0,\n (fun (n : ℕ) => exp (c * log n ^ ((1 : ℝ) / 12))) =O[atTop] (fun n => (erdos_160.h n : ℝ)) →\n ∀ c > 0, (fun (n : ℕ... | [] | TheoremDB P9850 · Candidate awaiting source review | null | Source-reported assessment, retrieved 8 September 2026. Frontier Science has not independently reviewed this status.
Open. No signed Lean verification is attached to this problem or its selected resolution.
This record is in the source backlog; its novelty and unresolved status have not passed source review. | null | published | incremental | hard | [
"source-candidate",
"theoremdb"
] | null | null | null | null | [
{
"kind": "website",
"citation": "TheoremDB P9850",
"url": "https://theoremdb.org/statements/P9850",
"note": null,
"sort_order": 0
},
{
"kind": "formalization",
"citation": "Source formal statement (not locally verified)",
"url": "https://theoremdb.org/statements/P9850",
"not... | {
"source_record": "P9850",
"imported_at": "2026-09-08",
"status_as_of": "2026-09-08",
"evidence_label": "Source candidate · not reviewed",
"note": "Full public directory statement. Source lane: backlog; source resolution: open. Source verification claims are attributed only; no local Lean verification has be... | {"created_at":"2026-08-17T20:03:56+00:00","display_standard":{"editorial_review":"submitter_supplied","formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":9850,"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": "2026-09-08T03:32:50.657Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"eecf0e44-afeb-4287-b694-4878e96e096b","format_issues":null,"number":3464} |
1.0.0 | problem:theoremdb-p9852 | problem | theoremdb-p9852 | Reed's omega, delta, and chi conjecture for finite graphs | {
"main": "math",
"sub": "combinatorics",
"slug": "combinatorics",
"name": "Combinatorics",
"original_discipline": "mathematics"
} | [] | [
{
"code": "05",
"basis": "community-map",
"evidence": "Inferred from community field: combinatorics"
}
] | Let $G$ be a finite simple graph. Let $\chi(G)$ denote the chromatic number of $G$, let $\omega(G)$ denote the clique number of $G$ (the size of the largest complete subgraph of $G$), and let $\Delta(G)$ denote the maximum degree of $G$. Prove that
$$
\chi(G) \leq \left\lceil \frac{\omega(G) + \Delta(G) + 1}{2} \right... | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P9852",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P9852",
"citation": null,
"doi": null,
"arxiv_id": null,
"license": "Selected packets label their content CC0-1.0; original references retain their own terms",
"notes": "Selected problem... | theorem reed_omega_delta_chi_conjecture_for_finite_graphs :
∀ {V : Type} [Fintype V] [DecidableEq V] (G : SimpleGraph V) [DecidableRel G.Adj],
let χ := G.chromaticNumber
let ω := G.cliqueNum
let Δ := G.maxDegree
2 * χ ≤ ω + Δ + 2 | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem reed_omega_delta_chi_conjecture_for_finite_graphs :\n ∀ {V : Type} [Fintype V] [DecidableEq V] (G : SimpleGraph V) [DecidableRel G.Adj],\n let χ := G.chromaticNumber\n let ω := G.cliqueNum\n let Δ := G.maxDegree\n 2 * χ ≤ ω + Δ + 2",
"target": null,
... | [] | TheoremDB P9852 · Candidate awaiting source review | null | Source-reported assessment, retrieved 8 September 2026. Frontier Science has not independently reviewed this status.
Open. No signed Lean verification is attached to this problem or its selected resolution.
This record is in the source backlog; its novelty and unresolved status have not passed source review. | null | published | incremental | hard | [
"source-candidate",
"theoremdb"
] | null | null | null | null | [
{
"kind": "website",
"citation": "TheoremDB P9852",
"url": "https://theoremdb.org/statements/P9852",
"note": null,
"sort_order": 0
},
{
"kind": "formalization",
"citation": "Source formal statement (not locally verified)",
"url": "https://theoremdb.org/statements/P9852",
"not... | {
"source_record": "P9852",
"imported_at": "2026-09-08",
"status_as_of": "2026-09-08",
"evidence_label": "Source candidate · not reviewed",
"note": "Full public directory statement. Source lane: backlog; source resolution: open. Source verification claims are attributed only; no local Lean verification has be... | {"created_at":"2026-08-17T20:03:57+00:00","display_standard":{"editorial_review":"submitter_supplied","formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":9852,"public_status":{"evidence":{"hasFormalization":false,"hasInformalProof":false,"hasRecordedWork":false,"solutionRecord":null},"lean":{... | [
"formalization-not-compiled-during-export",
"current-status-unassessed",
"source-full-text-not-present"
] | [] | {
"created_at": "2026-09-08T03:32:15.935Z",
"updated_at": "2026-09-08T03:32:15.935Z",
"published_at": "2026-09-08T03:33:05.595Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"96fbb204-72db-4f08-9776-f080a58e7908","format_issues":null,"number":6561} |
1.0.0 | problem:theoremdb-p9854 | problem | theoremdb-p9854 | Voronovskaja-type asymptotic formula for Bézier–Bernstein operators with shape parameter | {
"main": "math",
"sub": "analysis",
"slug": "analysis",
"name": "Analysis",
"original_discipline": "mathematics"
} | [] | [] | Fix a real number $\alpha$ with $\alpha > 0$ and $\alpha \neq 1$. For each $n \in \mathbb{N}$, the Bernstein basis polynomial of degree $n$ and index $j$ is
$$
p_{n,j}(x) = \binom{n}{j} x^j (1-x)^{n-j}.
$$
For $0 \leq k \leq n$, define the cumulative sum
$$
J_{n,k}(x) = \sum_{j=k}^{n} p_{n,j}(x),
$$
and set $J_... | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P9854",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P9854",
"citation": null,
"doi": null,
"arxiv_id": null,
"license": "Selected packets label their content CC0-1.0; original references retain their own terms",
"notes": "Selected problem... | theorem voronovskaja_theorem.bezier_bernstein_operators.variants.eventually_smooth
(α : ℝ) (hα_pos : 0 < α) (hα : α ≠ 1) :
let limitFormula : (ℝ → ℝ) → ℝ → ℝ := answer(sorry)
∀ᶠ m : ℕ in atTop,
∀ (f : ℝ → ℝ) (x : ℝ), x ∈ I → ContDiffOn ℝ m f I →
Tendsto (fun n : ℕ => Real.sqrt n * (bezierBerns... | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem voronovskaja_theorem.bezier_bernstein_operators.variants.eventually_smooth\n (α : ℝ) (hα_pos : 0 < α) (hα : α ≠ 1) :\n let limitFormula : (ℝ → ℝ) → ℝ → ℝ := answer(sorry)\n ∀ᶠ m : ℕ in atTop,\n ∀ (f : ℝ → ℝ) (x : ℝ), x ∈ I → ContDiffOn ℝ m f I →\n ... | [] | TheoremDB P9854 · Candidate awaiting source review | null | Source-reported assessment, retrieved 8 September 2026. Frontier Science has not independently reviewed this status.
Open. No signed Lean verification is attached to this problem or its selected resolution.
This record is in the source backlog; its novelty and unresolved status have not passed source review. | null | published | incremental | hard | [
"source-candidate",
"theoremdb"
] | null | null | null | null | [
{
"kind": "website",
"citation": "TheoremDB P9854",
"url": "https://theoremdb.org/statements/P9854",
"note": null,
"sort_order": 0
},
{
"kind": "formalization",
"citation": "Source formal statement (not locally verified)",
"url": "https://theoremdb.org/statements/P9854",
"not... | {
"source_record": "P9854",
"imported_at": "2026-09-08",
"status_as_of": "2026-09-08",
"evidence_label": "Source candidate · not reviewed",
"note": "Full public directory statement. Source lane: backlog; source resolution: open. Source verification claims are attributed only; no local Lean verification has be... | {"created_at":"2026-08-17T20:03:59+00:00","display_standard":{"editorial_review":"submitter_supplied","formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":9854,"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": "2026-09-08T03:33:07.441Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"bf5fe70d-d84f-436e-9c24-001157e4aed4","format_issues":null,"number":6960} |
1.0.0 | problem:theoremdb-p9856 | problem | theoremdb-p9856 | Zero area of the Mandelbrot set boundary | {
"main": "math",
"sub": "dynamical systems",
"slug": "dynamical-systems",
"name": "Dynamical systems",
"original_discipline": "mathematics"
} | [] | [
{
"code": "37",
"basis": "community-map",
"evidence": "Inferred from community field: dynamical-systems"
}
] | The Mandelbrot set is the set of all complex parameters $c$ for which the orbit of $0$ under iteration of the map $z \mapsto z^2 + c$ remains bounded; equivalently, it is the set $\{c \in \mathbb{C} : \forall k \in \mathbb{N}, \|z_k\| \leq 2\}$ where $z_0 = 0$ and $z_{k+1} = z_k^2 + c$. The frontier (topological bounda... | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P9856",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P9856",
"citation": null,
"doi": null,
"arxiv_id": null,
"license": "Selected packets label their content CC0-1.0; original references retain their own terms",
"notes": "Selected problem... | theorem volume_frontier_mandelbrotSet_eq_zero : volume (frontier mandelbrotSet) = 0 | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem volume_frontier_mandelbrotSet_eq_zero : volume (frontier mandelbrotSet) = 0",
"target": null,
"source_url": "https://theoremdb.org/statements/P9856",
"format": "declaration-or-fragment",
"sha256": "603496b8337fdade80cb9ea222903a701898301ab211c31c1b55d... | [] | TheoremDB P9856 · Candidate awaiting source review | null | Source-reported assessment, retrieved 8 September 2026. Frontier Science has not independently reviewed this status.
Open. No signed Lean verification is attached to this problem or its selected resolution.
This record is in the source backlog; its novelty and unresolved status have not passed source review. | null | published | incremental | hard | [
"source-candidate",
"theoremdb"
] | null | null | null | null | [
{
"kind": "website",
"citation": "TheoremDB P9856",
"url": "https://theoremdb.org/statements/P9856",
"note": null,
"sort_order": 0
},
{
"kind": "formalization",
"citation": "Source formal statement (not locally verified)",
"url": "https://theoremdb.org/statements/P9856",
"not... | {
"source_record": "P9856",
"imported_at": "2026-09-08",
"status_as_of": "2026-09-08",
"evidence_label": "Source candidate · not reviewed",
"note": "Full public directory statement. Source lane: backlog; source resolution: open. Source verification claims are attributed only; no local Lean verification has be... | {"created_at":"2026-08-17T20:04:01+00:00","display_standard":{"editorial_review":"submitter_supplied","formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":9856,"public_status":{"evidence":{"hasFormalization":false,"hasInformalProof":false,"hasRecordedWork":false,"solutionRecord":null},"lean":{... | [
"formalization-not-compiled-during-export",
"current-status-unassessed",
"source-full-text-not-present"
] | [] | {
"created_at": "2026-09-08T03:32:15.935Z",
"updated_at": "2026-09-08T03:32:15.935Z",
"published_at": "2026-09-08T03:33:07.613Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"df864523-0e3a-4289-bcef-a6f0e45b090c","format_issues":null,"number":6997} |
1.0.0 | problem:theoremdb-p9858 | problem | theoremdb-p9858 | Strict lower bound for the van der Waerden number W(3, 33) | {
"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 · P9858",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P9858",
"citation": null,
"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_33_lower : answer(sorry) ↔ W 3 33 > 1063 | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem W_3_33_lower : answer(sorry) ↔ W 3 33 > 1063",
"target": null,
"source_url": "https://theoremdb.org/statements/P9858",
"format": "declaration-or-fragment",
"sha256": "dcc3e24faafbb3356fb397d8b60fb1f5a17bed617fad45667f3ca7db0b5bbc58",
"license": "S... | [] | TheoremDB P9858 · Candidate awaiting source review | null | Source-reported assessment, retrieved 8 September 2026. Frontier Science has not independently reviewed this status.
Open. No signed Lean verification is attached to this problem or its selected resolution.
This record is in the source backlog; its novelty and unresolved status have not passed source review. | null | published | incremental | hard | [
"source-candidate",
"theoremdb"
] | null | null | null | null | [
{
"kind": "website",
"citation": "TheoremDB P9858",
"url": "https://theoremdb.org/statements/P9858",
"note": null,
"sort_order": 0
},
{
"kind": "formalization",
"citation": "Source formal statement (not locally verified)",
"url": "https://theoremdb.org/statements/P9858",
"not... | {
"source_record": "P9858",
"imported_at": "2026-09-08",
"status_as_of": "2026-09-08",
"evidence_label": "Source candidate · not reviewed",
"note": "Full public directory statement. Source lane: backlog; source resolution: open. Source verification claims are attributed only; no local Lean verification has be... | {"created_at":"2026-08-17T20:04:02+00:00","display_standard":{"editorial_review":"submitter_supplied","formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":9858,"public_status":{"evidence":{"hasFormalization":false,"hasInformalProof":false,"hasRecordedWork":false,"solutionRecord":null},"lean":{... | [
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"source-full-text-not-present"
] | [] | {
"created_at": "2026-09-08T03:32:15.935Z",
"updated_at": "2026-09-08T03:32:15.935Z",
"published_at": "2026-09-08T03:33:06.093Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"2e012c9a-938c-431c-aee9-1d8eb25d6620","format_issues":null,"number":6669} |
1.0.0 | problem:theoremdb-p9860 | problem | theoremdb-p9860 | Well Total Domination of Triangle-Free Graphs with Bounded Induced Path Size | {
"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. Suppose that $G$ is triangle-free, meaning $G$ contains no three distinct vertices $a, b, c$ with edges $ab$, $bc$, and $ca$. Suppose further that the size of a largest induced path in $G$ is at most $4$. A subset $S \subseteq V(G)$ is an *induced p... | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P9860",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P9860",
"citation": null,
"doi": null,
"arxiv_id": null,
"license": "Selected packets label their content CC0-1.0; original references retain their own terms",
"notes": "Selected problem... | theorem conjecture314 [Nontrivial α] (G : SimpleGraph α) [DecidableRel G.Adj]
(hG : G.Connected)
(hTriFree : ∀ a b c : α, G.Adj a b → G.Adj b c → G.Adj c a → False)
(hPath : largestInducedPathSize G ≤ 4) :
IsWellTotallyDominated G | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem conjecture314 [Nontrivial α] (G : SimpleGraph α) [DecidableRel G.Adj]\n (hG : G.Connected)\n (hTriFree : ∀ a b c : α, G.Adj a b → G.Adj b c → G.Adj c a → False)\n (hPath : largestInducedPathSize G ≤ 4) :\n IsWellTotallyDominated G",
"target": null,
... | [] | TheoremDB P9860 · Candidate awaiting source review | null | Source-reported assessment, retrieved 8 September 2026. Frontier Science has not independently reviewed this status.
Open. No signed Lean verification is attached to this problem or its selected resolution.
This record is in the source backlog; its novelty and unresolved status have not passed source review. | null | published | incremental | hard | [
"source-candidate",
"theoremdb"
] | null | null | null | null | [
{
"kind": "website",
"citation": "TheoremDB P9860",
"url": "https://theoremdb.org/statements/P9860",
"note": null,
"sort_order": 0
},
{
"kind": "formalization",
"citation": "Source formal statement (not locally verified)",
"url": "https://theoremdb.org/statements/P9860",
"not... | {
"source_record": "P9860",
"imported_at": "2026-09-08",
"status_as_of": "2026-09-08",
"evidence_label": "Source candidate · not reviewed",
"note": "Full public directory statement. Source lane: backlog; source resolution: open. Source verification claims are attributed only; no local Lean verification has be... | {"created_at":"2026-08-17T20:04:02+00:00","display_standard":{"editorial_review":"submitter_supplied","formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":9860,"public_status":{"evidence":{"hasFormalization":false,"hasInformalProof":false,"hasRecordedWork":false,"solutionRecord":null},"lean":{... | [
"formalization-not-compiled-during-export",
"current-status-unassessed",
"source-full-text-not-present"
] | [] | {
"created_at": "2026-09-08T03:32:15.935Z",
"updated_at": "2026-09-08T03:32:15.935Z",
"published_at": "2026-09-08T03:33:07.477Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"2c441a75-5c97-48ec-a422-a029239f5e63","format_issues":null,"number":6968} |
1.0.0 | problem:theoremdb-p9862 | problem | theoremdb-p9862 | Erdős's triangle-forcing edge density problem | {
"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. Suppose that for every subset $V' \subseteq V(G)$ with $|V'| \geq n/2$, the induced subgraph $G[V']$ has more than $n^2/50$ edges. Must $G$ contain a triangle? | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P9862",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P9862",
"citation": null,
"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_128 :
answer(sorry) ↔ ∀ (V : Type) [Fintype V] (G : SimpleGraph V),
(∀ V' : Set V, 2 * V'.ncard + 1 ≥ Fintype.card V →
50 * (G.induce V').edgeSet.ncard > Fintype.card V ^ 2) → ¬ G.CliqueFree 3 | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem erdos_128 :\n answer(sorry) ↔ ∀ (V : Type) [Fintype V] (G : SimpleGraph V),\n (∀ V' : Set V, 2 * V'.ncard + 1 ≥ Fintype.card V →\n 50 * (G.induce V').edgeSet.ncard > Fintype.card V ^ 2) → ¬ G.CliqueFree 3",
"target": null,
"source_url": "https:... | [] | TheoremDB P9862 · Candidate awaiting source review | null | Source-reported assessment, retrieved 8 September 2026. Frontier Science has not independently reviewed this status.
Open. No signed Lean verification is attached to this problem or its selected resolution.
This record is in the source backlog; its novelty and unresolved status have not passed source review. | null | published | incremental | hard | [
"source-candidate",
"theoremdb"
] | null | null | null | null | [
{
"kind": "website",
"citation": "TheoremDB P9862",
"url": "https://theoremdb.org/statements/P9862",
"note": null,
"sort_order": 0
},
{
"kind": "formalization",
"citation": "Source formal statement (not locally verified)",
"url": "https://theoremdb.org/statements/P9862",
"not... | {
"source_record": "P9862",
"imported_at": "2026-09-08",
"status_as_of": "2026-09-08",
"evidence_label": "Source candidate · not reviewed",
"note": "Full public directory statement. Source lane: backlog; source resolution: open. Source verification claims are attributed only; no local Lean verification has be... | {"created_at":"2026-08-17T20:04:03+00:00","display_standard":{"editorial_review":"submitter_supplied","formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":9862,"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": "2026-09-08T03:32:56.622Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"93ba2f77-eb7e-4a0c-b53c-32af4d037a5e","format_issues":null,"number":4780} |
1.0.0 | problem:theoremdb-p9864 | problem | theoremdb-p9864 | Asymptotic upper bound for sums of $k$ $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 integers $k \geq 1$, $m \geq 1$, and $x \geq 1$, let $f_{k,m}(x)$ denote the number of integers $n \leq x$ that can be expressed as the sum of $m$ nonnegative $k$th powers; that is, $n = v_1^k + v_2^k + \cdots + v_m^k$ for some nonnegative integers $v_1, \ldots, v_m$. For each $k > 2$, does $f_{k,k}(x) = o(x)$ as $... | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P9864",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P9864",
"citation": null,
"doi": null,
"arxiv_id": null,
"license": "Selected packets label their content CC0-1.0; original references retain their own terms",
"notes": "Selected problem... | theorem erdos_323.variants.k_gt_2 :
answer(sorry) ↔ ∀ k > 2, (fun (x : ℕ) ↦ (f k k x : ℝ)) =o[atTop] (fun (x : ℕ) ↦ (x : ℝ)) | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem erdos_323.variants.k_gt_2 :\n answer(sorry) ↔ ∀ k > 2, (fun (x : ℕ) ↦ (f k k x : ℝ)) =o[atTop] (fun (x : ℕ) ↦ (x : ℝ))",
"target": null,
"source_url": "https://theoremdb.org/statements/P9864",
"format": "declaration-or-fragment",
"sha256": "07a47a5... | [] | TheoremDB P9864 · Candidate awaiting source review | null | Source-reported assessment, retrieved 8 September 2026. Frontier Science has not independently reviewed this status.
Open. No signed Lean verification is attached to this problem or its selected resolution.
This record is in the source backlog; its novelty and unresolved status have not passed source review. | null | published | incremental | hard | [
"source-candidate",
"theoremdb"
] | null | null | null | null | [
{
"kind": "website",
"citation": "TheoremDB P9864",
"url": "https://theoremdb.org/statements/P9864",
"note": null,
"sort_order": 0
},
{
"kind": "formalization",
"citation": "Source formal statement (not locally verified)",
"url": "https://theoremdb.org/statements/P9864",
"not... | {
"source_record": "P9864",
"imported_at": "2026-09-08",
"status_as_of": "2026-09-08",
"evidence_label": "Source candidate · not reviewed",
"note": "Full public directory statement. Source lane: backlog; source resolution: open. Source verification claims are attributed only; no local Lean verification has be... | {"created_at":"2026-08-17T20:04:04+00:00","display_standard":{"editorial_review":"submitter_supplied","formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":9864,"public_status":{"evidence":{"hasFormalization":false,"hasInformalProof":false,"hasRecordedWork":false,"solutionRecord":null},"lean":{... | [
"formalization-contains-placeholder",
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"current-status-unassessed",
"source-full-text-not-present"
] | [] | {
"created_at": "2026-09-08T03:32:15.935Z",
"updated_at": "2026-09-08T03:32:15.935Z",
"published_at": "2026-09-08T03:32:50.437Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"bedb08c2-28c4-4b6c-a584-f4901956414c","format_issues":null,"number":3420} |
1.0.0 | problem:theoremdb-p9866 | problem | theoremdb-p9866 | Existence of a SIC-POVM in Dimension 75 | {
"main": "math",
"sub": "algebra",
"slug": "algebra",
"name": "Algebra",
"original_discipline": "mathematics"
} | [] | [] | Let $\mathbb{C}^{75}$ denote the 75-dimensional complex Hilbert space with its standard inner product. A state vector in $\mathbb{C}^{75}$ is a unit vector, i.e., a vector $\psi$ with $\|\psi\| = 1$. For two state vectors $\varphi, \psi$, their squared overlap is the quantity $|\langle \varphi, \psi \rangle|^2$, where ... | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P9866",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P9866",
"citation": null,
"doi": null,
"arxiv_id": null,
"license": "Selected packets label their content CC0-1.0; original references retain their own terms",
"notes": "Selected problem... | theorem hasSICPOVM_75 : answer(sorry) ↔ HasSICPOVM 75 | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem hasSICPOVM_75 : answer(sorry) ↔ HasSICPOVM 75",
"target": null,
"source_url": "https://theoremdb.org/statements/P9866",
"format": "declaration-or-fragment",
"sha256": "ff750cbfba31672c0fca7faa62e965221ce7d93f7edab3db32cd75326dd28783",
"license": "... | [] | TheoremDB P9866 · Candidate awaiting source review | null | Source-reported assessment, retrieved 8 September 2026. Frontier Science has not independently reviewed this status.
Open. No signed Lean verification is attached to this problem or its selected resolution.
This record is in the source backlog; its novelty and unresolved status have not passed source review. | null | published | incremental | hard | [
"source-candidate",
"theoremdb"
] | null | null | null | null | [
{
"kind": "website",
"citation": "TheoremDB P9866",
"url": "https://theoremdb.org/statements/P9866",
"note": null,
"sort_order": 0
},
{
"kind": "formalization",
"citation": "Source formal statement (not locally verified)",
"url": "https://theoremdb.org/statements/P9866",
"not... | {
"source_record": "P9866",
"imported_at": "2026-09-08",
"status_as_of": "2026-09-08",
"evidence_label": "Source candidate · not reviewed",
"note": "Full public directory statement. Source lane: backlog; source resolution: open. Source verification claims are attributed only; no local Lean verification has be... | {"created_at":"2026-08-17T20:04:06+00:00","display_standard":{"editorial_review":"submitter_supplied","formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":9866,"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": "2026-09-08T03:32:58.537Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"46e24813-340c-462b-9a11-4499265c5327","format_issues":null,"number":5203} |
1.0.0 | problem:theoremdb-p9868 | problem | theoremdb-p9868 | Erdős–Hajnal problem on countable chromatic number of small induced subgraphs | {
"main": "math",
"sub": "combinatorics",
"slug": "combinatorics",
"name": "Combinatorics",
"original_discipline": "mathematics"
} | [] | [
{
"code": "05",
"basis": "community-map",
"evidence": "Inferred from community field: combinatorics"
}
] | Let $\omega$ be an ordinal. Determine whether there exists a simple graph $G$ with vertex set $V$ such that the cardinality of $V$ is $\aleph_{\omega+1}$, the chromatic number of $G$ is $\aleph_1$, and for every subset $W \subseteq V$ with cardinality $\aleph_\omega$, the chromatic number of the induced subgraph on $W$... | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P9868",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P9868",
"citation": null,
"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.parts.ii :
answer(sorry) ↔ ∀ (ω : Ordinal),
∃ (V : Type u) (G : SimpleGraph V), #V = ℵ_ (ω + 1) ∧ G.chromaticCardinal = ℵ₁ ∧
∀ (W : Set V) (_ : #W = ℵ_ ω), (G.induce W).chromaticCardinal ≤ ℵ₀ | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem erdos_918.parts.ii :\n answer(sorry) ↔ ∀ (ω : Ordinal),\n ∃ (V : Type u) (G : SimpleGraph V), #V = ℵ_ (ω + 1) ∧ G.chromaticCardinal = ℵ₁ ∧\n ∀ (W : Set V) (_ : #W = ℵ_ ω), (G.induce W).chromaticCardinal ≤ ℵ₀",
"target": null,
"source_url": "https:/... | [] | TheoremDB P9868 · Candidate awaiting source review | null | Source-reported assessment, retrieved 8 September 2026. Frontier Science has not independently reviewed this status.
Open. No signed Lean verification is attached to this problem or its selected resolution.
This record is in the source backlog; its novelty and unresolved status have not passed source review. | null | published | incremental | hard | [
"source-candidate",
"theoremdb"
] | null | null | null | null | [
{
"kind": "website",
"citation": "TheoremDB P9868",
"url": "https://theoremdb.org/statements/P9868",
"note": null,
"sort_order": 0
},
{
"kind": "formalization",
"citation": "Source formal statement (not locally verified)",
"url": "https://theoremdb.org/statements/P9868",
"not... | {
"source_record": "P9868",
"imported_at": "2026-09-08",
"status_as_of": "2026-09-08",
"evidence_label": "Source candidate · not reviewed",
"note": "Full public directory statement. Source lane: backlog; source resolution: open. Source verification claims are attributed only; no local Lean verification has be... | {"created_at":"2026-08-17T20:04:07+00:00","display_standard":{"editorial_review":"submitter_supplied","formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":9868,"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": "2026-09-08T03:32:56.678Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"9cd38f8e-4c38-4c0f-86e4-cfed332bfdb3","format_issues":null,"number":4791} |
1.0.0 | problem:theoremdb-p9870 | problem | theoremdb-p9870 | Improving the lower bound in the minimum overlap problem | {
"main": "math",
"sub": "combinatorics",
"slug": "combinatorics",
"name": "Combinatorics",
"original_discipline": "mathematics"
} | [] | [
{
"code": "05",
"basis": "community-map",
"evidence": "Inferred from community field: combinatorics"
}
] | For each integer $n\ge 1$, partition $\{1,\ldots,2n\}$ into disjoint sets $A$ and $B$ with $|A|=|B|=n$. Define
$$
r_{A,B}(k)=\#\{(a,b)\in A\times B:a-b=k\},\qquad M(n)=\min_{A,B}\max_{k\in\mathbb Z}r_{A,B}(k).
$$
Find an explicit real constant $c>0.379005$ and prove that
$$
c\le \liminf_{n\to\infty}\frac{M(n)}{n}... | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P9870",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P9870",
"citation": null,
"doi": null,
"arxiv_id": null,
"license": "Selected packets label their content CC0-1.0; original references retain their own terms",
"notes": "Selected problem... | theorem erdos_36.variants.lower:
∃ (c : ℝ), 0.379005 < c ∧ c ≤ atTop.liminf MinOverlapQuotient ∧ c = answer(sorry) | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem erdos_36.variants.lower:\n ∃ (c : ℝ), 0.379005 < c ∧ c ≤ atTop.liminf MinOverlapQuotient ∧ c = answer(sorry)",
"target": null,
"source_url": "https://theoremdb.org/statements/P9870",
"format": "declaration-or-fragment",
"sha256": "3bfa87969cca64e41... | [] | TheoremDB P9870 · Candidate awaiting source review | null | Source-reported assessment, retrieved 8 September 2026. Frontier Science has not independently reviewed this status.
Open. No signed Lean verification is attached to this problem or its selected resolution.
This record is in the source backlog; its novelty and unresolved status have not passed source review. | null | published | incremental | hard | [
"source-candidate",
"theoremdb"
] | null | null | null | null | [
{
"kind": "website",
"citation": "TheoremDB P9870",
"url": "https://theoremdb.org/statements/P9870",
"note": null,
"sort_order": 0
},
{
"kind": "formalization",
"citation": "Source formal statement (not locally verified)",
"url": "https://theoremdb.org/statements/P9870",
"not... | {
"source_record": "P9870",
"imported_at": "2026-09-08",
"status_as_of": "2026-09-08",
"evidence_label": "Source candidate · not reviewed",
"note": "Full public directory statement. Source lane: backlog; source resolution: open. Source verification claims are attributed only; no local Lean verification has be... | {"created_at":"2026-08-17T20:04:09+00:00","display_standard":{"editorial_review":"submitter_supplied","formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":9870,"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": "2026-09-08T03:32:59.680Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"ce241fd5-860f-4b91-842f-52430feb3808","format_issues":null,"number":5447} |
1.0.0 | problem:theoremdb-p9872 | problem | theoremdb-p9872 | Irrationality of subseries of the reciprocals of Mersenne numbers | {
"main": "math",
"sub": "number theory",
"slug": "number-theory",
"name": "Number theory",
"original_discipline": "mathematics"
} | [] | [
{
"code": "11",
"basis": "community-map",
"evidence": "Inferred from community field: number-theory"
}
] | Let $\mathbb{N}$ denote the set of positive integers. For an infinite set $A \subseteq \mathbb{N}$, consider the series
$$
\sum_{n \in A} \frac{1}{2^n - 1}.
$$
Is this sum irrational for every infinite set $A \subseteq \mathbb{N}$? | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P9872",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P9872",
"citation": null,
"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_257 : answer(sorry) ↔ ∀ (A : Set ℕ), A.Infinite →
Irrational (∑' n : A, (1 : ℝ) / (2 ^ n.1 - 1)) | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem erdos_257 : answer(sorry) ↔ ∀ (A : Set ℕ), A.Infinite →\n Irrational (∑' n : A, (1 : ℝ) / (2 ^ n.1 - 1))",
"target": null,
"source_url": "https://theoremdb.org/statements/P9872",
"format": "declaration-or-fragment",
"sha256": "174d34a71f52ab89ca2a5... | [] | TheoremDB P9872 · Candidate awaiting source review | null | Source-reported assessment, retrieved 8 September 2026. Frontier Science has not independently reviewed this status.
Open. No signed Lean verification is attached to this problem or its selected resolution.
This record is in the source backlog; its novelty and unresolved status have not passed source review. | null | published | incremental | hard | [
"source-candidate",
"theoremdb"
] | null | null | null | null | [
{
"kind": "website",
"citation": "TheoremDB P9872",
"url": "https://theoremdb.org/statements/P9872",
"note": null,
"sort_order": 0
},
{
"kind": "formalization",
"citation": "Source formal statement (not locally verified)",
"url": "https://theoremdb.org/statements/P9872",
"not... | {
"source_record": "P9872",
"imported_at": "2026-09-08",
"status_as_of": "2026-09-08",
"evidence_label": "Source candidate · not reviewed",
"note": "Full public directory statement. Source lane: backlog; source resolution: open. Source verification claims are attributed only; no local Lean verification has be... | {"created_at":"2026-08-17T20:04:09+00:00","display_standard":{"editorial_review":"submitter_supplied","formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":9872,"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": "2026-09-08T03:33:01.408Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"fd1e5b2f-45e4-459b-b40c-341cbabcbedf","format_issues":null,"number":5724} |
1.0.0 | problem:theoremdb-p9874 | problem | theoremdb-p9874 | Asymptotic constant in Erdős's divisor sum problem for irreducible 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 $\tau(m)$ denote the number of positive divisors of a positive integer $m$. For a polynomial $f \in \mathbb{Z}[X]$ and a real number $x \geq 0$, define
$$
S_f(x) = \sum_{n = 0}^{\lfloor x \rfloor} \tau(\lfloor f(n) \rfloor),
$$
where $\lfloor \cdot \rfloor$ denotes the floor function, and we only sum over values ... | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P9874",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P9874",
"citation": null,
"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_975 : answer(sorry) ↔
∀ f : ℤ[X], f.natDegree ≠ 0 → Irreducible f → (∀ᶠ n in atTop, 1 ≤ f.eval n) →
∃ c > (0 : ℝ), Tendsto (fun x ↦ Erdos975Sum f x / (x * log x)) atTop (𝓝 c) | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem erdos_975 : answer(sorry) ↔\n ∀ f : ℤ[X], f.natDegree ≠ 0 → Irreducible f → (∀ᶠ n in atTop, 1 ≤ f.eval n) →\n ∃ c > (0 : ℝ), Tendsto (fun x ↦ Erdos975Sum f x / (x * log x)) atTop (𝓝 c)",
"target": null,
"source_url": "https://theoremdb.org/statements/P... | [] | TheoremDB P9874 · Candidate awaiting source review | null | Source-reported assessment, retrieved 8 September 2026. Frontier Science has not independently reviewed this status.
Open. No signed Lean verification is attached to this problem or its selected resolution.
This record is in the source backlog; its novelty and unresolved status have not passed source review. | null | published | incremental | hard | [
"source-candidate",
"theoremdb"
] | null | null | null | null | [
{
"kind": "website",
"citation": "TheoremDB P9874",
"url": "https://theoremdb.org/statements/P9874",
"note": null,
"sort_order": 0
},
{
"kind": "formalization",
"citation": "Source formal statement (not locally verified)",
"url": "https://theoremdb.org/statements/P9874",
"not... | {
"source_record": "P9874",
"imported_at": "2026-09-08",
"status_as_of": "2026-09-08",
"evidence_label": "Source candidate · not reviewed",
"note": "Full public directory statement. Source lane: backlog; source resolution: open. Source verification claims are attributed only; no local Lean verification has be... | {"created_at":"2026-08-17T20:04:10+00:00","display_standard":{"editorial_review":"submitter_supplied","formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":9874,"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": "2026-09-08T03:32:49.976Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"d4d64480-21b6-47c9-958e-f91ff58112d5","format_issues":null,"number":3325} |
1.0.0 | problem:theoremdb-p9876 | problem | theoremdb-p9876 | Asymptotic growth of the Erdős–Selfridge 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"
}
] | Let $A : \mathbb{N} \to \mathbb{N}$ be a sequence satisfying the following conditions: $A(0) = 1$; the sequence is strictly increasing; and for every $j \in \mathbb{N}$, the value $A(j+1)$ is the least natural number that is greater than $A(j)$ and cannot be expressed as a sum of consecutive terms from among $A(0), A(1... | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P9876",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P9876",
"citation": null,
"doi": null,
"arxiv_id": null,
"license": "Selected packets label their content CC0-1.0; original references retain their own terms",
"notes": "Selected problem... | theorem erdos_359.variants.isGoodFor_1_asymptotic (A : ℕ → ℕ) (hA : IsGoodFor A 1) :
(fun k ↦ (A k : ℝ)) ~[atTop] (fun k ↦ k * (k : ℝ).log / (k : ℝ).log.log) | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem erdos_359.variants.isGoodFor_1_asymptotic (A : ℕ → ℕ) (hA : IsGoodFor A 1) :\n (fun k ↦ (A k : ℝ)) ~[atTop] (fun k ↦ k * (k : ℝ).log / (k : ℝ).log.log)",
"target": null,
"source_url": "https://theoremdb.org/statements/P9876",
"format": "declaration-or-... | [] | TheoremDB P9876 · Candidate awaiting source review | null | Source-reported assessment, retrieved 8 September 2026. Frontier Science has not independently reviewed this status.
Open. No signed Lean verification is attached to this problem or its selected resolution.
This record is in the source backlog; its novelty and unresolved status have not passed source review. | null | published | incremental | hard | [
"source-candidate",
"theoremdb"
] | null | null | null | null | [
{
"kind": "website",
"citation": "TheoremDB P9876",
"url": "https://theoremdb.org/statements/P9876",
"note": null,
"sort_order": 0
},
{
"kind": "formalization",
"citation": "Source formal statement (not locally verified)",
"url": "https://theoremdb.org/statements/P9876",
"not... | {
"source_record": "P9876",
"imported_at": "2026-09-08",
"status_as_of": "2026-09-08",
"evidence_label": "Source candidate · not reviewed",
"note": "Full public directory statement. Source lane: backlog; source resolution: open. Source verification claims are attributed only; no local Lean verification has be... | {"created_at":"2026-08-17T20:04:11+00:00","display_standard":{"editorial_review":"submitter_supplied","formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":9876,"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": "2026-09-08T03:32:50.162Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"28f3bbfd-3132-4a68-90a2-4f1b9e134f4a","format_issues":null,"number":3364} |
1.0.0 | problem:theoremdb-p9878 | problem | theoremdb-p9878 | Erdős's Two Powers of Two Variant | {
"main": "math",
"sub": "number theory",
"slug": "number-theory",
"name": "Number theory",
"original_discipline": "mathematics"
} | [] | [
{
"code": "11",
"basis": "community-map",
"evidence": "Inferred from community field: number-theory"
}
] | Let $n$ be an odd integer greater than 1. Determine whether there always exist natural numbers $k$, $l$, and $m$ such that $k$ is squarefree and $n = k + 2^{l} + 2^{m}$. | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P9878",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P9878",
"citation": null,
"doi": null,
"arxiv_id": null,
"license": "Selected packets label their content CC0-1.0; original references retain their own terms",
"notes": "Selected problem... | theorem erdos_11.variants.two_pow_two (n : ℕ) (hn : Odd n) (hn' : 1 < n) :
∃ k l m : ℕ , Squarefree k ∧ n = k + 2^l + 2^m | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem erdos_11.variants.two_pow_two (n : ℕ) (hn : Odd n) (hn' : 1 < n) :\n ∃ k l m : ℕ , Squarefree k ∧ n = k + 2^l + 2^m",
"target": null,
"source_url": "https://theoremdb.org/statements/P9878",
"format": "declaration-or-fragment",
"sha256": "e2d6d632dc... | [] | TheoremDB P9878 · Candidate awaiting source review | null | Source-reported assessment, retrieved 8 September 2026. Frontier Science has not independently reviewed this status.
Open. No signed Lean verification is attached to this problem or its selected resolution.
This record is in the source backlog; its novelty and unresolved status have not passed source review. | null | published | incremental | hard | [
"source-candidate",
"theoremdb"
] | null | null | null | null | [
{
"kind": "website",
"citation": "TheoremDB P9878",
"url": "https://theoremdb.org/statements/P9878",
"note": null,
"sort_order": 0
},
{
"kind": "formalization",
"citation": "Source formal statement (not locally verified)",
"url": "https://theoremdb.org/statements/P9878",
"not... | {
"source_record": "P9878",
"imported_at": "2026-09-08",
"status_as_of": "2026-09-08",
"evidence_label": "Source candidate · not reviewed",
"note": "Full public directory statement. Source lane: backlog; source resolution: open. Source verification claims are attributed only; no local Lean verification has be... | {"created_at":"2026-08-17T20:04:11+00:00","display_standard":{"editorial_review":"submitter_supplied","formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":9878,"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": "2026-09-08T03:32:56.626Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"a3ad50f0-2cdb-4d0c-9caa-0b98b4261c92","format_issues":null,"number":4781} |
1.0.0 | problem:theoremdb-p9880 | problem | theoremdb-p9880 | The Ramsey number $R(5,5)$ | {
"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 $k$ and $\ell$, the graph Ramsey number $R(k,\ell)$ is the smallest natural number $n$ such that every simple graph on $n$ vertices contains either a clique of size $k$ or an independent set of size $\ell$ (equivalently, its complement contains a clique of size $\ell$). It is known that $43 \le R(... | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P9880",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P9880",
"citation": null,
"doi": null,
"arxiv_id": null,
"license": "Selected packets label their content CC0-1.0; original references retain their own terms",
"notes": "Selected problem... | theorem ramsey_number_five_five :
R(5, 5) = answer(sorry) | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem ramsey_number_five_five :\n R(5, 5) = answer(sorry)",
"target": null,
"source_url": "https://theoremdb.org/statements/P9880",
"format": "declaration-or-fragment",
"sha256": "2b66ef873cb553363b0703a029a9b0e0f77b65d550f15068f093f89b4d805d97",
"li... | [] | TheoremDB P9880 · Candidate awaiting source review | null | Source-reported assessment, retrieved 8 September 2026. Frontier Science has not independently reviewed this status.
Open. No signed Lean verification is attached to this problem or its selected resolution.
This record is in the source backlog; its novelty and unresolved status have not passed source review. | null | published | incremental | hard | [
"source-candidate",
"theoremdb"
] | null | null | null | null | [
{
"kind": "website",
"citation": "TheoremDB P9880",
"url": "https://theoremdb.org/statements/P9880",
"note": null,
"sort_order": 0
},
{
"kind": "formalization",
"citation": "Source formal statement (not locally verified)",
"url": "https://theoremdb.org/statements/P9880",
"not... | {
"source_record": "P9880",
"imported_at": "2026-09-08",
"status_as_of": "2026-09-08",
"evidence_label": "Source candidate · not reviewed",
"note": "Full public directory statement. Source lane: backlog; source resolution: open. Source verification claims are attributed only; no local Lean verification has be... | {"created_at":"2026-08-17T20:04:12+00:00","display_standard":{"editorial_review":"submitter_supplied","formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":9880,"public_status":{"evidence":{"hasFormalization":false,"hasInformalProof":false,"hasRecordedWork":false,"solutionRecord":null},"lean":{... | [
"formalization-contains-placeholder",
"formalization-not-compiled-during-export",
"current-status-unassessed",
"source-full-text-not-present"
] | [] | {
"created_at": "2026-09-08T03:32:15.935Z",
"updated_at": "2026-09-08T03:32:15.935Z",
"published_at": "2026-09-08T03:33:06.728Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"ae90efa0-3995-455b-9097-8548790aae01","format_issues":null,"number":6805} |
1.0.0 | problem:theoremdb-p9882 | problem | theoremdb-p9882 | Erdős Problem 14, Part (i): Lower Bound for Non-Uniquely 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"
}
] | Let $A$ be a subset of the natural numbers $\mathbb{N}$. Define $B$ to be the set of integers that can be represented in exactly one way as the sum of two elements from $A$. For each positive integer $N$, let $C(N)$ denote the number of integers in $\{1, 2, \ldots, N\}$ that are not in $B$ (that is, integers that are e... | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P9882",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P9882",
"citation": null,
"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_14.parts.i :
answer(sorry) ↔ ∀ A, ∀ ε > 0, nonUniqueSumCount A ≫ almostSquareRoot ε | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem erdos_14.parts.i :\n answer(sorry) ↔ ∀ A, ∀ ε > 0, nonUniqueSumCount A ≫ almostSquareRoot ε",
"target": null,
"source_url": "https://theoremdb.org/statements/P9882",
"format": "declaration-or-fragment",
"sha256": "c7cc809e1ef60d19139960e61878656ac3... | [] | TheoremDB P9882 · Candidate awaiting source review | null | Source-reported assessment, retrieved 8 September 2026. Frontier Science has not independently reviewed this status.
Open. No signed Lean verification is attached to this problem or its selected resolution.
This record is in the source backlog; its novelty and unresolved status have not passed source review. | null | published | incremental | hard | [
"source-candidate",
"theoremdb"
] | null | null | null | null | [
{
"kind": "website",
"citation": "TheoremDB P9882",
"url": "https://theoremdb.org/statements/P9882",
"note": null,
"sort_order": 0
},
{
"kind": "formalization",
"citation": "Source formal statement (not locally verified)",
"url": "https://theoremdb.org/statements/P9882",
"not... | {
"source_record": "P9882",
"imported_at": "2026-09-08",
"status_as_of": "2026-09-08",
"evidence_label": "Source candidate · not reviewed",
"note": "Full public directory statement. Source lane: backlog; source resolution: open. Source verification claims are attributed only; no local Lean verification has be... | {"created_at":"2026-08-17T20:04:13+00:00","display_standard":{"formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":9882,"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": "2026-09-08T03:32:55.148Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"6eeaccda-42e9-4140-ae6a-a7dd6fb7efb7","format_issues":null,"number":4457} |
1.0.0 | problem:theoremdb-p9884 | problem | theoremdb-p9884 | Strong sunflower-type bound for sets with bounded total prime factor count | {
"main": "math",
"sub": "combinatorics",
"slug": "combinatorics",
"name": "Combinatorics",
"original_discipline": "mathematics"
} | [] | [
{
"code": "05",
"basis": "community-map",
"evidence": "Inferred from community field: combinatorics"
}
] | Let $r \geq 3$ be an integer. For a positive integer $n$, let $\Omega(n)$ denote the total number of prime factors of $n$ counted with multiplicity. For a finite set $A$ of positive integers and a positive integer $k$, say that $A$ has all elements with $\Omega$-value $k$ if every element $a \in A$ satisfies $a \geq 1$... | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P9884",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P9884",
"citation": null,
"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.sunflower_strong {r : ℕ} (hr : 3 ≤ r) :
∃ c_r > (0 : ℝ),
∀ k : ℕ, ∀ A : Finset ℕ,
AllBigOmega k A →
NoConstantPairwiseGcdCoprimeSubsets r A →
(A.card : ℝ) ≤ c_r ^ k | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem erdos_535.variants.sunflower_strong {r : ℕ} (hr : 3 ≤ r) :\n ∃ c_r > (0 : ℝ),\n ∀ k : ℕ, ∀ A : Finset ℕ,\n AllBigOmega k A →\n NoConstantPairwiseGcdCoprimeSubsets r A →\n (A.card : ℝ) ≤ c_r ^ k",
"target": null,
"source_url": "htt... | [] | TheoremDB P9884 · Candidate awaiting source review | null | Source-reported assessment, retrieved 8 September 2026. Frontier Science has not independently reviewed this status.
Open. No signed Lean verification is attached to this problem or its selected resolution.
This record is in the source backlog; its novelty and unresolved status have not passed source review. | null | published | incremental | hard | [
"source-candidate",
"theoremdb"
] | null | null | null | null | [
{
"kind": "website",
"citation": "TheoremDB P9884",
"url": "https://theoremdb.org/statements/P9884",
"note": null,
"sort_order": 0
},
{
"kind": "formalization",
"citation": "Source formal statement (not locally verified)",
"url": "https://theoremdb.org/statements/P9884",
"not... | {
"source_record": "P9884",
"imported_at": "2026-09-08",
"status_as_of": "2026-09-08",
"evidence_label": "Source candidate · not reviewed",
"note": "Full public directory statement. Source lane: backlog; source resolution: open. Source verification claims are attributed only; no local Lean verification has be... | {"created_at":"2026-08-17T20:04:15+00:00","display_standard":{"editorial_review":"submitter_supplied","formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":9884,"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": "2026-09-08T03:33:06.122Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"bfd8ecd4-063e-4e1f-9399-0eb5a5e017b8","format_issues":null,"number":6675} |
1.0.0 | problem:theoremdb-p9886 | problem | theoremdb-p9886 | The Elliott–Halberstam conjecture on the distribution of primes in arithmetic progressions | {
"main": "math",
"sub": "number theory",
"slug": "number-theory",
"name": "Number theory",
"original_discipline": "mathematics"
} | [] | [
{
"code": "11",
"basis": "community-map",
"evidence": "Inferred from community field: number-theory"
}
] | For a positive integer $x$ and a positive integer $q$, let $\pi(x)$ denote the prime-counting function, let $\varphi(q)$ denote Euler's totient function, and let $\pi(x; q, a)$ denote the number of primes $p \le x$ such that $p \equiv a \pmod q$. Define the error term
$$
E(x; q) = \max_{\substack{1 \le a \le q \\ \gcd... | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P9886",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P9886",
"citation": null,
"doi": null,
"arxiv_id": null,
"license": "Selected packets label their content CC0-1.0; original references retain their own terms",
"notes": "Selected problem... | theorem elliott_halberstam (θ : ℝ) (hθ : θ < 1) (A : ℝ) (hA : 0 < A) :
∃ C > (0 : ℝ), ∀ x : ℕ, 2 < x →
∑ q ∈ Finset.Icc 1 ⌊(x : ℝ) ^ θ⌋₊, E x q ≤ C * x / Real.log x ^ A | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem elliott_halberstam (θ : ℝ) (hθ : θ < 1) (A : ℝ) (hA : 0 < A) :\n ∃ C > (0 : ℝ), ∀ x : ℕ, 2 < x →\n ∑ q ∈ Finset.Icc 1 ⌊(x : ℝ) ^ θ⌋₊, E x q ≤ C * x / Real.log x ^ A",
"target": null,
"source_url": "https://theoremdb.org/statements/P9886",
"format"... | [] | TheoremDB P9886 · Candidate awaiting source review | null | Source-reported assessment, retrieved 8 September 2026. Frontier Science has not independently reviewed this status.
Open. No signed Lean verification is attached to this problem or its selected resolution.
This record is in the source backlog; its novelty and unresolved status have not passed source review. | null | published | incremental | hard | [
"source-candidate",
"theoremdb"
] | null | null | null | null | [
{
"kind": "website",
"citation": "TheoremDB P9886",
"url": "https://theoremdb.org/statements/P9886",
"note": null,
"sort_order": 0
},
{
"kind": "formalization",
"citation": "Source formal statement (not locally verified)",
"url": "https://theoremdb.org/statements/P9886",
"not... | {
"source_record": "P9886",
"imported_at": "2026-09-08",
"status_as_of": "2026-09-08",
"evidence_label": "Source candidate · not reviewed",
"note": "Full public directory statement. Source lane: backlog; source resolution: open. Source verification claims are attributed only; no local Lean verification has be... | {"created_at":"2026-08-17T20:04:16+00:00","display_standard":{"formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":9886,"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": "2026-09-08T03:33:06.458Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"d135817f-8833-49c7-9087-c775fbf15dec","format_issues":null,"number":6747} |
1.0.0 | problem:theoremdb-p9888 | problem | theoremdb-p9888 | The value of the minimal covering density function at radius two | {
"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 $x \in \mathbb{F}_2^n$, the Hamming norm $\|x\|$ is the number of nonzero coordinates of $x$. For natural numbers $n$ and $r$, the Hamming ball $H_n(r)$ is the set $\{x \in \mathbb{F}_2^n : \|x\| \leq r\}$. A linear subsp... | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P9888",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P9888",
"citation": null,
"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.f_two_eq_one : answer(sorry) ↔ f 2 = 1 | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem green_40.f_two_eq_one : answer(sorry) ↔ f 2 = 1",
"target": null,
"source_url": "https://theoremdb.org/statements/P9888",
"format": "declaration-or-fragment",
"sha256": "805267b4ece7046637d3518b168318797e02d371a1d66f3de4e057ddcf19131b",
"license":... | [] | TheoremDB P9888 · Candidate awaiting source review | null | Source-reported assessment, retrieved 8 September 2026. Frontier Science has not independently reviewed this status.
Open. No signed Lean verification is attached to this problem or its selected resolution.
This record is in the source backlog; its novelty and unresolved status have not passed source review. | null | published | incremental | hard | [
"source-candidate",
"theoremdb"
] | null | null | null | null | [
{
"kind": "website",
"citation": "TheoremDB P9888",
"url": "https://theoremdb.org/statements/P9888",
"note": null,
"sort_order": 0
},
{
"kind": "formalization",
"citation": "Source formal statement (not locally verified)",
"url": "https://theoremdb.org/statements/P9888",
"not... | {
"source_record": "P9888",
"imported_at": "2026-09-08",
"status_as_of": "2026-09-08",
"evidence_label": "Source candidate · not reviewed",
"note": "Full public directory statement. Source lane: backlog; source resolution: open. Source verification claims are attributed only; no local Lean verification has be... | {"created_at":"2026-08-17T20:04:17+00:00","display_standard":{"editorial_review":"submitter_supplied","formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":9888,"public_status":{"evidence":{"hasFormalization":false,"hasInformalProof":false,"hasRecordedWork":false,"solutionRecord":null},"lean":{... | [
"formalization-contains-placeholder",
"formalization-not-compiled-during-export",
"current-status-unassessed",
"source-full-text-not-present"
] | [] | {
"created_at": "2026-09-08T03:32:15.935Z",
"updated_at": "2026-09-08T03:32:15.935Z",
"published_at": "2026-09-08T03:33:06.848Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"10ab9b15-66ef-4178-8a7a-a675841995a0","format_issues":null,"number":6831} |
1.0.0 | problem:theoremdb-p9890 | problem | theoremdb-p9890 | Asymptotic supremum of minimal divisor-sum representations | {
"main": "math",
"sub": "number theory",
"slug": "number-theory",
"name": "Number theory",
"original_discipline": "mathematics"
} | [] | [
{
"code": "11",
"basis": "community-map",
"evidence": "Inferred from community field: number-theory"
}
] | For each positive integer $n$, define $f(n)$ to be the smallest positive integer $m$ such that $n$ can be written as the sum of the $k$ smallest divisors of $m$ for some positive integer $k$. Does there exist a set $A \subseteq \mathbb{N}$ of natural density $1$ such that the limit superior of $f(n)/n$ as $n \to \infty... | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P9890",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P9890",
"citation": null,
"doi": null,
"arxiv_id": null,
"license": "Selected packets label their content CC0-1.0; original references retain their own terms",
"notes": "Selected problem... | theorem erdos_1054.parts.iii : answer(sorry) ↔ ∃ (A : Set ℕ), A.HasDensity 1 ∧
atTop.limsup (fun n ↦ (f n : EReal) / n) = ⊤ | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem erdos_1054.parts.iii : answer(sorry) ↔ ∃ (A : Set ℕ), A.HasDensity 1 ∧\n atTop.limsup (fun n ↦ (f n : EReal) / n) = ⊤",
"target": null,
"source_url": "https://theoremdb.org/statements/P9890",
"format": "declaration-or-fragment",
"sha256": "797e08c6... | [] | TheoremDB P9890 · Candidate awaiting source review | null | Source-reported assessment, retrieved 8 September 2026. Frontier Science has not independently reviewed this status.
Open. No signed Lean verification is attached to this problem or its selected resolution.
This record is in the source backlog; its novelty and unresolved status have not passed source review. | null | published | incremental | hard | [
"source-candidate",
"theoremdb"
] | null | null | null | null | [
{
"kind": "website",
"citation": "TheoremDB P9890",
"url": "https://theoremdb.org/statements/P9890",
"note": null,
"sort_order": 0
},
{
"kind": "formalization",
"citation": "Source formal statement (not locally verified)",
"url": "https://theoremdb.org/statements/P9890",
"not... | {
"source_record": "P9890",
"imported_at": "2026-09-08",
"status_as_of": "2026-09-08",
"evidence_label": "Source candidate · not reviewed",
"note": "Full public directory statement. Source lane: backlog; source resolution: open. Source verification claims are attributed only; no local Lean verification has be... | {"created_at":"2026-08-17T20:04:17+00:00","display_standard":{"editorial_review":"submitter_supplied","formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":9890,"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": "2026-09-08T03:32:50.405Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"f3dfbea9-d2ce-4a10-a0e6-c0d474110096","format_issues":null,"number":3413} |
1.0.0 | problem:theoremdb-p9892 | problem | theoremdb-p9892 | Partition of the natural numbers into two permutation-avoidant sets | {
"main": "math",
"sub": "combinatorics",
"slug": "combinatorics",
"name": "Combinatorics",
"original_discipline": "mathematics"
} | [] | [
{
"code": "05",
"basis": "community-map",
"evidence": "Inferred from community field: combinatorics"
}
] | Determine whether there exists a partition of the natural numbers $\mathbb{N}$ into two complementary sets $A$ and $B$ such that each set admits a bijection from $\mathbb{N}$ whose image sequence, when viewed as a sequence of natural numbers in the inherited order, contains no monotone 3-term arithmetic progression. Th... | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P9892",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P9892",
"citation": null,
"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_197 :
answer(sorry) ↔ ∃ A B : Set ℕ, IsCompl A B ∧
(∃ f : ℕ ≃ A, ¬HasMonotoneAP (Subtype.val ∘ f) 3) ∧
(∃ g : ℕ ≃ B, ¬HasMonotoneAP (Subtype.val ∘ g) 3) | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem erdos_197 :\n answer(sorry) ↔ ∃ A B : Set ℕ, IsCompl A B ∧\n (∃ f : ℕ ≃ A, ¬HasMonotoneAP (Subtype.val ∘ f) 3) ∧\n (∃ g : ℕ ≃ B, ¬HasMonotoneAP (Subtype.val ∘ g) 3)",
"target": null,
"source_url": "https://theoremdb.org/statements/P9892",
"fo... | [] | TheoremDB P9892 · Candidate awaiting source review | null | Source-reported assessment, retrieved 8 September 2026. Frontier Science has not independently reviewed this status.
Open. No signed Lean verification is attached to this problem or its selected resolution.
This record is in the source backlog; its novelty and unresolved status have not passed source review. | null | published | incremental | hard | [
"source-candidate",
"theoremdb"
] | null | null | null | null | [
{
"kind": "website",
"citation": "TheoremDB P9892",
"url": "https://theoremdb.org/statements/P9892",
"note": null,
"sort_order": 0
},
{
"kind": "formalization",
"citation": "Source formal statement (not locally verified)",
"url": "https://theoremdb.org/statements/P9892",
"not... | {
"source_record": "P9892",
"imported_at": "2026-09-08",
"status_as_of": "2026-09-08",
"evidence_label": "Source candidate · not reviewed",
"note": "Full public directory statement. Source lane: backlog; source resolution: open. Source verification claims are attributed only; no local Lean verification has be... | {"created_at":"2026-08-17T20:04:18+00:00","display_standard":{"editorial_review":"submitter_supplied","formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":9892,"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": "2026-09-08T03:33:04.764Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"ec19d28b-e8c4-4410-a022-d00cc513243f","format_issues":null,"number":6381} |
1.0.0 | problem:theoremdb-p9894 | problem | theoremdb-p9894 | Existence of a SIC-POVM in dimension 68 | {
"main": "math",
"sub": "algebra",
"slug": "algebra",
"name": "Algebra",
"original_discipline": "mathematics"
} | [] | [] | For a positive integer $d$, a state vector in $\mathbb{C}^d$ is a vector in the $d$-dimensional complex Euclidean space with the standard $L^2$ norm. A state vector is normalized if its norm equals $1$. For two state vectors $\varphi$ and $\psi$ in $\mathbb{C}^d$, their squared overlap is the squared magnitude of their... | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P9894",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P9894",
"citation": null,
"doi": null,
"arxiv_id": null,
"license": "Selected packets label their content CC0-1.0; original references retain their own terms",
"notes": "Selected problem... | theorem hasSICPOVM_68 : answer(sorry) ↔ HasSICPOVM 68 | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem hasSICPOVM_68 : answer(sorry) ↔ HasSICPOVM 68",
"target": null,
"source_url": "https://theoremdb.org/statements/P9894",
"format": "declaration-or-fragment",
"sha256": "5fe65238e33df406d5406aa3398f714764a2e89d463881120b4c6cd48426f39c",
"license": "... | [] | TheoremDB P9894 · Candidate awaiting source review | null | Source-reported assessment, retrieved 8 September 2026. Frontier Science has not independently reviewed this status.
Open. No signed Lean verification is attached to this problem or its selected resolution.
This record is in the source backlog; its novelty and unresolved status have not passed source review. | null | published | incremental | hard | [
"source-candidate",
"theoremdb"
] | null | null | null | null | [
{
"kind": "website",
"citation": "TheoremDB P9894",
"url": "https://theoremdb.org/statements/P9894",
"note": null,
"sort_order": 0
},
{
"kind": "formalization",
"citation": "Source formal statement (not locally verified)",
"url": "https://theoremdb.org/statements/P9894",
"not... | {
"source_record": "P9894",
"imported_at": "2026-09-08",
"status_as_of": "2026-09-08",
"evidence_label": "Source candidate · not reviewed",
"note": "Full public directory statement. Source lane: backlog; source resolution: open. Source verification claims are attributed only; no local Lean verification has be... | {"created_at":"2026-08-17T20:04:20+00:00","display_standard":{"editorial_review":"submitter_supplied","formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":9894,"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": "2026-09-08T03:32:58.513Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"ec3a6baa-b5bd-4dcf-bd78-ab4e24155f43","format_issues":null,"number":5198} |
1.0.0 | problem:theoremdb-p9896 | problem | theoremdb-p9896 | Powerful Values of Mersenne Numbers | {
"main": "math",
"sub": "number theory",
"slug": "number-theory",
"name": "Number theory",
"original_discipline": "mathematics"
} | [] | [
{
"code": "11",
"basis": "community-map",
"evidence": "Inferred from community field: number-theory"
}
] | A positive integer $m$ is called powerful if for every prime $p$ dividing $m$, the square $p^2$ also divides $m$. For a sequence $a : \mathbb{N} \to \mathbb{N}$, we say that $a$ is eventually not powerful if there are only finitely many natural numbers $n$ for which $a(n)$ is powerful. Consider the sequence defined by ... | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P9896",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P9896",
"citation": null,
"doi": null,
"arxiv_id": null,
"license": "Selected packets label their content CC0-1.0; original references retain their own terms",
"notes": "Selected problem... | theorem erdos_936.variants.two_pow_sub_one :
answer(sorry) ↔ EventuallyNotPowerful (2 ^ · - 1) | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem erdos_936.variants.two_pow_sub_one :\n answer(sorry) ↔ EventuallyNotPowerful (2 ^ · - 1)",
"target": null,
"source_url": "https://theoremdb.org/statements/P9896",
"format": "declaration-or-fragment",
"sha256": "0e484340611055705a88f3812752428b22470c... | [] | TheoremDB P9896 · Candidate awaiting source review | null | Source-reported assessment, retrieved 8 September 2026. Frontier Science has not independently reviewed this status.
Open. No signed Lean verification is attached to this problem or its selected resolution.
This record is in the source backlog; its novelty and unresolved status have not passed source review. | null | published | incremental | hard | [
"source-candidate",
"theoremdb"
] | null | null | null | null | [
{
"kind": "website",
"citation": "TheoremDB P9896",
"url": "https://theoremdb.org/statements/P9896",
"note": null,
"sort_order": 0
},
{
"kind": "formalization",
"citation": "Source formal statement (not locally verified)",
"url": "https://theoremdb.org/statements/P9896",
"not... | {
"source_record": "P9896",
"imported_at": "2026-09-08",
"status_as_of": "2026-09-08",
"evidence_label": "Source candidate · not reviewed",
"note": "Full public directory statement. Source lane: backlog; source resolution: open. Source verification claims are attributed only; no local Lean verification has be... | {"created_at":"2026-08-17T20:04:22+00:00","display_standard":{"formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":9896,"public_status":{"evidence":{"hasFormalization":false,"hasInformalProof":false,"hasRecordedWork":false,"solutionRecord":null},"lean":{"kernelAccepted":false,"label":"","reaso... | [
"formalization-contains-placeholder",
"formalization-not-compiled-during-export",
"current-status-unassessed",
"source-full-text-not-present"
] | [] | {
"created_at": "2026-09-08T03:32:15.935Z",
"updated_at": "2026-09-08T03:32:15.935Z",
"published_at": "2026-09-08T03:33:05.027Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"e9fe9096-d85a-4395-895b-491265190692","format_issues":null,"number":6437} |
1.0.0 | problem:theoremdb-p9898 | problem | theoremdb-p9898 | Asymptotic growth of shifted prime factor maxima | {
"main": "math",
"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 non-negative integers $n$ and $k$, let $v(n,k)$ denote the number of prime factors of $n+k$ that do not divide $n+i$ for any $i$ with $0 \leq i < k$. For each fixed non-negative integer $l$, define $v_l(n) = \sup_{k \geq l} v(n,k)$, where the supremum is taken over all $k \geq l$. Prove that for every fixed $l$, th... | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P9898",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P9898",
"citation": null,
"doi": null,
"arxiv_id": null,
"license": "Selected packets label their content CC0-1.0; original references retain their own terms",
"notes": "Selected problem... | theorem erdos_889.variants.general :
∀ l, Tendsto (v_l l) atTop (𝓝 ⊤) | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem erdos_889.variants.general :\n ∀ l, Tendsto (v_l l) atTop (𝓝 ⊤)",
"target": null,
"source_url": "https://theoremdb.org/statements/P9898",
"format": "declaration-or-fragment",
"sha256": "d179fff74b9630d330eda52b9823786aa3b1c252224523ce28265eb1cfbcd... | [] | TheoremDB P9898 · Candidate awaiting source review | null | Source-reported assessment, retrieved 8 September 2026. Frontier Science has not independently reviewed this status.
Open. No signed Lean verification is attached to this problem or its selected resolution.
This record is in the source backlog; its novelty and unresolved status have not passed source review. | null | published | incremental | hard | [
"source-candidate",
"theoremdb"
] | null | null | null | null | [
{
"kind": "website",
"citation": "TheoremDB P9898",
"url": "https://theoremdb.org/statements/P9898",
"note": null,
"sort_order": 0
},
{
"kind": "formalization",
"citation": "Source formal statement (not locally verified)",
"url": "https://theoremdb.org/statements/P9898",
"not... | {
"source_record": "P9898",
"imported_at": "2026-09-08",
"status_as_of": "2026-09-08",
"evidence_label": "Source candidate · not reviewed",
"note": "Full public directory statement. Source lane: backlog; source resolution: open. Source verification claims are attributed only; no local Lean verification has be... | {"created_at":"2026-08-17T20:04:22+00:00","display_standard":{"editorial_review":"submitter_supplied","formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":9898,"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": "2026-09-08T03:32:50.153Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"4e4ef673-2ffd-4d64-be15-c75f9a64a6de","format_issues":null,"number":3362} |
1.0.0 | problem:theoremdb-p9900 | problem | theoremdb-p9900 | Non-existence of monochromatic quantum graphs with ten vertices and four colours | {
"main": "math",
"sub": "combinatorics",
"slug": "combinatorics",
"name": "Combinatorics",
"original_discipline": "mathematics"
} | [] | [
{
"code": "05",
"basis": "community-map",
"evidence": "Inferred from community field: combinatorics"
}
] | Let $N$ be a positive even integer. A quantum graph with $N$ vertices and $D$ colours over a coefficient domain $\alpha$ is specified by a weight function $W$ assigning a value in $\alpha$ to every pair consisting of an unordered edge of the complete graph $K_N$ and a colour from $\{1,\dots,D\}$. For each vertex colour... | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P9900",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P9900",
"citation": null,
"doi": null,
"arxiv_id": null,
"license": "Selected packets label their content CC0-1.0; original references retain their own terms",
"notes": "Selected problem... | theorem eqSystem10_no_solution_d4 :
answer(sorry) ↔
¬ ∃ W : WeightsN 10 4 ℂ, EqSystemN 10 4 W | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem eqSystem10_no_solution_d4 :\n answer(sorry) ↔\n ¬ ∃ W : WeightsN 10 4 ℂ, EqSystemN 10 4 W",
"target": null,
"source_url": "https://theoremdb.org/statements/P9900",
"format": "declaration-or-fragment",
"sha256": "1cda358912491324e3b8e1394456e7f... | [] | TheoremDB P9900 · Candidate awaiting source review | null | Source-reported assessment, retrieved 8 September 2026. Frontier Science has not independently reviewed this status.
Open. No signed Lean verification is attached to this problem or its selected resolution.
This record is in the source backlog; its novelty and unresolved status have not passed source review. | null | published | incremental | hard | [
"source-candidate",
"theoremdb"
] | null | null | null | null | [
{
"kind": "website",
"citation": "TheoremDB P9900",
"url": "https://theoremdb.org/statements/P9900",
"note": null,
"sort_order": 0
},
{
"kind": "formalization",
"citation": "Source formal statement (not locally verified)",
"url": "https://theoremdb.org/statements/P9900",
"not... | {
"source_record": "P9900",
"imported_at": "2026-09-08",
"status_as_of": "2026-09-08",
"evidence_label": "Source candidate · not reviewed",
"note": "Full public directory statement. Source lane: backlog; source resolution: open. Source verification claims are attributed only; no local Lean verification has be... | {"created_at":"2026-08-17T20:04:22+00:00","display_standard":{"editorial_review":"submitter_supplied","formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":9900,"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": "2026-09-08T03:33:04.371Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"8338e4ca-767f-49e3-990a-cc297273b4a2","format_issues":null,"number":6296} |
1.0.0 | problem:theoremdb-p9902 | problem | theoremdb-p9902 | Asymptotic upper bound for the Erdős–Selfridge 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"
}
] | Let $\mathbb{N}$ denote the set of non-negative integers. A sequence $A : \mathbb{N} \to \mathbb{N}$ is called good for $n$ if it satisfies the following three conditions: $A(0) = n$; the sequence is strictly monotone, meaning $A(j) < A(j+1)$ for all $j$; and for every $j \in \mathbb{N}$, the value $A(j+1)$ is the leas... | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P9902",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P9902",
"citation": null,
"doi": null,
"arxiv_id": null,
"license": "Selected packets label their content CC0-1.0; original references retain their own terms",
"notes": "Selected problem... | theorem erdos_359.parts.ii (A : ℕ → ℕ) (hA : IsGoodFor A 1) (c : ℝ) (hc : 0 < c):
atTop.Tendsto (fun k ↦ A k / (k : ℝ) ^ (1 + c)) (nhds 0) | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem erdos_359.parts.ii (A : ℕ → ℕ) (hA : IsGoodFor A 1) (c : ℝ) (hc : 0 < c):\n atTop.Tendsto (fun k ↦ A k / (k : ℝ) ^ (1 + c)) (nhds 0)",
"target": null,
"source_url": "https://theoremdb.org/statements/P9902",
"format": "declaration-or-fragment",
"sha... | [] | TheoremDB P9902 · Candidate awaiting source review | null | Source-reported assessment, retrieved 8 September 2026. Frontier Science has not independently reviewed this status.
Open. No signed Lean verification is attached to this problem or its selected resolution.
This record is in the source backlog; its novelty and unresolved status have not passed source review. | null | published | incremental | hard | [
"source-candidate",
"theoremdb"
] | null | null | null | null | [
{
"kind": "website",
"citation": "TheoremDB P9902",
"url": "https://theoremdb.org/statements/P9902",
"note": null,
"sort_order": 0
},
{
"kind": "formalization",
"citation": "Source formal statement (not locally verified)",
"url": "https://theoremdb.org/statements/P9902",
"not... | {
"source_record": "P9902",
"imported_at": "2026-09-08",
"status_as_of": "2026-09-08",
"evidence_label": "Source candidate · not reviewed",
"note": "Full public directory statement. Source lane: backlog; source resolution: open. Source verification claims are attributed only; no local Lean verification has be... | {"created_at":"2026-08-17T20:04:24+00:00","display_standard":{"editorial_review":"submitter_supplied","formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":9902,"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": "2026-09-08T03:32:50.451Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"44ccdbbf-1354-4829-bf95-210abe62ccd3","format_issues":null,"number":3422} |
1.0.0 | problem:theoremdb-p9904 | problem | theoremdb-p9904 | Non-uniqueness of point sets minimizing distinct distances | {
"main": "math",
"sub": "geometry and topology",
"slug": "geometry-and-topology",
"name": "Geometry and topology",
"original_discipline": "mathematics"
} | [] | [] | For a finite set $A\subset\mathbb{R}^2$, let $d(A)$ denote the number of distinct Euclidean distances between pairs of distinct points of $A$. For each positive integer $n$, define $m(n)$ to be the minimum value of $d(A)$ over all sets $A\subset\mathbb{R}^2$ with $|A|=n$. A set $A$ with $|A|=n$ and $d(A)=m(n)$ is calle... | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P9904",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P9904",
"citation": null,
"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_91 :
(∀ᶠ n : ℕ in atTop, ¬ UniqueMinimizer n) ↔ answer(sorry) | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem erdos_91 :\n (∀ᶠ n : ℕ in atTop, ¬ UniqueMinimizer n) ↔ answer(sorry)",
"target": null,
"source_url": "https://theoremdb.org/statements/P9904",
"format": "declaration-or-fragment",
"sha256": "4656af29dc1e3d1d123b02bbc1a69bd63c69c3389235c297c87a97f4... | [] | TheoremDB P9904 · Candidate awaiting source review | null | Source-reported assessment, retrieved 8 September 2026. Frontier Science has not independently reviewed this status.
Open. No signed Lean verification is attached to this problem or its selected resolution.
This record is in the source backlog; its novelty and unresolved status have not passed source review. | null | published | incremental | hard | [
"source-candidate",
"theoremdb"
] | null | null | null | null | [
{
"kind": "website",
"citation": "TheoremDB P9904",
"url": "https://theoremdb.org/statements/P9904",
"note": null,
"sort_order": 0
},
{
"kind": "formalization",
"citation": "Source formal statement (not locally verified)",
"url": "https://theoremdb.org/statements/P9904",
"not... | {
"source_record": "P9904",
"imported_at": "2026-09-08",
"status_as_of": "2026-09-08",
"evidence_label": "Source candidate · not reviewed",
"note": "Full public directory statement. Source lane: backlog; source resolution: open. Source verification claims are attributed only; no local Lean verification has be... | {"created_at":"2026-08-17T20:04:25+00:00","display_standard":{"editorial_review":"submitter_supplied","formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":9904,"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": "2026-09-08T03:33:04.450Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"ba38a1fc-b358-436d-ac94-231940e5b28f","format_issues":null,"number":6312} |
1.0.0 | problem:theoremdb-p9906 | problem | theoremdb-p9906 | Asymptotic order of the totient descent time to a prime | {
"main": "math",
"sub": "number theory",
"slug": "number-theory",
"name": "Number theory",
"original_discipline": "mathematics"
} | [] | [
{
"code": "11",
"basis": "community-map",
"evidence": "Inferred from community field: number-theory"
}
] | For each positive integer $n$, define $c(n)$ to be the minimum number of iterations of the map $n \mapsto \varphi(n) + 1$ needed to reach a prime number, where $\varphi$ denotes Euler's totient function. That is, $c(n)$ is the least non-negative integer $i$ such that the $i$-fold iterate $(\varphi(\cdot) + 1)^{[i]}(n)$... | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P9906",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P9906",
"citation": null,
"doi": null,
"arxiv_id": null,
"license": "Selected packets label their content CC0-1.0; original references retain their own terms",
"notes": "Selected problem... | theorem erdos_409.parts.i.isTheta (c : ℕ → ℕ)
(h : ∀ n > 0, IsLeast { i | (φ · + 1)^[i] n |>.Prime } (c n)) :
(fun n => (c n : ℝ)) =Θ[atTop] (answer(sorry) : ℕ → ℝ) | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem erdos_409.parts.i.isTheta (c : ℕ → ℕ)\n (h : ∀ n > 0, IsLeast { i | (φ · + 1)^[i] n |>.Prime } (c n)) :\n (fun n => (c n : ℝ)) =Θ[atTop] (answer(sorry) : ℕ → ℝ)",
"target": null,
"source_url": "https://theoremdb.org/statements/P9906",
"format": "dec... | [] | TheoremDB P9906 · Candidate awaiting source review | null | Source-reported assessment, retrieved 8 September 2026. Frontier Science has not independently reviewed this status.
Open. No signed Lean verification is attached to this problem or its selected resolution.
This record is in the source backlog; its novelty and unresolved status have not passed source review. | null | published | incremental | hard | [
"source-candidate",
"theoremdb"
] | null | null | null | null | [
{
"kind": "website",
"citation": "TheoremDB P9906",
"url": "https://theoremdb.org/statements/P9906",
"note": null,
"sort_order": 0
},
{
"kind": "formalization",
"citation": "Source formal statement (not locally verified)",
"url": "https://theoremdb.org/statements/P9906",
"not... | {
"source_record": "P9906",
"imported_at": "2026-09-08",
"status_as_of": "2026-09-08",
"evidence_label": "Source candidate · not reviewed",
"note": "Full public directory statement. Source lane: backlog; source resolution: open. Source verification claims are attributed only; no local Lean verification has be... | {"created_at":"2026-08-17T20:04:26+00:00","display_standard":{"editorial_review":"submitter_supplied","formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":9906,"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": "2026-09-08T03:32:50.378Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"4a5f37b5-a91e-434e-8f84-ce38fe4dfce5","format_issues":null,"number":3407} |
1.0.0 | problem:theoremdb-p9908 | problem | theoremdb-p9908 | Eventual Periodicity of Greedy Sum-Free Sequence Differences | {
"main": "math",
"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 = \{a_1 < \cdots < a_k\}$ be a finite set of integers. Extend it to an infinite sequence $\overline{A} = \{a_1 < a_2 < \cdots\}$ by defining, for each $n \geq k$, the term $a_{n+1}$ to be the least integer exceeding $a_n$ that cannot be expressed as $a_i + a_j$ with $i, j \leq n$. Let $d_i = a_{i+1} - a_i$ denot... | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P9908",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P9908",
"citation": null,
"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_341 :
answer(sorry) ↔
∀ (a : ℕ → ℤ),
(∀ᶠ n in atTop,
IsLeast { x | a n < x ∧ x ∉ { a i + a j | (i ≤ n) (j ≤ n) } } (a (n + 1))) →
let d := fun i ↦ a (i + 1) - a i
∃ p > 0, ∀ᶠ m in atTop, d (m + p) = d m | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem erdos_341 :\n answer(sorry) ↔\n ∀ (a : ℕ → ℤ),\n (∀ᶠ n in atTop,\n IsLeast { x | a n < x ∧ x ∉ { a i + a j | (i ≤ n) (j ≤ n) } } (a (n + 1))) →\n let d := fun i ↦ a (i + 1) - a i\n ∃ p > 0, ∀ᶠ m in atTop, d (m + p) = d m",
"t... | [] | TheoremDB P9908 · Candidate awaiting source review | null | Source-reported assessment, retrieved 8 September 2026. Frontier Science has not independently reviewed this status.
Open. No signed Lean verification is attached to this problem or its selected resolution.
This record is in the source backlog; its novelty and unresolved status have not passed source review. | null | published | incremental | hard | [
"source-candidate",
"theoremdb"
] | null | null | null | null | [
{
"kind": "website",
"citation": "TheoremDB P9908",
"url": "https://theoremdb.org/statements/P9908",
"note": null,
"sort_order": 0
},
{
"kind": "formalization",
"citation": "Source formal statement (not locally verified)",
"url": "https://theoremdb.org/statements/P9908",
"not... | {
"source_record": "P9908",
"imported_at": "2026-09-08",
"status_as_of": "2026-09-08",
"evidence_label": "Source candidate · not reviewed",
"note": "Full public directory statement. Source lane: backlog; source resolution: open. Source verification claims are attributed only; no local Lean verification has be... | {"created_at":"2026-08-17T20:04:27+00:00","display_standard":{"formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":9908,"public_status":{"evidence":{"hasFormalization":false,"hasInformalProof":false,"hasRecordedWork":false,"solutionRecord":null},"lean":{"kernelAccepted":false,"label":"","reaso... | [
"formalization-contains-placeholder",
"formalization-not-compiled-during-export",
"current-status-unassessed",
"source-full-text-not-present"
] | [] | {
"created_at": "2026-09-08T03:32:15.935Z",
"updated_at": "2026-09-08T03:32:15.935Z",
"published_at": "2026-09-08T03:32:56.788Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"a5ff82b2-e417-455a-929d-2caba69163de","format_issues":null,"number":4815} |
1.0.0 | problem:theoremdb-p9910 | problem | theoremdb-p9910 | Erdős's problem on divisibility of factorials by squared shifted factorials | {
"main": "math",
"sub": "number theory",
"slug": "number-theory",
"name": "Number theory",
"original_discipline": "mathematics"
} | [] | [
{
"code": "11",
"basis": "community-map",
"evidence": "Inferred from community field: number-theory"
}
] | For an integer $k \geq 2$, let $S_k$ denote the set of natural numbers $n$ such that $\big((n+k)!\big)^2$ divides $(2n)!$. Determine whether $S_k$ is infinite for every $k \geq 2$. | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P9910",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P9910",
"citation": null,
"doi": null,
"arxiv_id": null,
"license": "Selected packets label their content CC0-1.0; original references retain their own terms",
"notes": "Selected problem... | theorem erdos_727 : answer(sorry) ↔ ∀ k ≥ 2,
Set.Infinite {n : ℕ | (Nat.factorial (n + k)) ^ 2 ∣ Nat.factorial (2 * n)} | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem erdos_727 : answer(sorry) ↔ ∀ k ≥ 2,\n Set.Infinite {n : ℕ | (Nat.factorial (n + k)) ^ 2 ∣ Nat.factorial (2 * n)}",
"target": null,
"source_url": "https://theoremdb.org/statements/P9910",
"format": "declaration-or-fragment",
"sha256": "bda8117129b7... | [] | TheoremDB P9910 · Candidate awaiting source review | null | Source-reported assessment, retrieved 8 September 2026. Frontier Science has not independently reviewed this status.
Open. No signed Lean verification is attached to this problem or its selected resolution.
This record is in the source backlog; its novelty and unresolved status have not passed source review. | null | published | incremental | hard | [
"source-candidate",
"theoremdb"
] | null | null | null | null | [
{
"kind": "website",
"citation": "TheoremDB P9910",
"url": "https://theoremdb.org/statements/P9910",
"note": null,
"sort_order": 0
},
{
"kind": "formalization",
"citation": "Source formal statement (not locally verified)",
"url": "https://theoremdb.org/statements/P9910",
"not... | {
"source_record": "P9910",
"imported_at": "2026-09-08",
"status_as_of": "2026-09-08",
"evidence_label": "Source candidate · not reviewed",
"note": "Full public directory statement. Source lane: backlog; source resolution: open. Source verification claims are attributed only; no local Lean verification has be... | {"created_at":"2026-08-17T20:04:29+00:00","display_standard":{"editorial_review":"submitter_supplied","formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":9910,"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": "2026-09-08T03:32:56.460Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"c6de7b19-7c61-4e72-9df0-3fc9dd7f4097","format_issues":null,"number":4745} |
1.0.0 | problem:theoremdb-p9912 | problem | theoremdb-p9912 | 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 positive integers $h$ and $k$, define
$$
S'(h, k) = \sum_{j=1}^{k-1} (-1)^{j + 1 + \lfloor hj/k \rfloor},
$$
where $\lfloor x \rfloor$ denotes the greatest integer less than or equal to $x$. For a positive integer $k$, define
$$
S(k) = \sum_{h=1}^{k-1} S'(h, k).
$$
Let $k$ be an odd prime with $k > 5$. Prove th... | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P9912",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P9912",
"citation": null,
"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_1 (k : ℕ) (hprim : k.Prime) (hodd : Odd k) (hgt : k > 5) : k < S k | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem conjecture_4_1 (k : ℕ) (hprim : k.Prime) (hodd : Odd k) (hgt : k > 5) : k < S k",
"target": null,
"source_url": "https://theoremdb.org/statements/P9912",
"format": "declaration-or-fragment",
"sha256": "1c0a1889d4ed3ffa373689d146d7b007674697547fcb81fe4... | [] | TheoremDB P9912 · Candidate awaiting source review | null | Source-reported assessment, retrieved 8 September 2026. Frontier Science has not independently reviewed this status.
Open. No signed Lean verification is attached to this problem or its selected resolution.
This record is in 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/P9912",
"not... | {
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"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"8fd6cb0d-c8bc-4ae4-a4a1-abd58ea88e56","format_issues":null,"number":6096} |
1.0.0 | problem:theoremdb-p9914 | problem | theoremdb-p9914 | Existence of a Four-Dimensional Euler Brick | {
"main": "math",
"sub": "number theory",
"slug": "number-theory",
"name": "Number theory",
"original_discipline": "mathematics"
} | [] | [
{
"code": "11",
"basis": "community-map",
"evidence": "Inferred from community field: number-theory"
}
] | An Euler hyperbrick in dimension $n$ is a collection of $n$ positive integers (called side lengths) such that for every pair of distinct indices $i$ and $j$, the sum of the squares of the corresponding side lengths is a perfect square. Does there exist an Euler hyperbrick in dimension $4$? That is, do there exist posit... | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P9914",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P9914",
"citation": null,
"doi": null,
"arxiv_id": null,
"license": "Selected packets label their content CC0-1.0; original references retain their own terms",
"notes": "Selected problem... | theorem four_dim_euler_brick_existence :
answer(sorry) ↔ ∃ sides : Fin 4 → ℕ+, IsEulerHyperBrick 4 sides | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem four_dim_euler_brick_existence :\n answer(sorry) ↔ ∃ sides : Fin 4 → ℕ+, IsEulerHyperBrick 4 sides",
"target": null,
"source_url": "https://theoremdb.org/statements/P9914",
"format": "declaration-or-fragment",
"sha256": "7933601096d69972591aecfaa72... | [] | TheoremDB P9914 · Candidate awaiting source review | null | Source-reported assessment, retrieved 8 September 2026. Frontier Science has not independently reviewed this status.
Open. No signed Lean verification is attached to this problem or its selected resolution.
This record is in the source backlog; its novelty and unresolved status have not passed source review. | null | published | incremental | hard | [
"source-candidate",
"theoremdb"
] | null | null | null | null | [
{
"kind": "website",
"citation": "TheoremDB P9914",
"url": "https://theoremdb.org/statements/P9914",
"note": null,
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{
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"citation": "Source formal statement (not locally verified)",
"url": "https://theoremdb.org/statements/P9914",
"not... | {
"source_record": "P9914",
"imported_at": "2026-09-08",
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"evidence_label": "Source candidate · not reviewed",
"note": "Full public directory statement. Source lane: backlog; source resolution: open. Source verification claims are attributed only; no local Lean verification has be... | {"created_at":"2026-08-17T20:04:30+00:00","display_standard":{"editorial_review":"submitter_supplied","formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":9914,"public_status":{"evidence":{"hasFormalization":false,"hasInformalProof":false,"hasRecordedWork":false,"solutionRecord":null},"lean":{... | [
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"updated_at": "2026-09-08T03:32:15.935Z",
"published_at": "2026-09-08T03:32:58.445Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"1a931d7a-c13d-42f0-83bd-76073ecc4de6","format_issues":null,"number":5184} |
1.0.0 | problem:theoremdb-p9916 | problem | theoremdb-p9916 | Existence of a universal limsup constant for Rademacher multiplicative functions | {
"main": "math",
"sub": "number theory",
"slug": "number-theory",
"name": "Number theory",
"original_discipline": "mathematics"
} | [] | [
{
"code": "11",
"basis": "community-map",
"evidence": "Inferred from community field: number-theory"
}
] | A function $f : \mathbb{N} \to \mathbb{R}$ defined on a probability space is called Rademacher multiplicative if it satisfies the following conditions: $f(1) = 1$; for each prime $p$, the value $f(p)$ is chosen independently and uniformly from $\{-1, 1\}$; for every square-free integer $n = p_1 \cdots p_r$ with distinc... | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P9916",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P9916",
"citation": null,
"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_520 :
answer(sorry) ↔ ∃ c > 0, ∀ (Ω : Type) [MeasureSpace Ω] [IsProbabilityMeasure (ℙ : Measure Ω)]
(f : ℕ → Ω → ℝ), IsRademacherMultiplicative f →
∀ᵐ ω, limsup (fun N ↦ ∑ m ≤ N, f m ω / sqrt (N * log (log N))) atTop = c | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem erdos_520 :\n answer(sorry) ↔ ∃ c > 0, ∀ (Ω : Type) [MeasureSpace Ω] [IsProbabilityMeasure (ℙ : Measure Ω)]\n (f : ℕ → Ω → ℝ), IsRademacherMultiplicative f →\n ∀ᵐ ω, limsup (fun N ↦ ∑ m ≤ N, f m ω / sqrt (N * log (log N))) atTop = c",
"target": null,... | [] | TheoremDB P9916 · Candidate awaiting source review | null | Source-reported assessment, retrieved 8 September 2026. Frontier Science has not independently reviewed this status.
Open. No signed Lean verification is attached to this problem or its selected resolution.
This record is in the source backlog; its novelty and unresolved status have not passed source review. | null | published | incremental | hard | [
"source-candidate",
"theoremdb"
] | null | null | null | null | [
{
"kind": "website",
"citation": "TheoremDB P9916",
"url": "https://theoremdb.org/statements/P9916",
"note": null,
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"citation": "Source formal statement (not locally verified)",
"url": "https://theoremdb.org/statements/P9916",
"not... | {
"source_record": "P9916",
"imported_at": "2026-09-08",
"status_as_of": "2026-09-08",
"evidence_label": "Source candidate · not reviewed",
"note": "Full public directory statement. Source lane: backlog; source resolution: open. Source verification claims are attributed only; no local Lean verification has be... | {"created_at":"2026-08-17T20:04:31+00:00","display_standard":{"editorial_review":"submitter_supplied","formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":9916,"public_status":{"evidence":{"hasFormalization":false,"hasInformalProof":false,"hasRecordedWork":false,"solutionRecord":null},"lean":{... | [
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] | [] | {
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"published_at": "2026-09-08T03:32:58.584Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"a7b10982-12b4-4a21-8173-4386b916830d","format_issues":null,"number":5213} |
1.0.0 | problem:theoremdb-p9918 | problem | theoremdb-p9918 | The Herzog–Schönheim conjecture on exact coverings of groups | {
"main": "math",
"sub": "algebra",
"slug": "algebra",
"name": "Algebra",
"original_discipline": "mathematics"
} | [] | [] | Let $G$ be a group with more than one element. Let $\iota$ be a finite type with more than one element. An exact covering of $G$ indexed by $\iota$ consists of a family of subgroups $H_i \leq G$ and a family of representatives $g_i \in G$ for $i \in \iota$ such that: each $H_i$ is nonempty; the left cosets $g_i H_i$ ar... | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P9918",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P9918",
"citation": null,
"doi": null,
"arxiv_id": null,
"license": "Selected packets label their content CC0-1.0; original references retain their own terms",
"notes": "Selected problem... | theorem herzog_schonheim {G : Type*} [Group G] (hG : 1 < ENat.card G) {ι : Type*} [Fintype ι]
(hι : 1 < Fintype.card ι) (P : Group.ExactCovering G ι) :
∃ i j, i ≠ j ∧ (P.parts i).index = (P.parts j).index | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem herzog_schonheim {G : Type*} [Group G] (hG : 1 < ENat.card G) {ι : Type*} [Fintype ι]\n (hι : 1 < Fintype.card ι) (P : Group.ExactCovering G ι) :\n ∃ i j, i ≠ j ∧ (P.parts i).index = (P.parts j).index",
"target": null,
"source_url": "https://theoremdb.o... | [] | TheoremDB P9918 · Candidate awaiting source review | null | Source-reported assessment, retrieved 8 September 2026. Frontier Science has not independently reviewed this status.
Open. No signed Lean verification is attached to this problem or its selected resolution.
This record is in the source backlog; its novelty and unresolved status have not passed source review. | null | published | incremental | hard | [
"source-candidate",
"theoremdb"
] | null | null | null | null | [
{
"kind": "website",
"citation": "TheoremDB P9918",
"url": "https://theoremdb.org/statements/P9918",
"note": null,
"sort_order": 0
},
{
"kind": "formalization",
"citation": "Source formal statement (not locally verified)",
"url": "https://theoremdb.org/statements/P9918",
"not... | {
"source_record": "P9918",
"imported_at": "2026-09-08",
"status_as_of": "2026-09-08",
"evidence_label": "Source candidate · not reviewed",
"note": "Full public directory statement. Source lane: backlog; source resolution: open. Source verification claims are attributed only; no local Lean verification has be... | {"created_at":"2026-08-17T20:04:32+00:00","display_standard":{"formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":9918,"public_status":{"evidence":{"hasFormalization":false,"hasInformalProof":false,"hasRecordedWork":false,"solutionRecord":null},"lean":{"kernelAccepted":false,"label":"","reaso... | [
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"created_at": "2026-09-08T03:32:15.935Z",
"updated_at": "2026-09-08T03:32:15.935Z",
"published_at": "2026-09-08T03:33:06.542Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"3c3cafc1-abbd-4928-87f9-b12b3a498160","format_issues":null,"number":6765} |
1.0.0 | problem:theoremdb-p9920 | problem | theoremdb-p9920 | Maximal Star in a Decreasing Family | {
"main": "math",
"sub": "combinatorics",
"slug": "combinatorics",
"name": "Combinatorics",
"original_discipline": "mathematics"
} | [] | [
{
"code": "05",
"basis": "community-map",
"evidence": "Inferred from community field: combinatorics"
}
] | Let $\alpha$ be a nonempty finite type. A family $F$ of finite subsets of $\alpha$ is called decreasing if for all finite subsets $A, B \subseteq \alpha$, whenever $B \subseteq A$ and $A \in F$, then $B \in F$. A family $G$ of finite subsets of $\alpha$ is called intersecting if for all $A, B \in G$, we have $A \cap B ... | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P9920",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P9920",
"citation": null,
"doi": null,
"arxiv_id": null,
"license": "Selected packets label their content CC0-1.0; original references retain their own terms",
"notes": "Selected problem... | theorem exists_maximal_star :
∀ F : Finset (Finset α), Decreasing F →
∃ x : α, ∀ G, G ⊆ F → Intersecting G → G.card ≤ { A ∈ F | x ∈ A }.card | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem exists_maximal_star :\n ∀ F : Finset (Finset α), Decreasing F →\n ∃ x : α, ∀ G, G ⊆ F → Intersecting G → G.card ≤ { A ∈ F | x ∈ A }.card",
"target": null,
"source_url": "https://theoremdb.org/statements/P9920",
"format": "declaration-or-fragment... | [] | TheoremDB P9920 · Candidate awaiting source review | null | Source-reported assessment, retrieved 8 September 2026. Frontier Science has not independently reviewed this status.
Open. No signed Lean verification is attached to this problem or its selected resolution.
This record is in the source backlog; its novelty and unresolved status have not passed source review. | null | published | incremental | hard | [
"source-candidate",
"theoremdb"
] | null | null | null | null | [
{
"kind": "website",
"citation": "TheoremDB P9920",
"url": "https://theoremdb.org/statements/P9920",
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},
{
"kind": "formalization",
"citation": "Source formal statement (not locally verified)",
"url": "https://theoremdb.org/statements/P9920",
"not... | {
"source_record": "P9920",
"imported_at": "2026-09-08",
"status_as_of": "2026-09-08",
"evidence_label": "Source candidate · not reviewed",
"note": "Full public directory statement. Source lane: backlog; source resolution: open. Source verification claims are attributed only; no local Lean verification has be... | {"created_at":"2026-08-17T20:04:33+00:00","display_standard":{"formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":9920,"public_status":{"evidence":{"hasFormalization":false,"hasInformalProof":false,"hasRecordedWork":false,"solutionRecord":null},"lean":{"kernelAccepted":false,"label":"","reaso... | [
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"created_at": "2026-09-08T03:32:15.935Z",
"updated_at": "2026-09-08T03:32:15.935Z",
"published_at": "2026-09-08T03:33:03.665Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"8299bee5-df68-4876-8247-d72ab424fe82","format_issues":null,"number":6143} |
1.0.0 | problem:theoremdb-p9922 | problem | theoremdb-p9922 | Irrationality of π raised to the power e | {
"main": "math",
"sub": "analysis",
"slug": "analysis",
"name": "Analysis",
"original_discipline": "mathematics"
} | [] | [] | Let $ \pi $ denote the ratio of a circle's circumference to its diameter, and let $ e = \exp(1) $ denote Euler's number, where $ \exp $ is the real exponential function. Determine whether $ \pi^e $ is irrational. | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P9922",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P9922",
"citation": null,
"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_pi_to_e :
answer(sorry) ↔ Irrational (π ^ e) | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem irrational_pi_to_e :\n answer(sorry) ↔ Irrational (π ^ e)",
"target": null,
"source_url": "https://theoremdb.org/statements/P9922",
"format": "declaration-or-fragment",
"sha256": "3d078121999c96e2821ececb63bc400aa6bb9096a6a90fa4e9fa1e9f07d3f039",
... | [] | TheoremDB P9922 · Candidate awaiting source review | null | Source-reported assessment, retrieved 8 September 2026. Frontier Science has not independently reviewed this status.
Open. No signed Lean verification is attached to this problem or its selected resolution.
This record is in 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/P9922",
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},
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"citation": "Source formal statement (not locally verified)",
"url": "https://theoremdb.org/statements/P9922",
"not... | {
"source_record": "P9922",
"imported_at": "2026-09-08",
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"evidence_label": "Source candidate · not reviewed",
"note": "Full public directory statement. Source lane: backlog; source resolution: open. Source verification claims are attributed only; no local Lean verification has be... | {"created_at":"2026-08-17T20:04:34+00:00","display_standard":{"editorial_review":"submitter_supplied","formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":9922,"public_status":{"evidence":{"hasFormalization":false,"hasInformalProof":false,"hasRecordedWork":false,"solutionRecord":null},"lean":{... | [
"formalization-contains-placeholder",
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"created_at": "2026-09-08T03:32:15.935Z",
"updated_at": "2026-09-08T03:32:15.935Z",
"published_at": "2026-09-08T03:33:01.840Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"e05ff740-b9e8-4f30-b4db-4ccd84ea84ad","format_issues":null,"number":5746} |
1.0.0 | problem:theoremdb-p9924 | problem | theoremdb-p9924 | Upper bound for 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 $n$, let $h(n)$ denote the largest integer $m$ such that the following holds: for every set $A$ of $n$ integers, there exists a subset $B \subseteq A$ with $|B| \geq m$ such that any equality of subset sums of $B$ forces the summands to have the same cardinality. That is, if $a_1 + \cdots + a_... | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P9924",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P9924",
"citation": null,
"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.isBigO_cube_root_linearithmic :
(fun n ↦ (subsetSumThreshold n : ℝ)) =O[atTop]
fun n ↦ (n * Real.log n) ^ ((1 : ℝ) / 3) | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem erdos_789.variants.isBigO_cube_root_linearithmic :\n (fun n ↦ (subsetSumThreshold n : ℝ)) =O[atTop]\n fun n ↦ (n * Real.log n) ^ ((1 : ℝ) / 3)",
"target": null,
"source_url": "https://theoremdb.org/statements/P9924",
"format": "declaration-or-frag... | [] | TheoremDB P9924 · Candidate awaiting source review | null | Source-reported assessment, retrieved 8 September 2026. Frontier Science has not independently reviewed this status.
Open. No signed Lean verification is attached to this problem or its selected resolution.
This record is in the source backlog; its novelty and unresolved status have not passed source review. | null | published | incremental | hard | [
"source-candidate",
"theoremdb"
] | null | null | null | null | [
{
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"citation": "TheoremDB P9924",
"url": "https://theoremdb.org/statements/P9924",
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},
{
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"citation": "Source formal statement (not locally verified)",
"url": "https://theoremdb.org/statements/P9924",
"not... | {
"source_record": "P9924",
"imported_at": "2026-09-08",
"status_as_of": "2026-09-08",
"evidence_label": "Source candidate · not reviewed",
"note": "Full public directory statement. Source lane: backlog; source resolution: open. Source verification claims are attributed only; no local Lean verification has be... | {"created_at":"2026-08-17T20:04:35+00:00","display_standard":{"editorial_review":"submitter_supplied","formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":9924,"public_status":{"evidence":{"hasFormalization":false,"hasInformalProof":false,"hasRecordedWork":false,"solutionRecord":null},"lean":{... | [
"formalization-not-compiled-during-export",
"current-status-unassessed",
"source-full-text-not-present"
] | [] | {
"created_at": "2026-09-08T03:32:15.935Z",
"updated_at": "2026-09-08T03:32:15.935Z",
"published_at": "2026-09-08T03:33:07.362Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"324472fe-e3c0-457b-85e8-f486c31ed1ba","format_issues":null,"number":6943} |
1.0.0 | problem:theoremdb-p9926 | problem | theoremdb-p9926 | Boundedness of the Number of Representations of Integers as Sums of Two Cubes | {
"main": "math",
"sub": "number theory",
"slug": "number-theory",
"name": "Number theory",
"original_discipline": "mathematics"
} | [] | [
{
"code": "11",
"basis": "community-map",
"evidence": "Inferred from community field: number-theory"
}
] | For a set $A \subseteq \mathbb{N}$ and $n \in \mathbb{N}$, define the representation counting function $r_A(n)$ to be the number of ordered pairs $(a_1, a_2) \in A \times A$ such that $a_1 + a_2 = n$. Let $A$ be the set of perfect cubes, that is, $A = \{k^3 : k \in \mathbb{N}\}$. Determine whether there exists a natura... | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P9926",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P9926",
"citation": null,
"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_829 :
answer(sorry) ↔
∃ C : ℕ, (fun n : ℕ => (sumRep cubes n : ℝ)) =O[atTop]
(fun n : ℕ => (Real.log n) ^ C) | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem erdos_829 :\n answer(sorry) ↔\n ∃ C : ℕ, (fun n : ℕ => (sumRep cubes n : ℝ)) =O[atTop]\n (fun n : ℕ => (Real.log n) ^ C)",
"target": null,
"source_url": "https://theoremdb.org/statements/P9926",
"format": "declaration-or-fragment",
"sha... | [] | TheoremDB P9926 · Candidate awaiting source review | null | Source-reported assessment, retrieved 8 September 2026. Frontier Science has not independently reviewed this status.
Open. No signed Lean verification is attached to this problem or its selected resolution.
This record is in the source backlog; its novelty and unresolved status have not passed source review. | null | published | incremental | hard | [
"source-candidate",
"theoremdb"
] | null | null | null | null | [
{
"kind": "website",
"citation": "TheoremDB P9926",
"url": "https://theoremdb.org/statements/P9926",
"note": null,
"sort_order": 0
},
{
"kind": "formalization",
"citation": "Source formal statement (not locally verified)",
"url": "https://theoremdb.org/statements/P9926",
"not... | {
"source_record": "P9926",
"imported_at": "2026-09-08",
"status_as_of": "2026-09-08",
"evidence_label": "Source candidate · not reviewed",
"note": "Full public directory statement. Source lane: backlog; source resolution: open. Source verification claims are attributed only; no local Lean verification has be... | {"created_at":"2026-08-17T20:04:36+00:00","display_standard":{"editorial_review":"submitter_supplied","formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":9926,"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": "2026-09-08T03:32:50.735Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"2920e9c8-41f1-4df1-91e9-4a4d2b493bf8","format_issues":null,"number":3481} |
1.0.0 | problem:theoremdb-p9928 | problem | theoremdb-p9928 | Powerfulness of factorial minus one | {
"main": "math",
"sub": "number theory",
"slug": "number-theory",
"name": "Number theory",
"original_discipline": "mathematics"
} | [] | [
{
"code": "11",
"basis": "community-map",
"evidence": "Inferred from community field: number-theory"
}
] | A positive integer $m$ is called powerful if for every prime $p$ dividing $m$, the square $p^2$ also divides $m$. Determine whether $n! - 1$ is powerful for only finitely many positive integers $n$. | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P9928",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P9928",
"citation": null,
"doi": null,
"arxiv_id": null,
"license": "Selected packets label their content CC0-1.0; original references retain their own terms",
"notes": "Selected problem... | theorem erdos_936.variants.factorial_sub_one :
answer(sorry) ↔ EventuallyNotPowerful (·! - 1) | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem erdos_936.variants.factorial_sub_one :\n answer(sorry) ↔ EventuallyNotPowerful (·! - 1)",
"target": null,
"source_url": "https://theoremdb.org/statements/P9928",
"format": "declaration-or-fragment",
"sha256": "e48f0cecda7d6755287b0d51b5f68de3bc99155... | [] | TheoremDB P9928 · Candidate awaiting source review | null | Source-reported assessment, retrieved 8 September 2026. Frontier Science has not independently reviewed this status.
Open. No signed Lean verification is attached to this problem or its selected resolution.
This record is in the source backlog; its novelty and unresolved status have not passed source review. | null | published | incremental | hard | [
"source-candidate",
"theoremdb"
] | null | null | null | null | [
{
"kind": "website",
"citation": "TheoremDB P9928",
"url": "https://theoremdb.org/statements/P9928",
"note": null,
"sort_order": 0
},
{
"kind": "formalization",
"citation": "Source formal statement (not locally verified)",
"url": "https://theoremdb.org/statements/P9928",
"not... | {
"source_record": "P9928",
"imported_at": "2026-09-08",
"status_as_of": "2026-09-08",
"evidence_label": "Source candidate · not reviewed",
"note": "Full public directory statement. Source lane: backlog; source resolution: open. Source verification claims are attributed only; no local Lean verification has be... | {"created_at":"2026-08-17T20:04:37+00:00","display_standard":{"formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":9928,"public_status":{"evidence":{"hasFormalization":false,"hasInformalProof":false,"hasRecordedWork":false,"solutionRecord":null},"lean":{"kernelAccepted":false,"label":"","reaso... | [
"formalization-contains-placeholder",
"formalization-not-compiled-during-export",
"current-status-unassessed",
"source-full-text-not-present"
] | [] | {
"created_at": "2026-09-08T03:32:15.935Z",
"updated_at": "2026-09-08T03:32:15.935Z",
"published_at": "2026-09-08T03:33:05.031Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"aa9f8d19-9f7c-46a6-a490-0f5e2e1940bd","format_issues":null,"number":6438} |
1.0.0 | problem:theoremdb-p9930 | problem | theoremdb-p9930 | Graceful Tree Conjecture | {
"main": "math",
"sub": "combinatorics",
"slug": "combinatorics",
"name": "Combinatorics",
"original_discipline": "mathematics"
} | [] | [
{
"code": "05",
"basis": "community-map",
"evidence": "Inferred from community field: combinatorics"
}
] | Let $T = (V, E)$ be a tree with $m = |E|$ edges. A graceful labeling of $T$ is an injective function $f : V \to \{0, 1, \ldots, m\}$ such that the set of absolute differences $\{|f(u) - f(v)| : \{u,v\} \in E\}$ equals $\{1, 2, \ldots, m\}$. Does every tree admit a graceful labeling? | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P9930",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P9930",
"citation": null,
"doi": null,
"arxiv_id": null,
"license": "Selected packets label their content CC0-1.0; original references retain their own terms",
"notes": "Selected problem... | theorem graceful_tree_conjecture {V : Type*} [Fintype V] [DecidableEq V]
(T : SimpleGraph V) [DecidableRel T.Adj] (hT : T.IsTree) :
let m := T.edgeFinset.card
∃ f : V → ℕ,
Function.Injective f ∧
(∀ v, f v ≤ m) ∧
T.edgeFinset.image (fun e =>
e.lift ⟨fun u v => Int.natAbs ((f u : ℤ) ... | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem graceful_tree_conjecture {V : Type*} [Fintype V] [DecidableEq V]\n (T : SimpleGraph V) [DecidableRel T.Adj] (hT : T.IsTree) :\n let m := T.edgeFinset.card\n ∃ f : V → ℕ,\n Function.Injective f ∧\n (∀ v, f v ≤ m) ∧\n T.edgeFinset.image (fun e =... | [] | TheoremDB P9930 · Candidate awaiting source review | null | Source-reported assessment, retrieved 8 September 2026. Frontier Science has not independently reviewed this status.
Open. No signed Lean verification is attached to this problem or its selected resolution.
This record is in the source backlog; its novelty and unresolved status have not passed source review. | null | published | incremental | hard | [
"source-candidate",
"theoremdb"
] | null | null | null | null | [
{
"kind": "website",
"citation": "TheoremDB P9930",
"url": "https://theoremdb.org/statements/P9930",
"note": null,
"sort_order": 0
},
{
"kind": "formalization",
"citation": "Source formal statement (not locally verified)",
"url": "https://theoremdb.org/statements/P9930",
"not... | {
"source_record": "P9930",
"imported_at": "2026-09-08",
"status_as_of": "2026-09-08",
"evidence_label": "Source candidate · not reviewed",
"note": "Full public directory statement. Source lane: backlog; source resolution: open. Source verification claims are attributed only; no local Lean verification has be... | {"created_at":"2026-08-17T20:04:38+00:00","display_standard":{"editorial_review":"submitter_supplied","formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":9930,"public_status":{"evidence":{"hasFormalization":false,"hasInformalProof":false,"hasRecordedWork":false,"solutionRecord":null},"lean":{... | [
"formalization-not-compiled-during-export",
"current-status-unassessed",
"source-full-text-not-present"
] | [] | {
"created_at": "2026-09-08T03:32:15.935Z",
"updated_at": "2026-09-08T03:32:15.935Z",
"published_at": "2026-09-08T03:32:59.389Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"06a17ba6-4813-48de-af01-93e530d89313","format_issues":null,"number":5385} |
1.0.0 | problem:theoremdb-p9932 | problem | theoremdb-p9932 | Existence of a SIC-POVM in dimension 72 | {
"main": "math",
"sub": "algebra",
"slug": "algebra",
"name": "Algebra",
"original_discipline": "mathematics"
} | [] | [] | For a positive integer $d$, let $\mathbb{C}^d$ denote the $d$-dimensional complex Hilbert space with the standard inner product $\langle \phi, \psi \rangle = \sum_{i=0}^{d-1} \overline{\phi_i} \psi_i$. A state vector in $\mathbb{C}^d$ is a unit vector, i.e., a vector $\psi \in \mathbb{C}^d$ with $\|\psi\|_2 = 1$. For t... | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P9932",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P9932",
"citation": null,
"doi": null,
"arxiv_id": null,
"license": "Selected packets label their content CC0-1.0; original references retain their own terms",
"notes": "Selected problem... | theorem hasSICPOVM_72 : answer(sorry) ↔ HasSICPOVM 72 | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem hasSICPOVM_72 : answer(sorry) ↔ HasSICPOVM 72",
"target": null,
"source_url": "https://theoremdb.org/statements/P9932",
"format": "declaration-or-fragment",
"sha256": "bb18b10f8497101d698e81c9625333b45b51a5ddaa4b7cc218478b9062e1ef12",
"license": "... | [] | TheoremDB P9932 · Candidate awaiting source review | null | Source-reported assessment, retrieved 8 September 2026. Frontier Science has not independently reviewed this status.
Open. No signed Lean verification is attached to this problem or its selected resolution.
This record is in the source backlog; its novelty and unresolved status have not passed source review. | null | published | incremental | hard | [
"source-candidate",
"theoremdb"
] | null | null | null | null | [
{
"kind": "website",
"citation": "TheoremDB P9932",
"url": "https://theoremdb.org/statements/P9932",
"note": null,
"sort_order": 0
},
{
"kind": "formalization",
"citation": "Source formal statement (not locally verified)",
"url": "https://theoremdb.org/statements/P9932",
"not... | {
"source_record": "P9932",
"imported_at": "2026-09-08",
"status_as_of": "2026-09-08",
"evidence_label": "Source candidate · not reviewed",
"note": "Full public directory statement. Source lane: backlog; source resolution: open. Source verification claims are attributed only; no local Lean verification has be... | {"created_at":"2026-08-17T20:04:39+00:00","display_standard":{"editorial_review":"submitter_supplied","formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":9932,"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": "2026-09-08T03:32:58.532Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"9bf7795d-7d2d-46e9-8098-6b3743fc793d","format_issues":null,"number":5202} |
1.0.0 | problem:theoremdb-p9934 | problem | theoremdb-p9934 | Non-existence of monochromatic quantum graphs with six vertices over the reals | {
"main": "math",
"sub": "combinatorics",
"slug": "combinatorics",
"name": "Combinatorics",
"original_discipline": "mathematics"
} | [] | [
{
"code": "05",
"basis": "community-map",
"evidence": "Inferred from community field: combinatorics"
}
] | Let $N$ be a positive even integer and let $D$ be a positive integer. A quantum graph with $N$ vertices and $D$ colours consists of a complete graph on $N$ vertices together with a weight function $W$ that assigns to each edge a value in a coefficient domain, where each edge is also assigned one of $D$ colours. For eac... | unassessed | {
"key": "theoremdb",
"name": "TheoremDB · P9934",
"type": "problem-list",
"url": "https://theoremdb.org/statements/P9934",
"citation": null,
"doi": null,
"arxiv_id": null,
"license": "Selected packets label their content CC0-1.0; original references retain their own terms",
"notes": "Selected problem... | theorem eqSystem6_no_solution_ge3_real :
answer(sorry) ↔
∀ D : Nat, D ≥ 3 →
¬ ∃ W : WeightsN 6 D ℝ, EqSystemN 6 D W | null | null | not-recorded | not-recorded | null | [
{
"language": "Lean 4",
"code": "theorem eqSystem6_no_solution_ge3_real :\n answer(sorry) ↔\n ∀ D : Nat, D ≥ 3 →\n ¬ ∃ W : WeightsN 6 D ℝ, EqSystemN 6 D W",
"target": null,
"source_url": "https://theoremdb.org/statements/P9934",
"format": "declaration-or-fragment",
"sha256": "... | [] | TheoremDB P9934 · Candidate awaiting source review | null | Source-reported assessment, retrieved 8 September 2026. Frontier Science has not independently reviewed this status.
Open. No signed Lean verification is attached to this problem or its selected resolution.
This record is in the source backlog; its novelty and unresolved status have not passed source review. | null | published | incremental | hard | [
"source-candidate",
"theoremdb"
] | null | null | null | null | [
{
"kind": "website",
"citation": "TheoremDB P9934",
"url": "https://theoremdb.org/statements/P9934",
"note": null,
"sort_order": 0
},
{
"kind": "formalization",
"citation": "Source formal statement (not locally verified)",
"url": "https://theoremdb.org/statements/P9934",
"not... | {
"source_record": "P9934",
"imported_at": "2026-09-08",
"status_as_of": "2026-09-08",
"evidence_label": "Source candidate · not reviewed",
"note": "Full public directory statement. Source lane: backlog; source resolution: open. Source verification claims are attributed only; no local Lean verification has be... | {"created_at":"2026-08-17T20:04:40+00:00","display_standard":{"editorial_review":"submitter_supplied","formal_correspondence":"not_assessed"},"lean_verified":false,"problem_number":9934,"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": "2026-09-08T03:33:04.352Z",
"solved_at": null
} | 2026-10-10T16:07:49.246Z | {"database_id":"8fafe112-b00b-4df1-ab59-eb77d532f652","format_issues":null,"number":6292} |
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