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Erdős·erdos:730

Erdős Problem 730

problem formal record: solved source: solved F2 declared

No independent check recorded yet. A formal artifact, declaration or published object is attached, but no rebuild of it is recorded here.

Fidelity F2: The correspondence is declared through a Comparator challenge, an alignment table and written divergences.

number theory, binomial coefficients, base representations·Source

AI activity

How grades work
GPT-5.5 Pro

24 Jun 2026

Candidate full solution

candidate A3 V0 F2
Reasoning and sources

Autonomy

AI standalone; human involvement recorded as non-significant

Claimed on erdosproblems.com

1

Proof claims posted on erdosproblems.com, which says that listing a claim “is no guarantee of proof correctness”. The register records who claims what, with which systems, and links to each claim there. Nobody has examined them, and none counts in the register's totals.

Liam Price

a full proof claimed·15 Jul 2026·using GPT Pro·1 comment there

Proof

candidateA? V0

Fidelity

How fidelity is graded

F2 declared. The correspondence is declared through a Comparator challenge, an alignment table and written divergences.

declares divergences from its source reviewed by its authors only

Declared by the projects

2

As each project's formalization.yaml states it.

lean-proofs: formal Lean 4 proofs of solved Erdős problemsplby/lean-proofs/src/latest/ErdosProblems/Erdos730 · joined by names · no independent check

Read formalization.yaml

authors
Will Blair
method
agent
review
self-assessed
axioms
Classical.choice, Quot.sound, propext
sorry
0 unproved goals declared
results
3 main results named, checked with Comparator, with an alignment table
sources
Erdős Problem #730 (erdosproblems.com), after P. Erdős, R. L. Graham, I. Z. Ruzsa and E. G. Straus, 'On the prime factors of C(2n, n)', Math. Comp. 29 (1975) — background, authors n/a; Comment on Erdős Problem #730 asserting, with a one-paragraph gist, that GPT Pro proves infinitely many consecutive pairs (n, n+1) — formalizes, authors not-contacted; Closing derivation ('Proof Route Mapping') linked as a follow-up, reproducing the algebraic skeleton of the argument; the analytic sections exist only in a private document and are reconstructed in this formalisation — adapts, authors not-contacted; Bernt Lindström, 'Well distribution of Sidon sets in residue classes', J. Number Theory 69 (1998), 197–200 — adapts, authors n/a
related
Woett/Lean-files/blob/main/ErdosProblem154.lean — builds-on; AlexKontorovich/PrimeNumberTheoremAnd — builds-on; google-deepmind/formal-conjectures/pull/664 — other
divergences
#730: none in the statement — the Challenge inlines the Formal Conjectures set verbatim; the proof establishes the stronger consecutive-pair statement, available in the development as Erdos730.FullDensityCore.GoodParameter and the density theorems around it. #154: the Challenge states the sumset form Formal Conjectures records, which is the consequence proved here of Lindström's theorem for A itself, with IsSidon in the Formal Conjectures shape (two representations agree up to order). #94: an elementary bounded identity only; it does not prove the cubic distance-multiplicity theorem or the regular-polygon conjecture of that problem.
checked by
nobody independent of its authors yet
lean-proofs: formal Lean 4 proofs of solved Erdős problemswilliamjblair/lean-proofs · joined by artifact · Palomar · a project holding several results

Read formalization.yaml

authors
Will Blair
method
agent
review
self-assessed
axioms
Classical.choice, Quot.sound, propext
sorry
0 unproved goals declared
results
3 main results named, checked with Comparator, with an alignment table
sources
Erdős Problem #730 (erdosproblems.com), after P. Erdős, R. L. Graham, I. Z. Ruzsa and E. G. Straus, 'On the prime factors of C(2n, n)', Math. Comp. 29 (1975) — background, authors n/a; Comment on Erdős Problem #730 asserting, with a one-paragraph gist, that GPT Pro proves infinitely many consecutive pairs (n, n+1) — formalizes, authors not-contacted; Closing derivation ('Proof Route Mapping') linked as a follow-up, reproducing the algebraic skeleton of the argument; the analytic sections exist only in a private document and are reconstructed in this formalisation — adapts, authors not-contacted; Bernt Lindström, 'Well distribution of Sidon sets in residue classes', J. Number Theory 69 (1998), 197–200 — adapts, authors n/a
related
Woett/Lean-files/blob/main/ErdosProblem154.lean — builds-on; AlexKontorovich/PrimeNumberTheoremAnd — builds-on; google-deepmind/formal-conjectures/pull/664 — other
divergences
#730: none in the statement — the Challenge inlines the Formal Conjectures set verbatim; the proof establishes the stronger consecutive-pair statement, available in the development as Erdos730.FullDensityCore.GoodParameter and the density theorems around it. #154: the Challenge states the sumset form Formal Conjectures records, which is the consequence proved here of Lindström's theorem for A itself, with IsSidon in the Formal Conjectures shape (two representations agree up to order). #94: an elementary bounded identity only; it does not prove the cubic distance-multiplicity theorem or the regular-polygon conjecture of that problem.
checked by
Palomar

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Formal material

Formal statements · 1
Cited proofs · 1
Also known as · 3
  • https://www.erdosproblems.com/730
  • https://github.com/google-deepmind/formal-conjectures/blob/main/FormalConjectures/ErdosProblems/730.lean
  • FormalConjectures/ErdosProblems/730.lean

Cite this record

qed.bot, “Erdős Problem 730”, https://qed.bot/s/erdos-730, as of 30 Sep 2026.

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