Singmaster’s problem: new bound by Claude Opus 5.5
Awaiting review
Importance: major (4 of 5)The takeaway
Ben Whitmore deposited on Hexagon a paper proving unconditionally that the number N(t) of ways to write t as a binomial coefficient is O((log t)^(1/2+o(1))), improving the previous unconditional bound and beating the earlier bound conditional on Cramér’s conjecture.
Status
- Claim
Awaiting review
- Our reporting
- Medium confidence
- Verification
- Lean-verified
- Importance
- Major (4 of 5)
- Last verified
- 6 November 2026
Your AI and this story
- GPT-6 Astra189 days after its cutoff
- Claude Opus 5.5128 days after its cutoff
- Gemini 4 Argon219 days after its cutoff
- Grok 4.7158 days after its cutoff
None of these four assistants can know about it. The closest, Claude Opus 5.5, stops 128 days before it.
Key facts
- Hexagon 2611.00106, 5 Nov 2026; authorship ‘no-human-author-claimed’, human understanding ‘some’
- Previous best: N(t) = O(log t · logloglog t / (loglog t)^3) unconditionally; O((log t)^(2/3+ε)) conditional on Cramér’s conjecture
- New: N(t) = O((log t)^(1/2+o(1))) unconditionally; large m handled via a theorem of Matomäki et al.
- Provenance: ‘All of the proofs, the Lean 4 formalisation, and most of the text content of the paper were generated by Claude Opus 5.5.’
- Singmaster conjectured in 1971 that N(t) = O(1)
What happened
Hexagon 2611.001061, contributed by Ben Whitmore on 5 Nov 2026, claims a new unconditional upper bound for Singmaster’s function N(t). The record states that Claude Opus 5.5 generated all proofs and a Lean 4 formalization. Hexagon does not judge correctness; we did not check which statements the Lean files cover.
Why it matters
Singmaster’s problem is a well-known open problem in elementary number theory, and the claimed exponent 1/2 improves even on what was previously known under Cramér’s conjecture. The result is unrefereed and no expert comment was found on 6 Nov.
What is disputed or not yet verified
| Verification | Hexagon deposit with Lean 4 formalization (per provenance); unrefereed |
|---|
Sources
1 source from 1 site. Numbers match the chips in the text.
1 source: 1 primary
Primary
Changes
- Filed from the Hexagon record