Erdős Problem #387
Erdős and Graham asked whether with must always have a divisor that is close to , meaning bigger than a fixed constant times . Settled in both directions: true when is large enough as a function of , but false in general, since there are with small compared to having no such divisor.
- Result
- Disproved(see note)
- Status
- Resolved
- AI contribution
- AI co-developed
- Method
- Argument
- Field
- Number theory
- Posed by
- Paul Erdős, Ronald Graham
- Year posed
- —
- Years open
- —
- Solved
- 2026-05-20
- Model
- ChatGPT 5.5 Pro
- Vendor
- OpenAI
- Collaborators
- Hung M. Bui, Slava Naprienko, Kyle Pratt, Alexandru Zaharescu
- Verification
- Unreviewed
- Publication
- Preprint
- Significance
- 35 / 100
- Disclosed cost
- —
- Wikipedia
- No dedicated article
What was actually shown
false in general; the positive direction holds for k large relative to n
What the AI did
The disclosure is carefully scoped rather than blanket. The main ideas in the proof of Theorem 5.1 were developed in interactive sessions between the authors and ChatGPT 5.5 Pro, and some documents and code in the accompanying repository were generated with AI assistance. The authors separately used ChatGPT for literature searches and for spotting typos, and they state that all text in the paper is human-generated. The heavier half of the paper, a restricted covering problem attacked with sieve methods and exponential sum estimates, is presented as the authors' own.
Verification
arXiv preprint, not peer-reviewed. The authors credit the model with the main ideas of one theorem rather than the paper, so the bulk of the argument rests on ordinary refereeing.
Source
arXiv:2605.21221 - Binomial coefficients with divisors avoiding an interval