VibeMathedMath problems solved by AI

Erdos's Conjecture on Consecutive Integers Free of Certain Prime Factors

Let nkn_k be the least n>2kn > 2k such that (n−k)(n−k+1)⋯(n−1)(n-k)(n-k+1)\cdots(n-1) has no prime factor in (k,2k)(k, 2k). Erdos conjectured a superpolynomial lower bound; for all large kk, nk>elog⁡2k/(20log⁡log⁡k)n_k > e^{\log^2 k / (20 \log\log k)}.

Result
Proved
Status
Resolved
AI contribution
AI-discovered
Method
Argument
Field
Number theory
Posed by
Paul Erdos
Year posed
—
Years open
—
Solved
2026-06-18
Model
ChatGPT 5.5 Pro, Aristotle
Vendor
OpenAI / Harmonic
Collaborators
Wouter van Doorn, Quanyu Tang
Verification
Unreviewed
Publication
Preprint
Significance
15 / 100
Disclosed cost
—
Wikipedia
No dedicated article

What the AI did

The paper says the bound was conceived of by ChatGPT 5.5 Pro and that the original paper the model wrote remains publicly available, so the provenance is checkable rather than asserted. The supporting proofs were obtained by Harmonic's Aristotle and are stated to be fully self-contained.

Verification

Proofs produced by an automated theorem prover and described as fully self-contained; we have not independently checked them. arXiv preprint, not peer-reviewed.

Source

arXiv:2606.19863 - Consecutive integers free of certain prime factors

Discussion