Problem 682

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claims/: The 1 claim page of Problem 682, one per claimant's result; the problem's standing derives from them.

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Statement. Is it true that for almost all $n$ there exists some $m\in (p_n,p_{n+1})$ such that

$$p(m) \geq p_{n+1}-p_n,$$

where $p(m)$ denotes the least prime factor of $m$?

Status. Proved, on the site's label, which credits Gafni and Tao [GaTa25] for the affirmative answer: all but $O(X/(\log X)^2)$ of the prime gaps starting in $[X,2X]$ contain an integer whose least prime factor is at least the gap. A Lean proof of the statement in Alexeev's repository, not built here, is recorded beside the credit on the claim page.

Source. erdosproblems.com/682, accessed 2026-09-04. Cite as: T. F. Bloom, Erdős Problem #682, https://www.erdosproblems.com/682.

References.

• [GaTa25] A. Gafni and T. Tao, Rough numbers between consecutive primes. arXiv:2508.06463 (2025).

Formalization. Statement in formal-conjectures, at its commit of 22 September 2026, whose formal_proof attribute points to the proof in Alexeev's lean-proofs repository linked from the claim page; neither is built or audited here.

Progress

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Known Results

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Linked library material

These entries are derived from explicit links on library pages. They are navigation only and do not by themselves record mathematical progress.

• gafni_2025_rough_numbers_between_consecutive_primes