Problem 768

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claims/: The 1 claim page of Problem 768, one per claimant's result; the problem's standing derives from them.

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Statement. Let $A\subset\mathbb{N}$ be the set of $n$ such that for every prime $p\mid n$ there exists some $d\mid n$ with $d>1$ such that $d\equiv 1\pmod{p}$. Is it true that there exists some constant $c>0$ such that for all large $N$

$$\frac{\lvert A\cap [1,N]\rvert}{N}=\exp(-(c+o(1))\sqrt{\log N}\log\log N).$$

Status. The site's label is OPEN (page last edited 14 September 2025; proof-claim tab accessed 2026-10-06).

Source. erdosproblems.com/768, accessed 2026-09-04. Cite as: T. F. Bloom, Erdős Problem #768, https://www.erdosproblems.com/768.

Formalization. No formalized statement on the site. The claimant's Lean 4 development and Johan Land's independent development of Li's proof are linked from the claim page; neither was built or audited here.

Current assessment

The standing derives from the claim pages: the full claim Li 2026, an arXiv preprint of 23 June 2026 submitted to the site's proof-claim tab on 17 July 2026, where the tab records it as made using GPT-5.5 Pro, asserts that the asymptotic holds with $c=1/(2\sqrt{\log2})$, with a Lean 4 development the author says proves the main theorem. The site had not acted on it when the tab was accessed, no referee or named expert is recorded as having examined it, and nothing was built or audited here, so the claim is claimed and the problem's standing is claimed, proved.

The site's commentary records Erdős's own bounds: a lower bound $\exp(-c\sqrt{\log N}\log\log N)$ for some $c>0$ and an upper bound $\exp(-(1+o(1))\sqrt{\log N\log\log N})$, and his motive, that $\lvert A\cap[1,N]\rvert$ bounds the number of orders $n\le N$ of non-cyclic simple groups.

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.

• li_2026_resolution_erdos_problem_768_sylow_divisor
• li_2026_resolution_erdos_problem_768_sylow_divisor / theorem_1_1
• li_2026_resolution_erdos_problem_768_sylow_divisor / theorem_4_3
• li_2026_resolution_erdos_problem_768_sylow_divisor / theorem_8_3