Problem 697

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claims/: The 1 claim page of Problem 697, one per claimant's result; the problem's standing derives from them.

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Statement. Let $\delta(m,\alpha)$ denote the density of the set of integers which are divisible by some $d\equiv 1\pmod{m}$ with $1<d<\exp(m^\alpha)$. Does there exist some $\beta\in (1,\infty)$ such that

$$\lim_{m\to \infty}\delta(m,\alpha)$$

is $0$ if $\alpha<\beta$ and $1$ if $\alpha>\beta$?

Status. Proved. The derived standing, solved and proved, rests on Hall's accepted claim: the threshold exists and equals $1/\log2$.

Source. erdosproblems.com/697, accessed 2026-09-04. Cite as: T. F. Bloom, Erdős Problem #697, https://www.erdosproblems.com/697.

References.

• [Er79e] Erdős, Paul, Some unconventional problems in number theory. Astérisque (1979), 73-82.
• [Ha92] Hall, R. R., On some conjectures of Erdős in Astérisque, I. J. Number Theory 42 (1992), no. 3, 313-319.

Formalization. Statement in formal-conjectures.

Current assessment

The site's formulation of 2026-09-04 asks whether a threshold $\beta>1$ separates limit $0$ from limit $1$ for $\delta(m,\alpha)$. Hall's refereed paper answers yes with $\beta=1/\log2$, and the site's curator credits it; that is the accepted claim. Below the threshold the trivial bound $\delta(m,\alpha)<(m^\alpha+1)/m$ already gives the limit $0$ for $\alpha<1$, and Erdős writes on p. 81 of Er79e that he can prove the case $\alpha=1$; the same page poses the threshold question, which it calls related to the estimation of the divisor chains $H(n)$ of Problem 696.

Hall's paper is not held; no proof is compiled and no independent review is recorded, so the account rests on the publication record and the site's credit. A Lean formalization of Hall's theorem by Codex and GPT-5.6 Sol, in Boris Alexeev's repository of Lean proofs, is linked on the claim page; it has not been built here.

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.

• erdos_1979_unconventional_problems_number_theory_asterisque