PROVED
This has been solved in the affirmative.
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 $\alpha>\beta$?
It is trivial that\[\delta(m,\alpha)<\frac{m^\alpha+1}{m}\to 0\]if $\alpha <1$, and Erdős claims in
[Er79e] he could prove that the same is true for $\alpha=1$.
This was proved in the affirmative with $\beta=1/\log 2$ by Hall
[Ha92].
See also
[696].
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This page was last edited 20 December 2025.
Additional thanks to: Alfaiz
When referring to this problem, please use the original sources of Erdős. If you wish to acknowledge this website, the recommended citation format is:
T. F. Bloom, Erdős Problem #697, https://www.erdosproblems.com/697, accessed 2026-01-16
