DISPROVED (LEAN)
This has been solved in the negative and the proof verified in Lean.
Let $\delta_1(n,m)$ be the density of the set of integers with exactly one divisor in $(n,m)$. Is $\delta_1(n,m)$ unimodular for $m>n+1$ (i.e. increases until some $m$ then decreases thereafter)? For fixed $n$, where does $\delta_1(n,m)$ achieve its maximum?
This was asked by Erdős in 1986 at Oberwolfach.
Erdős proved that\[\delta_1(n,m) \ll \frac{1}{(\log n)^c}\]for all $m$, for some constant $c>0$. Sharper bounds (for various ranges of $n$ and $m$) were given by Ford
[Fo08].
Cambie has calculated that unimodularity fails even for $n=2$ and $n=3$. For example,\[\delta_1(3,6)= 0.35\quad \delta_1(3,7)\approx 0.33\quad \delta_1(3,8)\approx 0.3619.\]Furthermore, Cambie
[Ca25] has shown that, for fixed $n$, the sequence $\delta_1(n,m)$ has superpolynomially many local maxima $m$.
See also
[446].
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This page was last edited 04 November 2025. View history
Additional thanks to: Stijn Cambie
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 #692, https://www.erdosproblems.com/692, accessed 2026-04-11
