PROVED (LEAN)
This has been solved in the affirmative and the proof verified in Lean.
Call $x_1,x_2,\ldots \in (0,1)$ well-distributed if, for every $\epsilon>0$, if $k$ is sufficiently large then, for all $n>0$ and intervals $I\subseteq [0,1]$,\[\lvert \# \{ n<m\leq n+k : x_m\in I\} - \lvert I\rvert k\rvert < \epsilon k.\]Is it true that, for every $\alpha$, the sequence $\{ \alpha p_n\}$ is not well-distributed, if $p_n$ is the sequence of primes?
The notion of a well-distributed sequence was introduced by Hlawka and Petersen
[Hl55].
Erdős proved that, if $n_k$ is a lacunary sequence, then the sequence $\{ \alpha n_k\}$ is not well-distributed for almost all $\alpha$.
He also claimed in
[Er64b] to have proved that there exists an irrational $\alpha$ for which $\{\alpha p_n\}$ is not well-distributed. He later retracted this claim in
[Er85e], saying "The theorem is no doubt correct and perhaps will not be difficult to prove but I never was able to reconstruct my 'proof' which perhaps never existed ."
The existence of such an $\alpha$ was established by Champagne, Le, Liu, and Wooley
[CLLW24].
An internal model at OpenAI
[APSSV26] has proved that the sequence $\{\alpha p_n\}$ is not well-distributed for any $\alpha$, using the work of Banks, Freiberg, and Turnage-Butterbaugh
[BFT15] on consecutive primes with small gaps (which in turn builds on the work of Maynard and Tao on bounded gaps between primes).
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This page was last edited 01 April 2026. View history
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 #997, https://www.erdosproblems.com/997, accessed 2026-04-11
