Erdős Problem #997 — 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 $I\subseteq [0,1]$,…

Erdős asked whether, for every real $\alpha$, the fractional parts along the primes are never well-distributed; he had earlier claimed and then retracted an existential version. The universal assertion is now claimed in a manuscript by Alexeev, Putterman, Sawhney, Sellke, and Valiant.

Known results

• Erdős, 1964: claimed an irrational $\alpha$ with failure of well-distribution; he retracted the claim in 1985.
• Champagne, Lê, Liu, and Wooley, 2024: proved that an irrational, indeed transcendental, $\alpha$ with this failure exists.
• Vinogradov: proved ordinary equidistribution for every irrational $\alpha$, a weaker property.

March 2026 universal-proof claim

The manuscript claims that every real $\alpha$ yields arbitrarily long strings of consecutive primes whose fractional parts cluster, using Dirichlet approximation and Maynard–Tao–BFT results. It says the proof came entirely from an internal OpenAI model; the claim remains unverified.

Current status (as of September 2026): The existential case is established, while the universal statement is claimed in arXiv:2603.29961 but remains unverified.