Problem 997

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claims/: The 2 claim pages of Problem 997, one per claimant's result; the problem's standing derives from them.

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Statement. 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?

Status. PROVED (LEAN): Alexeev, Putterman, Sawhney, Sellke and Valiant [APSSV26] showed that $\{\alpha p_n\}$ is not well-distributed for every real $\alpha$, the accepted claim Alexeev, Putterman, Sawhney, Sellke and Valiant 2026, accepted on the site's label and Terence Tao's thread comment; no journal version of the preprint was found on 2026-10-07. The site's Lean qualification refers to a formalization that takes the Banks–Freiberg–Turnage-Butterbaugh theorem [BFT15] as an axiom; a later public development in Boris Alexeev's repository states an unconditional proof of the same statement; neither is built or audited here, so the claim page lists no formalized evidence. Champagne, Lê, Liu and Wooley [CLLW24] had earlier found one irrational $\alpha$ with this property, the partial claim Champagne, Lê, Liu and Wooley 2024.

Source. erdosproblems.com/997, accessed 2026-09-04. Cite as: T. F. Bloom, Erdős Problem #997, https://www.erdosproblems.com/997.

References.

• [APSSV26] B. Alexeev, M. Putterman, M. Sawhney, M. Sellke, and G. Valiant, Short proofs in combinatorics and number theory. arXiv:2603.29961 (2026).
• [BFT15] Banks, William D. and Freiberg, Tristan and Turnage-Butterbaugh, Caroline L., Consecutive primes in tuples. Acta Arith. (2015), 261-266.
• [CLLW24] J. Champagne, T. Le, Y.-R. Liu, and T. D. Wooley, Well-distribution modulo one and the primes. arXiv:2406.19491 (2024).
• [Er64b] Erdős, P., Problems and results on diophantine approximations. Compositio Math. (1964), 52-65.
• [Er85e] Erdős, P., Some problems and results in number theory. Number theory and combinatorics. Japan 1984 (Tokyo, Okayama and Kyoto, 1984) (1985), 65-87.
• [Hl55] Hlawka, Edmund, Zur formalen Theorie der Gleichverteilung in kompakten Gruppen. Rend. Circ. Mat. Palermo (2) (1955), 33-47.

Formalization. Statement in formal-conjectures (the revision of 2026-09-18, pinned in the link), marked research solved and pointing at a Lean 4 proof posted by Monticone, autoformalized by Aristotle, that assumes the Banks–Freiberg–Turnage-Butterbaugh theorem as an axiom; a later version of that file in Boris Alexeev's repository states an unconditional proof. The claim page records the pinned revisions and the qualifications.

Current assessment

Erdős wrote in [Er64b] that he could prove that there is an irrational $\alpha$ for which $(p_n\alpha)$ is not well distributed, and that it seemed very probable that $(p_n\alpha)$ is well distributed for no $\alpha$, which he could not show. In [Er85e] he retracted the first statement, saying he had never been able to reconstruct the proof, while holding the second as beyond doubt for every irrational $\alpha$. Champagne, Lê, Liu and Wooley [CLLW24] proved the existence statement in 2024, and Alexeev, Putterman, Sawhney, Sellke and Valiant [APSSV26] proved the conjecture for every real $\alpha$ in 2026; the site accepted the latter as the resolution.

Known Results

Theorem 1.1 of [CLLW24]: there is an irrational, indeed transcendental, $\alpha$ for which $(\alpha p_n)$ is not well-distributed modulo $1$, refereed in Proc. Amer. Math. Soc. 153 (2025), recorded on the partial claim page. Theorem 4.1 of [APSSV26]: for every real $\alpha$ the sequence $\{\alpha p_n\}$ is not well-distributed, proved by approximating $\alpha$ by a rational and taking from the Banks–Freiberg–Turnage-Butterbaugh theorem [BFT15] a run of consecutive primes in one residue class, recorded on the full claim page.

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_1985_problems_results_number_theory
• champagne_2024_well_distribution_modulo_one_primes
• champagne_2024_well_distribution_modulo_one_primes / lemma_2_1
• champagne_2024_well_distribution_modulo_one_primes / theorem_1_1
• erdos_1964_problems_results_diophantine_approximations
• alexeev_2026_short_proofs_combinatorics_number_theory
• alexeev_2026_short_proofs_combinatorics_number_theory / theorem_4_1
• alexeev_2026_short_proofs_combinatorics_number_theory / theorem_4_2