Well-distribution of prime multiples modulo one
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Can the fractional parts of αp, with p running through consecutive primes, be well-distributed uniformly over the starting index of a block? This is stronger than ordinary equidistribution and asks about every real α.
Why it matters and what counts as progress
Why it matters: Uniformity in the starting index is stronger than ordinary equidistribution, so a negative answer is compatible with classical prime equidistribution.
What counts as progress: Prove universal failure of well-distribution or find an α satisfying the stronger uniform-in-starting-index property.
How the problem developed
Paul Erdős Discussed the well-distribution of irrational multiples of the primes and claimed an example for which the sequence is not well-distributed; he later retracted that claim.
- 27 Jun 2024 · Counterexample [3]
J. Champagne, T. H. Lê, Y.-R. Liu and T. D. Wooley Constructed an irrational α for which (αp_n) is not well-distributed modulo one, while leaving open whether every irrational α has this failure.
- 31 Mar 2026 · Reported claim [4]
Boris Alexeev, Moe Putterman, Mehtaab Sawhney, Mark Sellke and Gregory Valiant Proved that (αp_n) is not well-distributed for every real α in the uniform-in-start sense; the paper attributes the proof entirely to an internal OpenAI model.
AI contribution
Short proofs in combinatorics and number theory
For every real α, the sequence of fractional parts of α times consecutive primes is not well-distributed in the uniform-in-starting-index interval sense. This does not say ordinary equidistribution fails.
Sources
- Erdős, Problems and results on diophantine approximations
- Alexeev and coauthors, Short Proofs in Combinatorics and Number Theory
- Champagne, Lê, Liu and Wooley, Well-distribution modulo one and the primes
- Alexeev et al., Short proofs in combinatorics and number theory
- Formal Conjectures, Erdős problem statement and later proof
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