Erdős Problem #997
PROVED LEANdatabase updated 2026-04-03
The statement as posed is true, and a proof exists. A solution has also been checked by machine, in Lean.
What the community database records, not our verdict. erdosproblems.com no longer shows statuses (why).
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 \(I\subseteq [0,1]\), \(\lvert \# \{ n < m\leq n+k : x_m\in I\} - \lvert I\rvert k\rvert < \epsilon k.\) The notion of a well-distributed sequence was introduced by Hlawka and Petersen [Hl55].
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 answer is yes, by [APSSV26, Section 4]; a Lean formalisation is available in [Mo26].
Statement as formalized in Formal Conjectures (Google DeepMind, Apache-2.0) · their label: research solved · 2 further variants / known partial results in the Lean file. The authoritative wording is on erdosproblems.com/997.Prizenone
Topicsanalysis, discrepancy, primes
Statement in LeanFormal Conjectures
Solution in Leanyes
Original pageerdosproblems.com/997
Attempts
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