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OP-8886OpenMathematics · general

Erdős #200 · primes / arithmetic progressions

Does the longest arithmetic progression of primes in {1,…,N}\{1,\ldots,N\} have length o(log⁡N)o(\log N)?

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Problem statement

Does the longest arithmetic progression of primes in {1,…,N}\{1,\ldots,N\} have length o(log⁡N)o(\log N)?

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Background

It follows from the prime number theorem that such a progression has length ≤(1+o(1))log⁡N\leq(1+o(1))\log N.

Progress & source assessment

The source labels this entry OPEN in the 2026-09-08 snapshot. Editorial progress notes are reproduced as source commentary, not independently verified results. Prize amounts and Lean badges at the source are not local awards or verification.

References
  1. [1]Thomas Bloom (ed.), Erdős Problems #200 link
  2. [2]Upstream Lean 4 statement for Erdős #200 linkExternal statement reference. Not compiled or verified by this platform; upstream files may contain open proof obligations.
  3. [3]Formal Conjectures — Erdős Problem 200 link

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