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OP-8854OpenCombinatorics

Erdős #141 · additive combinatorics / primes

Let k≥3k\geq 3. Are there kk consecutive primes in arithmetic progression?

MSC 05 · Combinatorics

Inferred from the community field; finer classification has not been reviewed.

Source reports OPENAs of 2026-09-08Source & evidence ↓
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Problem statement

Let k≥3k\geq 3. Are there kk consecutive primes in arithmetic progression?

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Background

Green and Tao [GrTa08] have proved that there must always exist some kk primes in arithmetic progression, but these need not be consecutive. Erdős called this conjecture 'completely hopeless at present'.

The existence of such progressions for small kk has been verified for k≤10k\leq 10, see the Wikipedia page. It is open, even for k=3k=3, whether there are infinitely many such progressions.

See also [219].

This is discussed in problem A6 of Guy's collection [Gu04].

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 #141 link
  2. [2]Upstream Lean 4 statement for Erdős #141 linkExternal statement reference. Not compiled or verified by this platform; upstream files may contain open proof obligations.
  3. [3]Formal Conjectures — Erdős Problem 141 link

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