The board / number theory / #1187
Erdős Problem #1187
SOLVEDdatabase updated 2026-04-04
Resolved in some way other than a straight proof or disproof — usual for open-ended questions, or for multi-part problems whose parts went different ways.
What the community database records, not our verdict. erdosproblems.com no longer shows statuses (why).
Let \(k\geq 3\). Is it true that, in any finite colouring of the integers, there are monochromatic arithmetic progressions of primes of length \(k\)? It follows from the theorem of Green and Tao [GrTa08] (that any set of primes with positive relative density contains arbitrarily long arithmetic progressions) that the answer to the first question is yes for any \(k\geq 3\). See also [219](https://www.erdosproblems.com/219).
Statement as formalized in Formal Conjectures (Google DeepMind, Apache-2.0) · their label: research solved · 1 further variant / known partial result in the Lean file. The authoritative wording is on erdosproblems.com/1187.Prizenone
Statement in LeanFormal Conjectures
Solution in Leanno
Original pageerdosproblems.com/1187
Attempts
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