The board / number theory / #645
Erdős Problem #645
PROVED LEANdatabase updated 2025-11-23
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).
If ℕ is \(2\)-coloured then there must exist a monochromatic three-term arithmetic progression \(x,x+d,x+2d\) such that \(d>x\). This was first proved by Brown and Landman [BrLa99], who in fact show that this is always possible with \(d>f(x)\) for any increasing function \(f\). This was formalized in Lean by Alexeev using Aristotle and ChatGPT.
Statement as formalized in Formal Conjectures (Google DeepMind, Apache-2.0) · their label: research solved. The authoritative wording is on erdosproblems.com/645.Prizenone
Statement in LeanFormal Conjectures
Solution in Leanyes
Original pageerdosproblems.com/645
Attempts
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