The board / number theory / #937
Erdős Problem #937
PROVED LEANdatabase updated 2026-08-23
The statement as posed is true, and a proof exists. A solution has also been checked by machine, in Lean.
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The four numbers \(a, a+d, a+2d, a+3d\) form a four-term arithmetic progression (\(d > 0\)) of pairwise coprime powerful numbers.
Are there infinitely many four-term arithmetic progressions of coprime powerful numbers? (A number \(n\) is *powerful* if \(p \mid n \to p^2 \mid n\); `Nat.Powerful`.) Erdős [Er76d] asked this; the answer is **yes**: Bajpai, Bennett and Chan [BBC24] proved that there are infinitely many four-term arithmetic progressions of pairwise coprime powerful numbers. (Without coprimality this is easy, and by a theorem of Fermat there are no four *squares* in arithmetic progression.)
Statement as formalized in Formal Conjectures (Google DeepMind, Apache-2.0) · their label: research solved. The authoritative wording is on erdosproblems.com/937.Prizenone
Topicsnumber theory, powerful
Statement in LeanFormal Conjectures
Solution in Leanyes
Original pageerdosproblems.com/937
Attempts
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