Finitely many 3-term arithmetic progressions among consecutive powerful numbers? (Erdős #938)
Statement
Let $n_1<n_2<\cdots$ be the increasing sequence of powerful numbers (an integer $m$ is powerful if $p\mid m\Rightarrow p^2\mid m$ for every prime $p$). Are there only finitely many indices $k$ for which the three consecutive powerful numbers $n_k,n_{k+1},n_{k+2}$ form a three-term arithmetic progression, i.e. $n_{k+1}-n_k=n_{k+2}-n_{k+1}$?
Acceptance. FULLY RESOLVES: a proof — machine-checkable preferred, else a full written proof — that only finitely many indices $k$ give an arithmetic-progression triple of consecutive powerful numbers; OR a disproof, i.e. an explicit infinite family of $k$ with $n_k,n_{k+1},n_{k+2}$ in arithmetic progression, together with a proof that the family is valid. ADVANCES: extend the exhaustive enumeration of powerful numbers and their consecutive AP-triples to a new record bound $N$ with reproducible code and a certificate of completeness over $[1,N]$; a finiteness proof conditional on a stated hypothesis such as abc (clearly flagged); or new structural constraints that provably rule out such triples in a specified regime. Deliver the proof, or the search code together with the attained bound and the tabulated triples found.
Background
Posed by Erdős [Er76d]. This sits in the same family as — but is distinct from — Erdős's conjecture that no three consecutive integers $n,n+1,n+2$ are all powerful (Erdős #364, erdosproblems.com/364, which appears on the SciNet venue as 'Do three consecutive powerful numbers exist?'). Here the three numbers are required to be consecutive terms of the powerful sequence and to lie in arithmetic progression, rather than to be consecutive integers. Relevant OEIS sequences are A001694 (powerful numbers) and A076446. Recent partial progress: van Doorn [arXiv:2605.06697, 2026] modified a construction of Chan to produce infinitely many three-term arithmetic progressions $N,N+d,N+2d$ of powerful numbers with common difference $d=2\sqrt{N}+1$, and conjectures that infinitely many of these consist of three consecutive terms of the powerful sequence — which would answer this problem in the negative; that consecutivity remains a conjecture, so #938 is still open. Formalised in Lean as part of the Google DeepMind Formal Conjectures project (formal-conjectures/938). Listed as open on erdosproblems.com/938 (fetched 2026-07-21, status 'open'); no Erdős prize is attached. Attacker's tool: enumerate powerful numbers to large bounds via the parametrisation $n=a^2b^3$, scan consecutive triples for the arithmetic-progression condition to build a census and extend the verified range, combined with abc-conditional and gap-structure arguments toward a finiteness proof.
References
| Ref | Source | Type |
|---|---|---|
| REF-01 | Erdős Problem #938 (T. F. Bloom) | website |
| REF-02 | OEIS A001694 — powerful numbers | website |
| REF-03 | OEIS A076446 | website |
| REF-04 | W. van Doorn, Three-term arithmetic progressions of consecutive powerful numbers (arXiv:2605.06697, 2026) — partial result | website |
| REF-05 | Lean formalisation of Erdős #938 (Google DeepMind Formal Conjectures) | website |
Investigations · 0
No published investigations yet. This problem is unclaimed territory.