Problem 937

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claims/: The 1 claim page of Problem 937, one per claimant's result; the problem's standing derives from them.

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Statement. Are there infinitely many four-term arithmetic progressions of coprime powerful numbers (i.e. if $p\mid n$ then $p^2\mid n$)?

Status. PROVED (LEAN).

Source. erdosproblems.com/937, accessed 2026-09-05. Cite as: T. F. Bloom, Erdős Problem #937, https://www.erdosproblems.com/937.

References.

• [BBC24] Bajpai, Prajeet and Bennett, Michael A. and Chan, Tsz Ho, Arithmetic progressions in squarefull numbers. Int. J. Number Theory 20 (2024), no. 1, 19-45. DOI: 10.1142/S1793042124500027.
• [BW24] Bennett, Michael A. and Walsh, P. G., Computing four-term arithmetic progressions of powerful numbers. INTEGERS 24A (2024), A3. DOI: 10.5281/zenodo.11352598.
• [Er76d] Erdős, P., Problems and results on number theoretic properties of consecutive integers and related questions. Proceedings of the Fifth Manitoba Conference on Numerical Mathematics (Univ. Manitoba, Winnipeg, Man., 1975) (1976), 25-44.

Formalization. The statement declaration in formal-conjectures is closed by sorry, but it is tagged as solved and carries a formal-proof attribute pointing at a Lean 4 proof in Boris Alexeev's repository (src/latest/ErdosProblems/Erdos937.lean, pinned to the commit the claim page links), which declares itself a formalization of the Bajpai–Bennett–Chan solution and proves that infinitely many pairs $(a,d)$ give pairwise coprime powerful $a,a+d,a+2d,a+3d$. The community database (teorth/erdosproblems) accordingly records the problem as proved in Lean. That proof has not been built or audited here, so it is recorded as the claimant's formalization link on the claim page and as no formalized evidence.

Current assessment

Erdős posed the question in 1975; the cited conference proceedings version appeared in 1976 [Er76d, p. 33]. Bajpai, Bennett, and Chan answered it affirmatively by constructing infinitely many primitive four-term progressions of squarefull numbers [BBC24]. Their construction proves the stronger pairwise-coprime property requested here, not merely $\gcd(N,d)=1$.

The standing derives from the accepted claim page Bajpai–Bennett–Chan 2023, whose evidence is the refereed publication and the site's credit. No forum proof claim and no other outside claim to settle the problem was found (search scope: the site's forum and proof-claim tab, 2026-10-06; the community database, teorth/erdosproblems, 2026-10-07); Bennett–Walsh [BW24], cited under Progress, quotes Erdős's question but only computes a smaller example. This page records checks of the examples by exact integer arithmetic, but no independent review of the full infinitude construction. The examples illustrate the theorem; the separate minimality question is described below.

Progress

The accepted manuscript is dated June 26, 2023. It replaced the much larger explicit example in arXiv v1 with a 190-digit example. Bennett and Walsh subsequently published a 111-digit record [BW24], which they describe as the smallest known example. They say only that it may be minimal and that a proof would need a careful analysis of the quadratic forms and of generators for the elliptic curves.

Known Results

The unconditional solution is Bajpai--Bennett--Chan, Proposition 5.2. It parametrizes three square terms, maps the fourth-term condition to an elliptic curve, uses a finite division-polynomial computation modulo $73$, and proves a $2$-adic parity condition. The Chinese remainder class $n\equiv404\pmod{16\cdot73}$ then supplies infinitely many progressions

$$x^2,\quad y^2,\quad z^2,\quad73^3w^2,$$

up to reversal. A direct gcd argument proves that every pair of terms is coprime. The result is also the $(m,k)=(4,2)$ case of Theorem 1.2.

The same paper proves restrictions on longer primitive progressions only under the $abc$ conjecture. Its Theorem 1.1 forces $\gcd(N,d)$ to grow outside three exceptional pairs; with the unconditional constructions this yields the conditional classification

$$A^\infty(2)=4,\qquad A^\infty(3)=3,\qquad A^\infty(k)=2\quad(k\geq4).$$

The manuscript's revised 190-digit example and Bennett--Walsh's later 111-digit record example have both been checked by exact integer arithmetic. These examples illustrate the theorem but do not establish the infinitude; that comes from the full elliptic-curve family above.

Linked library material

These entries are derived from explicit links on library pages. They are navigation only and do not by themselves record mathematical progress.

• bajpai_2024_arithmetic_progressions_squarefull_numbers
• bajpai_2024_arithmetic_progressions_squarefull_numbers / accepted_manuscript_example
• bajpai_2024_arithmetic_progressions_squarefull_numbers / corollary_1_3
• bajpai_2024_arithmetic_progressions_squarefull_numbers / lemma_2_1
• bajpai_2024_arithmetic_progressions_squarefull_numbers / lemma_2_2
• bajpai_2024_arithmetic_progressions_squarefull_numbers / lemma_3_1
• bajpai_2024_arithmetic_progressions_squarefull_numbers / proposition_5_1
• bajpai_2024_arithmetic_progressions_squarefull_numbers / proposition_5_2
• bajpai_2024_arithmetic_progressions_squarefull_numbers / theorem_1_1
• bajpai_2024_arithmetic_progressions_squarefull_numbers / theorem_1_2
• bennett_2024_computing_four_term_arithmetic_progressions_powerful_numbers
• bennett_2024_computing_four_term_arithmetic_progressions_powerful_numbers / section_4_record_example