Problem 364

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claims/: The 4 claim pages of Problem 364, one per claimant's result; the problem's standing derives from them.

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Statement. Are there any triples of consecutive positive integers all of which are powerful (i.e. if $p\mid n$ then $p^2\mid n$)?

Status. Verifiable: the site's label, meaning open but settled by a finite example if one exists (page last edited 13 April 2026; problem page, discussion thread and proof-claims tab read 2026-10-07). The site's proof-claims tab carries one partial proof claim, Sayim's exclusion of the two mixed shapes for the neighbors of a cube, recorded on its claim page as pending; no claim settles the question, and the frontmatter standing is derived from the claim pages.

Source. erdosproblems.com/364, accessed 2026-09-04. Cite as: T. F. Bloom, Erdős Problem #364, https://www.erdosproblems.com/364.

References.

• [Ch25] Chan, Tsz Ho, A note on three consecutive powerful numbers. Integers (2025), Paper No. A7, 7.
• [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.
• [MoWa86] Mollin, R. A. and Walsh, P. G., On powerful numbers. Internat. J. Math. Math. Sci. (1986), 801-806.
• [Sh25] She, Jialai, Nonexistence of consecutive powerful triplets around cubes with prime-square factors. Integers (2025), Paper No. A103, 9.

Formalization. Statement in formal-conjectures, pinned to the repository's revision of 2026-10-06, where the statement is tagged open and carries no formal proof.

Current assessment

The standing judges the site's formulation of 2026-09-04 above: whether three consecutive positive integers can all be powerful, the conjecture of Erdős, Mollin and Walsh that none can. The question is open. It is verifiable in the site's sense, since a single triple would settle it, and no finite computation can prove the conjecture; this page records that as a note, not as a claim. Pairs of consecutive powerful numbers are infinite, as Mahler answered Erdős from the Pell equation $x^2=8y^2+1$, and no four consecutive integers are all powerful, since one of them is $2$ modulo $4$. Erdős [Er76d] expected the answer to be no and, more strongly, that the $k$th powerful number $n_k$ satisfies $n_{k+2}-n_k>n_k^c$ for some constant $c>0$; the abc conjecture implies that only finitely many triples exist. By the site's commentary, the OEIS sequence A076445 shows that no triple starts below $7.38\times10^{28}$. The known partial results concern triples centered at a cube $x^3$ whose outer members are a prime power times a square or a cube. Two are accepted partial claims, refereed in Integers: Chan [Ch25] excludes the shape $x^3\mp1=p^3\cdot\square$ on both sides (claim page, card), and She [Sh25] excludes $x^3\mp1=p^2\cdot\text{cube}$ on both sides (claim page, card). Sayim's pending claim (claim page) excludes the two mixed combinations, so that all four combinations of these shapes are excluded if the claim holds. Ma's pending preprint of 2026-08-24 (claim page) extends Chan's shape to middle members that are $n$th powers for every $n\geq5$ whose prime factors are all $\equiv5\pmod 8$. None of this touches a triple whose middle member is not a perfect power.

Sayim's collected volume of 2026-10-04 (Zenodo 10.5281/zenodo.23127546), linked from the discussion thread, reports computations and reductions. Among them, no triple has a middle below $1.83\times10^{18}$, which is weaker than the OEIS bound, and the number of consecutive pairs has a lower bound. These settle no instance, so the volume has no claim page.

Search scope: the site's problem page, discussion thread and proof-claims tab, read 2026-10-07, the Zenodo records of Sayim's preprint and volume, and the arXiv record of Ma's preprint. Nothing on this page is independently reviewed.

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.

• chan_2025_note_three_consecutive_powerful_numbers
• cushing_2016_powerful_numbers_abc_conjecture
• cushing_2016_powerful_numbers_abc_conjecture / theorem_5_2
• she_2025_nonexistence_consecutive_powerful_triplets_around_cubes