Problem 364

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

Number theory
Powerful numbers

Wiki page

Status

Open

No claim settles this problem.

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 (Sayim, 2026) as pending; no claim settles the question, and the frontmatter standing is derived from the claim pages.

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.

Problem 364

Status

Proof claims

Discussion