Problem 366

Are there any $2$ -full $n$ such that $n+1$ is $3$ -full? That is, if $p\mid n$ then $p^2\mid n$ and if $p\mid n+1$ then $p^3\mid n+1$ .

Source: erdosproblems.com/366

Number theory
Powerful numbers

Wiki page

Status

Open

No claim settles this problem.

Verifiable, the site's label (VERIFIABLE, which the site explains as open but provable by a finite example; page last edited 16 July 2026, accessed 2026-10-07). No claim settles the question. The site's proof-claims tab carries one partial proof claim, Xeff's bound on any solution under Baker's explicit abc conjecture, which decides nothing and gets no claim page; the Current assessment records it with the reason.

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; p. 31. Library home: Erdos 1976.
ErGr80: Erdős, P. and Graham, R. L., Old and new problems and results in combinatorial number theory. Monographies de L'Enseignement Mathématique 28, Université de Genève (1980), 128 pp.; p. 68. Library home: Erdős and Graham 1980.
Go70: Golomb, S. W., Powerful numbers. Amer. Math. Monthly (1970), 848-855.
Gu04: Guy, Richard K., Unsolved problems in number theory. 3rd ed., Problem Books in Mathematics, Springer, New York (2004), xviii+437 pp.; section B16 "Powerful numbers. Squarefree numbers.", printed p. 106, which carries Erdős's questions on $k$ -full numbers, among them the question on consecutive full numbers that the site's commentary places there. Library home: Guy 2004.

Problem 366

Status

Proof claims

Discussion