OPEN
This is open, and cannot be resolved with a finite computation.
Do all pairs of consecutive
powerful numbers $n$ and $n+1$ come from solutions to
Pell equations? In other words, must either $n$ or $n+1$ be a square?
Is the number of such $n\leq x$ bounded by $(\log x)^{O(1)}$?
Disclaimer: The open status of this problem reflects the current belief of the owner of this website. There may be literature on this problem that I am unaware of, which may partially or completely solve the stated problem. Please do your own literature search before expending significant effort on solving this problem. If you find any relevant literature not mentioned here, please add this in a comment.
Erdős originally asked Mahler whether there are infinitely many pairs of consecutive powerful numbers, but Mahler immediately observed that the answer is yes from the infinitely many solutions to the Pell equation $x^2=2^3y^2+1$.
The list of $n$ such that $n$ and $n+1$ are both powerful is
A060355 in the OEIS.
The answer to the first question is no: Golomb
[Go70] observed that both $12167=23^3$ and $12168=2^33^213^2$ are powerful. Walker
[Wa76] proved that the equation\[7^3x^2=3^3y^2+1\]has infinitely many solutions, giving infinitely many counterexamples.
See also
[364].
This is discussed in problem B16 of Guy's collection
[Gu04].
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This page was last edited 31 October 2025. View history
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When referring to this problem, please use the original sources of Erdős. If you wish to acknowledge this website, the recommended citation format is:
T. F. Bloom, Erdős Problem #365, https://www.erdosproblems.com/365, accessed 2026-04-11
