Problem 365

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 $(\log x)^{O(1)}$ ?

Source: erdosproblems.com/365

Number theory
Powerful numbers

Wiki page

Status

Open

No claim settles this problem.

Open, in the site's label (OPEN; page last edited 31 October 2025), which attaches to the pair of questions. The site's commentary answers the first question no, crediting Golomb's counterexample [Go70] and Walker's infinite family [Wa76]; the corpus accepts both on their refereed publication as partial claims settling the part pell, on the claim pages Golomb 1970 and Walker 1976. The second question, the $(\log x)^{O(1)}$ bound on the count, is unsettled, so the problem stays open.

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 pp. 105--106, asks both questions of the page for Erdős's $2$ -full numbers $u_i^{(2)}$ , the second as whether the count of solutions with $u_i<x$ is less than $(\ln x)^c$ . Library home: Guy 2004.
Wa76: Walker, David T., Consecutive integer pairs of powerful numbers and related Diophantine equations. Fibonacci Quart. (1976), 111-116.

Problem 365

Status

Proof claims

Discussion