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

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].