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Do all pairs of consecutive powerful numbers come from solutions to Pell equations? Is the number of such pairs $\leq x$ 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 $x^2=8y^2+1$.