A note on powerful numbers in short intervals

Abstract We investigate uniform upper bounds for the number of powerful numbers in short intervals $(x, x + y]$ . We obtain unconditional upper bounds $O({y}/{\log y})$ and $O(\kern1.3pt y^{11/12})$ for all powerful numbers and $y^{1/2}$ -smooth powerful numbers, respectively. Conditional on the $abc$ -conjecture, we prove the bound $O({y}/{\log ^{1+\epsilon } y})$ for squarefull numbers and the bound $O(\kern1.3pt y^{(2 + \epsilon )/k})$ for k-full numbers when $k \ge 3$ . These bounds are related to Roth’s theorem on arithmetic progressions and the conjecture on the nonexistence of three consecutive squarefull numbers.