A NOTE ON POWERFUL NUMBERS IN SHORT INTERVALS

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