A note on powerful numbers in short intervals

Abstract

In this note, we are interested in obtaining uniform upper bounds for the number of powerful numbers in short intervals $(x,x+y]$. We obtain unconditional upper bounds $O(\frac{y}{\log y})$ and $O(y^{11/12})$ for all powerful numbers and $y^{1/2}$-smooth powerful numbers respectively. Conditional on the $abc$-conjecture, we prove the bound $O(\frac{y}{{\log }^{1+\epsilon }y})$ for squarefull numbers and the bound $O(y^{(2+\epsilon )/k})$ for $k$-full numbers when $k\ge 3$. They are related to Roth's theorem on arithmetic progressions and the conjecture on non-existence of three consecutive squarefull numbers.

7 pages, welcome any comments