1 results
A NOTE ON POWERFUL NUMBERS IN SHORT INTERVALS
- Part of
-
- Journal:
- Bulletin of the Australian Mathematical Society / Volume 108 / Issue 1 / August 2023
- Published online by Cambridge University Press:
- 22 September 2022, pp. 99-106
- Print publication:
- August 2023
-
- Article
-
- You have access
- HTML
- Export citation
-
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.
