Abstract
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.
Publisher
Cambridge University Press (CUP)
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