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arXiv · 2105.13568

Somewhat smooth numbers in short intervals

Andreas Weingartner
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Abstract

We use exponent pairs to establish the existence of many $x^a$-smooth numbers in short intervals $[x-x^b,x]$, when $a>1/2$. In particular, $b=1-a-a(1-a)^3$ is admissible. Assuming the exponent-pairs conjecture, one can take $b=(1-a)/2+\epsilon$. As an application, we show that $[x-x^{0.4872},x]$ contains many practical numbers when $x$ is large.

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Andreas Weingartner. 2021-05-28. Somewhat smooth numbers in short intervals. https://arxiv.org/abs/2105.13568

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