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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Weingartner, A. Somewhat smooth numbers in short intervals. Ramanujan J 60, 447–453 (2023). https://doi.org/10.1007/s11139-022-00552-w
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DOI: https://doi.org/10.1007/s11139-022-00552-w