Almost cubes and fourth powers in short intervals

Tsz Ho Chan

paper

Almost cubes and fourth powers in short intervals

arXiv:2503.21490

Abstract

In this paper, we study how short an interval $[x, x + x^{I^} ¸]$ contains an integer of the form $n_{1} n_{2} n_{3}$ and $m_{1} m_{2} m_{3} m_{4}$ with $n_{1} \approx n_{2} \approx n_{3}$ and $m_{1} \approx m_{2} \approx m_{3} \approx m_{4}$ . The new idea is to adopt a second moment method (usually used for almost all results) to deduce a result for all short intervals.

9 pages