Shortest Distance in Modular Hyperbola and Least Quadratic Nonresidue

Abstract

In this paper, we study how small a box contains at least two points from a modular hyperbola $xy\equiv c(\mathrm{mod}p)$. There are two such points in a square of side length $p^{1/4+\epsilon }$. Furthermore, it turns out that either there are two such points in a square of side length $p^{1/6+\epsilon }$ or the least quadratic nonresidue is less than $p^{1/(6\sqrt{e})+\epsilon }$.

arXiv comment5 pages