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 $x y \equiv c \pmod p$. 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 $p^{1/(6 \sqrt{e}) + \epsilon}$.