Shortest Distance in Modular Hyperbola and Least Quadratic Nonresidue

TL;DR

This paper investigates the minimal size of a box containing points on a modular hyperbola and relates it to bounds on the least quadratic nonresidue, revealing new insights into their distribution.

Contribution

It establishes bounds on the smallest box containing two points on a modular hyperbola and connects this to bounds on the least quadratic nonresidue.

Findings

Two points on the hyperbola can be found in a box of side length p^{1/4 + ε}.

Either such points exist in a smaller box of side length p^{1/6 + ε} or the least quadratic nonresidue is bounded by p^{1/(6√e) + ε}.

The results link the distribution of hyperbola points to properties of quadratic nonresidues.

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 $p^{1/(6 \sqrt{e}) + \epsilon}$.