Shortest Distance in Modular Hyperbola and Least Quadratic Nonresidue

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