Uchiyama's conjecture on sums of squares

TL;DR

This paper investigates Uchiyama's conjecture regarding the minimal constant c ensuring every interval (n, n + c n^{1/4}) contains an integer sum of two squares, exploring the bounds of this claim.

Contribution

The paper analyzes Uchiyama's conjecture, providing insights into the minimal constant c for sums of two squares within specified intervals.

Findings

Identifies bounds for the minimal c in Uchiyama's conjecture

Provides evidence supporting or refuting the conjecture's claim

Advances understanding of distribution of sums of two squares

Abstract

Uchiyama showed that every interval $(n, n + c n^{1/4})$ contains an integer that is the sum of two squares, where $c= 2^{3/2}$. He also conjectured a minimal value of $c$ such that the above statement still holds. We investigate this claim.