Uchiyama's conjecture on sums of 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.

To the Australian Test Cricket selectors, to whom I am indebted a slice of humble pie, after they ignored the strident remonstrances of TS Trudgian and included MR Marsh in the team