Hugh Montgomery, University of Michigan

Abstract: Suppose that $n$ is a given positive integer, let $s_1^2$ be the largest square not exceeding $n$, $s_2^2$ the largest square not exceeding $n - s_1^2$, and so on, so that $$ n = s_1^2 + s_2^2 + \cdots + s_r^2 $$ with $s_1 \ge s_2 \ge \cdots \ge s_r$. We call this a greedy sum of squares, and we consider whether the summands are all distinct. The initial summands are distinct because they decrease quickly in size, but the very last summands may be repeated, as in $8 = 4 + 4$ or $ = 1 + 1 + 1$. We analyze the local and global distribution of greedy sums of distinct squares.