Wednesday, 14 June 2000 at 1:30PM

**UBC, Math Annex 1102**

Hugh Montgomery, University of Michigan

`Greedy sums of distinct squares''

** 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.