On partitions into squares of distinct integers whose reciprocals sum to 1

TL;DR

This paper determines the precise largest integer (8542) that cannot be partitioned into squares of distinct positive integers with reciprocals summing to 1, extending Graham's 1963 conjecture about such partitions.

Contribution

The paper proves the exact bound for the existence of partitions into squares of distinct integers with reciprocal sum 1, confirming Graham's conjecture for large integers.

Findings

8542 is the largest integer with no such partition

All integers greater than 8542 can be partitioned as described

The result confirms Graham's conjecture for sufficiently large integers

Abstract

In 1963, Graham proved that all integers greater than 77 (but not 77 itself) can be partitioned into distinct positive integers whose reciprocals sum to 1. He further conjectured that for any sufficiently large integer, it can be partitioned into squares of distinct positive integers whose reciprocals sum to 1. In this study, we establish the exact bound for existence of such partitions by proving that 8542 is the largest integer with no such partition.