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

TL;DR

The paper resolves Graham's conjecture by proving that the largest integer not representable as a sum of squares of distinct positive integers whose reciprocals sum to $1$ is $8542$. It combines an efficient backtracking algorithm to find representations with a translation framework: Graham-style maps $f_0,f_1,f_2,f_3$, acting on $\\{21,39\}\$-avoiding representations, produce representations of $g_i(m)=4m+c$, enabling inductive propagation of representability via a complete set of $t$-translations. A general propagation theorem for $t$-representability is developed, and computational verification establishes a wide initial range of representable numbers, with a small exceptional set and subsequent extension to all larger numbers. The work also provides computational data and generalized bounds (e.g., the largest non-$6$-representable number is $15707$) and offers a practical framework for similar Egyptian-fraction partition problems, supported by supplementary representations and datasets.

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.

Theorems & Definitions (12)

• Theorem 1
• Lemma 2
• proof
• Lemma 3
• Lemma 4
• proof
• Lemma 5
• proof
• Theorem 6
• proof