On Sets of Integers where Each Pair Sums to a Square

TL;DR

This paper investigates sets of integers where every pair sums to a perfect square, confirming known results for smaller sets and discovering new solutions for larger sets up to size six, contributing to a classic number theory problem.

Contribution

It extends known solutions for sets of size six, providing new solution sets and confirming minimal results for size five, advancing understanding of sum-of-two-squares sets.

Findings

Confirmed minimal results for n=5 sets.

Discovered new solution sets for n=6.

Contributed to the unresolved problem D15 in Guy's book.

Abstract

We discuss the problem of finding distinct integer sets $\{x_1,x_2,...,x_n\}$ where each sum $x_i+x_j, i \ne j$ is a square, and $n \le 7$. We confirm minimal results of Lagrange and Nicolas for $n=5$ and for the related problem with triples. We provide new solution sets for $n=6$ to add to the single known set. This provides new information for problem D15 in Guy's {\it Unsolved Problems in Number Theory}