Two sets of integers such that all elements of the sumset of the two sets are perfect squares

TL;DR

This paper explores the problem of constructing two integer sets such that all pairwise sums are perfect squares, providing new multi-parameter solutions for specific set sizes and suggesting methods for generating more solutions.

Contribution

The paper introduces new multi-parameter solutions for the perfect square sumset problem for set sizes (3,3), (5,3), and (4,4), expanding known solution methods.

Findings

Multiple new solutions for (3,3), (5,3), and (4,4) cases.

Methodology for generating additional solutions.

Extension of known solution sets to larger parameters.

Abstract

This paper is concerned with the problem of finding two sets of integers, $\{a_1, a_2, \ldots$, $a_m\}$ and $\{b_1, b_2, \ldots, b_n\}$, such that all the $mn$ sums $a_i+b_j, i=1, \ldots, m, j=1, \ldots, n$, are perfect squares. A method is known for generating numerical examples of such sets when $m=2$ or 3 and $n$ is arbitrary. When both $m$ and $n$ exceed 2, only one two-parameter solution with $(m, n)=(4, 4)$ has been published. In this paper we obtain several multi-parameter solutions of the problem in three cases when $(m, n)$ is $(3, 3)$ or $(5, 3)$ or $(4, 4)$, and we indicate how more such solutions may be obtained.