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

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.