Quintuples of positive integers whose sums in pairs or in triples are squares

Abstract

This paper is concerned with the two diophantine problems of finding five positive integers such that their sums either in pairs or in triples are perfect squares. The known parametric solutions of these problems are very cumbersome and not given explicitly in the published literature. Moreover, these solutions lead to numerical examples involving quite large integers. In this paper we first prove that there is a one-to-one correspondence between quintuples whose pairwise sums are squares and quintuples whose sums in triples are squares. Next we obtain all quintuples, with three or four of the five integers being distinct, such that their sums in pairs or in triples are perfect squares. We also obtain parametrized families of quintuples of distinct polynomials of degrees 6, 10, and 12 such that their sums in pairs or in triples are squares. These solutions yield quintuples of positive integers that are much smaller than the solutions generated by the known parametric solutions.