Three integers whose sum, product and the sum of the products of the integers, taken two at a time, are perfect squares
Ajai Choudhry
TL;DR
The paper addresses the Diophantine problem of finding integers $a,b,c$ whose sum, product, and the sum of pairwise products are all perfect squares. It moves beyond Euler's large-number parametrizations by introducing a method that forces the elementary symmetric functions to be squares, using a cubic framework and a key lemma that each root can be written as a sum of two rational squares. The work yields multiple parametric families, including cases where exactly one, all, or none of $a,b,c$ is a square, and constructs triples where all three are squares or sums of two squares, supported by explicit polynomial expressions and infinite families. The approach provides numerous numerically small examples and demonstrates substantial gains in efficiency and tractability over classical methods, contributing new constructive insight into this classical problem and offering a rich set of small solutions for further study.
Abstract
Euler had considered the problem of finding three integers whose sum, product, and also the sum of the products of the integers, taken two at a time, are all perfect squares. Euler's methods of solving the problem lead to parametric solutions in terms of polynomials of high degrees and his numerical solutions consisted of very large integers. We obtain, by a new method, several parametric solutions given by polynomials of much smaller degrees and thus we get a number of numerically small solutions of the problem.