Three integers whose sum, product and the sum of the products of the integers, taken two at a time, are perfect squares

TL;DR

This paper presents a new method to find smaller parametric solutions for Euler's problem of three integers where the sum, product, and sum of pairwise products are all perfect squares, improving on previous high-degree polynomial solutions.

Contribution

The authors introduce a novel approach that yields smaller degree polynomial parametric solutions, resulting in more manageable and smaller numerical solutions for Euler's problem.

Findings

Multiple parametric solutions with smaller degree polynomials

Numerically smaller solutions compared to Euler's original solutions

Enhanced understanding of the structure of solutions for the problem

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.