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

2025/05/25 by Ajai Choudhry, Choudhry, Ajai
Mathematics · #11D25 #Advanced Mathematical Identities #Advanced Mathematical Theories #Analytic Number Theory Research #FOS: Mathematics #Number Theory (math.NT)

paper · pdf · doi:10.48550/arxiv.2505.19088

openalex publication_date 2025/05/25 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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.

Citations

• Quadratic diophantine equations with applications to quartic equations
• Rational points on elliptic curves

Related

• The problem of finding three numbers such that the sum or difference of any two of them is a square number
• A Diophantine System
• Square Eulerian Quadruples
• An arbitrary number of squares whose sum, on excluding any one of them, is also a square
• A solution to a problem of Fermat, on two numbers of which the sum is a square and the sum of their squares is a biquadrate, inspired by the Illustrious La Grange