The problem of finding three numbers such that the sum or difference of any two of them is a square number

TL;DR

This paper investigates the problem of identifying three numbers where the sum or difference of any two is a perfect square, presenting parametric solutions and computational results within a large range.

Contribution

The paper introduces new parametric solutions using elliptic curves and provides a comprehensive search for solutions up to 10^8.

Findings

Parametric solutions derived from elliptic curves.

Identification of the smallest solutions.

Complete solutions within the range of 10^8.

Abstract

Euler explored the problem of finding three numbers such that the sum or difference of any two of them is a perfect square. He discovered a parametric solution represented by polynomials of degree 18 and identified the smallest of these solutions. Additionally, we derive two parametric solutions using the elliptic curve method and find all solutions within the range of $10^8$.