An Ancient Diophantine Equation with applications to Numerical Curios and Geometric Series

TL;DR

This paper explores the Diophantine equation x^k - y^k = x - y for positive integers k, providing methods to generate solutions for k=4 and demonstrating applications in numerical curiosities and geometric series.

Contribution

It introduces a method to generate infinitely many solutions for k=4 and applies these solutions to create numerical curiosities and geometric series with unique properties.

Findings

Infinite solutions for k=4 generated

Numerical curiosities involving roots constructed

Examples of geometric series with special properties created

Abstract

In this paper we examine the diophantine equation $x^k-y^k=x-y$ where $k$ is a positive integer $\geq 2$, and consider its applications. While the complete solution of the equation $x^k-y^k=x-y$ in positive rational numbers is already known when $k=2$ or $3$, till now only one numerical solution of the equation in positive rational numbers has been published when $k=4$, and no nontrivial solution is known when $k \geq 5$. We describe a method of generating infinitely many positive rational solutions of the equation when $k=4$. We use the positive rational solutions of the equation with $k=2,\, 3$ or 4 to produce numerical curios involving square roots, cube roots and fourth roots, and as another application of these solutions, we show how to construct examples of geometric series with an interesting property.