An ancient diophantine equation with applications to numerical curios and geometric series
We examine the diophantine equation $a^k-b^k=a-b$, where $k$ is a positive integer $\geq 2$, and consider its applications. While the complete solution of the equation $a^k-b^k=a-b$ 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, we show how to construct examples of geometric series with an interesting property.