  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.
