Integer solutions to $x^2 + y^2 = z^2 - k$ for a fixed integer value $k$

Abstract

For a given integer $k$, general necessary and sufficient conditions for the existence of integer solutions to an equation of the form $x^2+y^2=z^2-k$ are given. It is shown that when there is a solution, there are infinitely many solutions. An elementary method for finding the solutions, when they exist, is described.