On the equation x ² + ry ² = z ²

Abstract

In this Article, we derive a method for classifying all positive integer solutions of the equation x 2 + ry 2 = z 2 , where r is a given positive rational . In order to simplify notation, such a solution with x = a , y = b and z = c will be denoted by the ordered triple ( a , b , c ) and, in all that follows, the term solution will be taken to mean a positive integer solution of the above equation. The method employed will be similar to that used in [1] to find all integer triangles containing an angle whose cosine is known.