On Parametric and Matrix Solutions to the Diophantine Equation x 2 + d y 2 − z 2 = 0 Where d Is a Positive Square‐Free Integer

Abstract

The well‐known matrix‐generated tree structure for Pythagorean triplets is extended to the primitive solutions of the Diophantine equation $x^2+\mathrm{d}{y}^2-z^2=0$ where $d$ is a positive square‐free integer. The proof is based on a parametrization of these solutions as well as on a dual version of the Fermat’s method of descent.