Solving Some Quadratic Diophantine Equations with Clifford Algebra

Abstract

In this work, the equivalence class representatives of integer solutions of the Diophantine equation of the type a_1x1 ² +...+ a_px_p ²= a_p+1x² _p+1 +...+a_p+qx² _p+q + a₁x² _n+1 (a_i>0, i=1,...,p+q, x_n+1≠0) are found using simple reflections of orthogonal vectors, manipulated using the Clifford algebra over orthogonal spaces R^p,q. These solutions are obtained from the application of a useful Lemma: given two different non-zero vectors of the same norm, we can map one onto the other, or its negative, by means of a simple reflection. With this Lemma, we extend and improve a previous work [1] concerning generalized Pythagorean numbers, which now can be obtained as a Corollary. We also show that our technique is promising for solving others Diophantine equations.