Solving Some Quadratic Diophantine Equations with Clifford Algebra

In this work, the equivalence class representatives of integer solutions of the Diophantine equation of the type ${{a_1x_1^2+ .,.,. + a_px_p^2 = a_{p+1}x^2_{p+1} + .,.,. +a_{p+q}x^2_{p+q} +a_1x^2_{n+1} (a_i > 0,i=1, .,.,.,,p+q,x_{n+1}neq0)}}${{a_1x_1^2+ .,.,. + a_px_p^2 = a_{p+1}x^2_{p+1} + .,.,. +a_{p+q}x^2_{p+q} +a_1x^2_{n+1} (a_i > 0,i=1, .,.,.,,p+q,x_{n+1}neq0)}} 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.