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(1)x(1)(2) + ... + a(p)x(p)(2) = a(p+1)x(p)(+1)(2) + ... + a(p+q)x(p)(+q)(2) + a(1)x(n)(+1)(2) (ai > 0,i = 1 ,..., p + q, x(n+1) not equal 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.