A Computational Method to Address a Diophantine Equation

The parameterization of the integer Pythagorean triples is well known, and parameterizations exist for other unique factorization domains, such as the Gaussian integers. Considering the quaternions, however, no such result has been established. In attempting to parameterize the quaternion Pythagorean triples, the importance of understanding the arithmetic properties of the quaternions has become clear. More particularly, a desire to computationally produce classes of quaternion Pythagorean triples led to investigations into the questions of quaternion square roots and of the factorization of quaternions into irreducibles. On the basis of these investigations, a program was written in order to generate the desired triples. This presentation will emphasize the computational, pattern-seeking approach used in the exploration of the aforementioned ideas, beginning with the development of the mentioned software, including an unanticipated result with connections to Jacobi’s Four-Squares Theorem, and concluding with a description of the progress made thereby up to this date.