All Solutions of the Diophantine Equation x 2 −

The main thrust of this article is to show how complete solutions of quadratic Diophantine equations can be given, for any positive discriminant, in terms of the continued fraction algorithm. This is in response to recent results by Zhang [4]-[6], wherein semi-simple continued fractions were introduced to generalize the well-known fact that solutions of quadratic Diophantine equations with values bounded above by √ D, for a given field radicand D, are convergents of the simple continued fraction expansion of √ D. We show here that Zhang's theory is not required since the continued fraction algorithm is a sufficient and more accurate mechanism for explaining how this all works, and that it actually contains a more refined version of these so-called semi-simple continued fractions. We do this by getting complete and explicit solutions for any quadratic Diophantine equation having arbitrary positive radicand D (corresponding to any real quadratic order) using only known theory, and we generalize the results of Zhang in the process. We show that solutions to these Diophantine equations arise as convergents of a continued fraction expansion of √ D obtained via the continued fraction algorithm. We develop the entire theory from the basics, since the works of Nagell [2] and Perron [3] are insufficient to explain completely the method of obtaining all solutions to quadratic Diophantine equations via the infrastructure, namely the interrelationship between continued fractions and ideals, thereby filling a gap in the literature in a structure-rich fashion from a modern viewpoint.