Lagrange, central norms, and quadratic Diophantine equations

Abstract

We consider the Diophantine equation of the form x2−Dy2=c, where c=±1,±2, and provide a generalization of results of Lagrange with elementary proofs using only basic properties of simple continued fractions. As a consequence, we achieve a completely general, simple, and elegant criterion for the central norm to be 2 in the simple continued fraction expansion of D.