Lagrange, central norms, and quadratic Diophantine equations (Q2570920)

The author considers Diophantine equations of the form \(x^2-Dy^2=c\), where \(c=\pm 1, \pm 2\) and provides a generalization of results of Lagrange. As a consequence, a completely general, simple and elegant criterion is achieved for the central norm to be \(2\) in the simple continued fraction expansion of \(\sqrt{D}\). The proofs are elementary, only some basic properties of simple continued fractions are used.