Lagrange, central norms, and quadratic Diophantine equations

As is often the case, some results get rediscovered over time. In particular, some rather striking results of Lagrange are often recreated. For instance, in [6], a result pertaining to the Pell equation for a prime discriminant was recast in the light of nonabelian cohomology groups. Yet, in [1], the authors acknowledged the fact that the result “has been discovered before,” and provided an elementary proof of it and two other results related to Lagrange. In this paper, we present complete generalizations of these results (see Theorems 3.1, 3.5, and 3.9 below), and do so with only elementary properties of the simple continued fraction expansions of general √ D. As a consequence, we obtain