LAGRANGE, CENTRAL NORMS, AND QUADRATIC DIOPHANTINE EQUATIONS

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.


Introduction
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 where x 2 0 − Dy 2 0 = 1 is the fundamental solution, is the (even) period length of the continued fraction expansion of √ D, and Q /2 is the central norm (see Theorem 3.11 below).

Notation and preliminaries
Herein, we will be concerned with the simple continued fraction expansions of √ D, where D is an integer that is not a perfect square.We denote this expansion by √ D = q 0 ; q 1 , q 2 ,..., q −1 ,2q 0 , ( where = ( √ D) is the period length, q 0 = √ D (the floor of √ D), and q 1 , q 2 ,..., q −1 is a palindrome.
The kth convergent of α for k ≥ 0 is given by where 1, and B −1 = 0.The complete quotients are given by (P k + √ D)/Q k , where P 0 = 0, Q 0 = 1, and for k ≥ 1, (2.5) We will also need the following facts (which can be found in most introductory texts in number theory, such as [3]; also, see [2] for a more advanced exposition): (2.6) Also, (2.8) (2.9) In particular, for any (2.10) Also, we will need the elementary facts that for any k ≥ 1, When is even, (2.12) , so by (2.5), where Q /2 is called the central norm, (via (2.9)).Furthermore, (2.15) R. A. Mollin 1041 In the next section, we will be considering what are typically called the standard Pell equations (2.16) and (2.18), given below.The fundamental solution of such an equation means the (unique) least positive integers (x, y) = (x 0 , y 0 ) satisfying it.The following result shows how all solutions of the Pell equations are determined from continued fractions.
Theorem 2.1.Suppose that = ( √ D) and k is any positive integer.Then if is even, all positive solutions of are given by whereas there are no solutions to If is odd, then all positive solutions of (2.16) are given by whereas all positive solutions of (2.18) are given by (2.20) Proof.This appears in many introductory number theory texts possessing an in-depth section on continued fractions.For instance, see [3, Corollary 5.3.3,page 249].
In the following (which we need in the next section), and all subsequent results, the notation for the A k , B k , Q k , and so forth apply to the above-developed notation for the continued fraction expansion of Theorem 2.2.Let D be a positive integer that is not a perfect square.Then = ( √

D) is even if and only if one of the following two conditions occurs.
(1) There exists a factorization D = ab with 1 < a < b such that the following equation has an integral solution (x, y): For the proof of all this, see [4].Lastly, we will require the following number-theoretic results.
Theorem 2.3.If c is an odd positive integer, then the following Jacobi symbol identities hold: (2.23) Proof.This may be found in introductory number theory texts.For instance, see [3, Theorem 4.2.1, page 197].

Central norms and Diophantine equations
The following extends Lagrange's [1, Theorem, page 181] to its greatest possible generality.In the proof, we use only elementary continued fraction results.The end product is that we reveal the underlying reason for the phenomenon in terms of the central norm being equal to 2.
Theorem 3.1.Let D > 2 be an integer that is not a perfect square.Also assume that (x 0 , y 0 ) is the fundamental solution of (2.16).Then the following are equivalent.
Note that in Theorem 3.1, all odd primes p dividing D must be of the form p ≡ ±1(mod 8), and 4 does not divide D.