On a Diophantine equation of the second degree (Q768391)

For solving an equation of the type \(x^2 - Dy^2 =\pm N\), where \(D\) and \(N\) are positive rational integers, one may use either the theory of quadratic forms or the theory of quadratic fields. \textit{T. Nagell} has shown (e. g. pp. 195--212 in [Introduction to number theory''. New York: John Wiley (1951; Zbl 0042.26702)]) how it is possible to determine all solutions in rational integers \(x\) and \(y\) independently of these theories and completely elementarily. His investigations have been continued by the author [Ark. Mat. 2, 1--23 (1952; Zbl 0047.04008)]. In this paper the following equation is considered: \[ Au^2 + Buv + Cv^2 = \varepsilon N,\tag{1} \] where \(A, B, C\) and \(N\) are rational integers, \(A > 0\), \(N > 0\), \(\varepsilon = \pm 1\) and where \(B^2 - 4AC = D\) is a positive integer which is not a perfect square. It is shown how it is possible to avoid the usual linear transformations and congruences in order to obtain all the integral solutions of (1). If \(u = t/A\) is a fractional number and \(v\) is an integer which satisfy (1), the number \(f(u, v) = (2Au + Bv + v \sqrt D)/2\) is called a solution of (1). The set of all solutions associated with each other forms a class of solutions of (1). If \(u\) and \(v\) are two integers satisfying (1), \(f(u,v)\) is called an integral solution of (1). It is proved that if one solution of a class \(K\) is an integral solution, then every solution of \(K\) is integral. If \(f(u,v)\) is the fundamental solution (in a sense defined) of (1), then one has the following inequality: \[ 0<v\leq \sqrt{(AN/D) (x_1-2\varepsilon)}, \] where \(v = 0\) also may be possible if \(\varepsilon = 1\), \((x_1, y_1)\) denoting the fundamental solution of \(x^2 - Dy^2 = 4\). If in \(f(u,v)\) the number \((2Au+Bv)v/N\) is an integer, the class is said to be quasi-ambiguous. Some theorems concerning such classes are proved. At last examples are given.