Discriminantal divisors and binary quadratic forms

Extract

An ancipital form is a form [a, b, c] in which b= 0 or b= a; these fall into pairs of associates: [a, 0, c] and [c, 0, a] (type 1), and [a, a, c] and [4c–a, 4c–a, c] (type 2). The set of discriminantal divisors of discriminant d is formed by choosing, from each pair of primitive associate ancipital forms of discriminant d, exactly one of the two leading coefficients. In this article we study representations of discriminantal divisors of a given discriminant by binary quadratic forms of that discriminant, previously studied by the author and by G. Pall. We are concerned here with discriminants d= 4^kpq, where k ≥ 1, p = 1, q = 3 (mod 4) are primes, and d = 4^kp, where k ≥ 1 and p is an odd prime. This investigation arose in connection with the search for integral solutions of x² –Dy² = – 1.