Class Numbers of Quadratic Fields Determined by Solvability of Diophantine Equations

In the literature there has been considerable attention given to the exploration of relationships between certain diophantine equations and class numbers of quadratic fields. In this paper we provide criteria for the insolvability of certain diophantine equations. This result is then used to determine when related real quadratic fields have class number bigger than 1. Moreover, based on criteria which we find for the solvability of a certain class of diophantine equations, we are able to determine when the class number of related imaginary quadratic fields is divisible by a given integer. Introduction. The primary aim of this paper is to investigate the relationship between solvability of diophantine equations and class numbers of quadratic fields. Most such investigations into real quadratic fields in the literature deal with Richaud-Degert (R-D)-typequadratic fields (see [4] and [14]); that is, those Q(Jn) where « is a square-free positive integer of the form « = I2 + r with integer / > 0, integer r dividing 4/ and -/ < r < /. The seminal paper in this regard is by Ankeny, Chowla, and Hasse [1]. However, many authors have studied such fields and considered generalizations thereof. Among them are: Azuhata [2], Kutsuna [8], Lang [9], Takeuchi [15], Yokoi [16]-[18], and the author [10]-[13]. In Section 1 we investigate a larger class of real quadratic fields than the (R-D)-types. We obtain conditions for the solvability of certain diophantine equations and use the result to determine nontriviality of the class numbers of these real quadratic fields. Moreover, we obtain as immediate consequences many of the above results in the literature. The connection between solvability of certain diophantine equations and the divisibility of the class number of imaginary quadratic fields by a given integer has been given much attention. Among such inquiries are: Cowles [3], Hongwen [7], Gross and Rohrlich [5], and the author [11] and [13]. In the second section we obtain sufficient conditions for a quadratic field (real or imaginary) to have the exponent of its class group divisible by a given integer /. This result is most readily applied to imaginary quadratic fields upon which we focus. We provide sufficient conditions (in elementary arithmetic terms) for the exponent of the class group of certain Received November 27, 1985. 1980 Mathematics Subject Classification. Primary 12A50, 12A25, 12A45; Secondary 10B05.