Ternary quadratic forms and the class numbers of imaginary quadratic fields

摘要

We establish a series of relations between the numbers of solutions of some positive definite ternary quadratic forms and the class numbers of the corresponding imaginary quadratic fields. In particular, we obtain an explicit formula for the ternary quadratic form , where p and q are odd primes. Let h(d) be the class number of , and for a ternary quadratic form Q, we use for the number of integral solutions (x, y, z) of the equation . We give two examples to illustrate our results: (i) For , if m is a square-free positive integer with , then (ii) for , if m is a square-free positive integer with , then