Class Numbers of Indefinite Binary Quadratic Forms and the Residual Indices of Integers Modulo p

Let h(d) be the class number of properly equivalent primitive binary quadratic forms ax²+bxy+cy² with discriminant d=b²-4ac. The behavior of h(5p²), where p runs over primes, is studied. It is easy to show that there are few discriminants of the form 5p² with large class numbers. In fact, one has the estimate where is an arbitrary constant number in (0;1/2). Assume that (x) is a positive function monotonically increasing for x and (x). If , then (assuming the validity of the extended Riemann hypothesis for certain Dedekind zeta-functions) it is proved that It is also proved that for an infinite set of p with one has the inequality where log_kp is the k-fold iterated logarithm (k is an arbitrary integer, k3). Results on mean values of h(5p²) are also obtained. Similar facts are true for the residual indices of an integer a2 modulo p: where o(a,p) is the order of a modulo p. Bibliography: 13 titles.