Congruences modulo 16 for the class numbers of complex quadratic fields

AbstractLet h(d) denote the class number of the quadratic field Q(√d) of discriminant d. A number of new determinations of h(d) modulo 16 are proved. For example, it is shown that if p and q are primes satisfying p ≡ q ≡ 5 (mod 8), (pq) = 1, then h(−8pq)≡4(mod16) ifaA+bBp=(−1)(b+B+4)412(mod16) ifaA+bBp=(−1)(b+B)4 where a and b are unique integers such that p = a2 + b2, a ≡ 1 (mod 4), b ≡ ((p − 1)2)! a (mod p), and A and B are the unique integers such that q = A2 + B2, A ≡ 1 (mod 4), B ≡ ((q − 1)2)! A (mod q)