Let 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), ( p q) = 1, then h(-8pq)≡ 4(mod16) if aA+bB p =(-1) (b+B+4) 4 12(mod16) if aA+bB p =(-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).

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Persistent URL dx.doi.org/10.1016/0022-314X(87)90060-6
Journal Journal of Number Theory
Citation
Hardy, K. (Kenneth), & Williams, K.S. (1987). Congruences modulo 16 for the class numbers of complex quadratic fields. Journal of Number Theory, 27(2), 178–195. doi:10.1016/0022-314X(87)90060-6