Let p ≡ 5 (mod 8) be a prime. Let h(p) denote the class number of the real quadratic field Q(√ p). It is well-known that h(p) ≡ 1 (mod 2). In this paper the residue of h(p) modulo 4 is determined.

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Journal Pacific Journal of Mathematics
Williams, K.S. (1981). The class number of q(√p) modulo 4, for p = 5 (mod 8) a prime. Pacific Journal of Mathematics, 92(1), 241–248.