CONGRUENCES RELATED TO THE ANKENY-ARTIN-CHOWLA CONJECTURE

摘要

Let p be an odd prime with p 1 (mod 4) and " = (t + u pp)/2 > 1 be the fundamental unit of the real quadratic field K = Q( pp) over the rationals. The Ankeny-Artin-Chowla conjecture asserts that p - u, which still remains unsolved. In this paper, we investigate various kinds of congruences equivalent to its negation p | u by making use of Dirichlet’s class number formula, the products of quadratic residues and non-residues modulo p and a special type of congruence for Bernoulli numbers.