Ankeny-Artin-Chowla congruence Guide, Meaning , Facts, Information and Description

In number theory, the Ankeny-Artin-Chowla congruence is a result published in 1951 by N.C. Ankeny, Emil Artin and S. Chowla. It concerns the class number h of a real quadratic field of discriminant d > 0. If the fundamental unit of the field is

ε = ½(t + u√d)

with integers t and u, it expresses in another form

ht/u modulo p

for any prime number p > 2 that divides d. In case p > 3 it states that

where m = d/p, χ is the Dirichlet character for the quadratic field. For p = 3 there is a factor (1 + m) multiplying the LHS. Here

represents the floor function of x.

A related result is that if p is congruent to one mod four, then

where B_n is the nth Bernoulli number.

There are some generalisations of these basic results, in the papers of the authors.

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