Ankeny–Artin–Chowla congruence

In number theory, the Ankeny–Artin–Chowla congruence is a result published in 1953 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

with integers t and u, it expresses in another form

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

where m = d p   and   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 d=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.