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
- 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 Bn is the nth Bernoulli number.
There are some generalisations of these basic results, in the papers of the authors.
- N. C. Ankeny, E. Artin, S.Chowla, The class-number of real quadratic number fields, Annals of Math. 56 (1953), 479-492