Related Searches
Definitions
Nearby Words

# 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

ε = ½(t + ud)

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

$-2\left\{mht over u\right\} = sum_\left\{0 < k < d\right\} \left\{chi\left(k\right) over k\right\}lfloor \left\{k/p\right\} rfloor mod p$

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

$lfloor xrfloor$

represents the floor function of x.

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

$\left\{u over t\right\}h equiv B_\left\{\left(p-1\right)/2\right\} mod p$

where Bn is the nth Bernoulli number.

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

## References

• N. C. Ankeny, E. Artin, S.Chowla, The class-number of real quadratic number fields, Annals of Math. 56 (1953), 479-492

Search another word or see Artinon Dictionary | Thesaurus |Spanish