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" + "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

:$lfloor\; x\; floor$

represents the floor function of "x".

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

:$\{u\; over\; t\}h\; equiv\; B\_\{(p-1)/2\}\; mod\; p$

where B_n 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