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
:
floor 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
:$lfloor\; x$
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.

Contents

References

References

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