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

:$\varepsilon = \frac$

with integers ''t'' and ''u'', it expresses in another form

:$\frac \pmod\;$

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

:$-2 \equiv \sum_ \lfloor \rfloor \pmod$

where $m = \frac\;$   and  $\chi\;$  is the Dirichlet character for the quadratic field. For ''p'' = 3 there is a factor (1 + ''m'') multiplying the LHS. Here

:$\lfloor x\rfloor$

represents the floor function of ''x''.

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

:$h \equiv B_ \pmod$

where ''B''_''n'' is the ''n''th Bernoulli number.

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

References

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Theorems in algebraic number theory

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