Ferrofluid

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

$\epsilon =\frac{t+u\sqrt{d}}{2}$

with integers t and u, it expresses in another form

$\frac{ht}{u}(\mathrm{mod}p)$

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

$-2\frac{mht}{u}\equiv \sum _{0<k<d}\frac{\chi (k)}{k}\lfloor k/p\rfloor (\mathrm{mod}p)$

where $m=\frac{d}{p}$   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

$\frac{u}{t}h\equiv B_{(p-1)/2}(\mathrm{mod}p)$

where B_n is the nth Bernoulli number.

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

References

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