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Concept# Ankeny–Artin–Chowla congruence

Summary

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{t + u \sqrt{d}}{2}
with integers t and u, it expresses in another form
:\frac{ht}{u} \pmod{p};
for any prime number p > 2 that divides d. In case p > 3 it states that
:-2{mht \over u} \equiv \sum_{0 < k < d} {\chi(k) \over k}\lfloor {k/p} \rfloor \pmod {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
:{u \over t}h \equiv

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