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| 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 (number theory)|class number]] ''h'' of a real [[quadratic field]] of [[discriminant]] ''d'' > 0. If the [[fundamental unit]] of the field is
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| :<math>\varepsilon = \frac{t + u \sqrt{d}}{2}</math>
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| with integers ''t'' and ''u'', it expresses in another form
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| :<math>\frac{ht}{u} \pmod{p}\;</math>
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| for any [[prime number]] ''p'' > 2 that divides ''d''. In case ''p'' > 3 it states that
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| :<math>-2{mht \over u} \equiv \sum_{0 < k < d} {\chi(k) \over k}\lfloor {k/p} \rfloor \pmod {p}</math>
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| where <math>m = \frac{d}{p}\;</math> and <math>\chi\;</math> is the [[Dirichlet character]] for the quadratic field. For ''p'' = 3 there is a factor (1 + ''m'') multiplying the [[Sides of an equation|LHS]]. Here
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| :<math>\lfloor x\rfloor</math> | |
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| represents the [[floor function]] of ''x''.
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| A related result is that if ''d=p'' is congruent to one mod four, then
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| :<math>{u \over t}h \equiv B_{(p-1)/2} \pmod{ p}</math> | |
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| where ''B''<sub>''n''</sub> is the ''n''th [[Bernoulli number]].
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| There are some generalisations of these basic results, in the papers of the authors.
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| ==References==
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| *{{citation
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| | last1 = Ankeny | first1 = N. C. | author1-link = Nesmith Ankeny
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| | last2 = Artin | first2 = E. | author2-link = Emil Artin
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| | last3 = Chowla | first3 = S. | author3-link = Sarvadaman Chowla
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| | doi = 10.2307/1969656
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| | journal = [[Annals of Mathematics]]
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| | mr = 0049948
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| | pages = 479–493
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| | series = Second Series
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| | title = The class-number of real quadratic number fields
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| | volume = 56
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| | year = 1952}}
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| {{DEFAULTSORT:Ankeny-Artin-Chowla congruence}}
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| [[Category:Algebraic number theory]]
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