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Mathematics of Computation

ISSN 1088-6842(online) ISSN 0025-5718(print)



Quadratic residue covers for certain real quadratic fields

Authors: R. A. Mollin and H. C. Williams
Journal: Math. Comp. 62 (1994), 885-897
MSC: Primary 11R11; Secondary 11Y40
MathSciNet review: 1218346
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Abstract: Let $ {\Delta _n}(a,b) = {(b{a^n} + (a - 1)/b)^2} + 4{a^n}$ with $ n \geq 1$ and $ b\vert a - 1$. If $ \mathcal{C}$ is a finite set of primes such that for each $ n \geq 1$ there exists some $ q \in \mathcal{C}$ for which the Legendre symbol $ ({\Delta _n}(a,b)/q) \ne - 1$, we call $ \mathcal{C}$ a quadratic residue cover (QRC) for the quadratic fields $ {K_n}(a,b) = Q(\sqrt {{\Delta _n}(a,b))} $. It is shown how the existence of a QRC for any a, b can be used to determine lower bounds on the class number of $ {K_n}(a,b)$ when $ {\Delta _n}(a,b)$ is the discriminant of $ {K_n}(a,b)$. Also, QRCs are computed for all $ 1 \leq a,b \leq 10000$.

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Article copyright: © Copyright 1994 American Mathematical Society

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