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Mathematics of Computation

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



Integer sequences having prescribed quadratic character

Authors: D. H. Lehmer, Emma Lehmer and Daniel Shanks
Journal: Math. Comp. 24 (1970), 433-451
MSC: Primary 10.03
MathSciNet review: 0271006
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Abstract: For the odd primes $ {p_1} = 3,$, $ {p_2} = 5, \cdots ,$ we determine integer sequences $ {N_p}$ such that the Legendre symbol $ ({N \mathord{\left/ {\vphantom {N {{p_i}}}} \right. \kern-\nulldelimiterspace} {{p_i}}}) = \pm 1$ for all $ {p_i} \leqq p$ for a prescribed array of signs $ \pm 1$; (i.e., for a prescribed quadratic character). We examine six quadratic characters having special interest and applications. We present tables of these $ {N_p}$ and examine some applications, particularly to questions concerning extreme values for the smallest primitive root (of a prime $ N$), the class number of the quadratic field $ R(\surd - N)$, the real Dirichlet $ L$ functions, and quadratic character sums.

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Keywords: Quadratic character, sieves, primitive roots, class number, Dirichlet $ L$ functions, quadratic character sums, pseudo-squares
Article copyright: © Copyright 1970 American Mathematical Society