Integer sequences having prescribed quadratic character

Lehmer, D. H.; Lehmer, Emma; Shanks, Daniel

doi:10.1090/S0025-5718-1970-0271006-X

Integer sequences having prescribed quadratic character
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by D. H. Lehmer, Emma Lehmer and Daniel Shanks PDF

Math. Comp. 24 (1970), 433-451 Request permission

Abstract:

For the odd primes ${p_1} = 3,$, ${p_2} = 5, \cdots ,$ we determine integer sequences ${N_p}$ such that the Legendre symbol $({N \left / {\vphantom {N {{p_i}}}} \right . {{p_i}}}) = \pm 1$ 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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