A problem concerning a character sum.(English)Zbl 0938.11061

Authors’ abstract: “Let $$p$$ be a prime congruent to $$-1$$ modulo $$4$$, $${n \choose p}$$ the Legendre symbol and $$S(k)=\sum_{n=1}^{p-1} n^k {n \choose p}$$. The problem of finding a prime $$p$$ such that $$S(3)>0$$ was one of the motivating forces behind the development of several of Shanks’ ideas for computing in algebraic number fields, although neither he nor D. H. and E. Lehmer were ever successful in finding such a $$p$$. In this paper we exhibit some techniques which were successful in producing, for each $$k$$ such that $$3 \leq k \leq 2000$$, a value for $$p$$ such that $$S(k)>0$$”.
Reviewer: J.Hinz (Marburg)

MSC:

 11Y40 Algebraic number theory computations 11Y99 Computational number theory 11L99 Exponential sums and character sums

Keywords:

character sum; Legendre symbol

LiDIA; SIMATH
Full Text:

References:

 [1] Ayoub R., J. London Math. Soc. 42 pp 152– (1967) · Zbl 0146.26804 [2] Bach E., Number theory (Halifax, 1994) pp 13– (1995) [3] Brillhart J., Math. Comp. 29 pp 620– (1975) [4] Elliott P. D.T. A., Invent. Math. 21 pp 319– (1973) · Zbl 0265.10022 [5] Elliott P. D.T. A., Probabilistic number theory, II: Central limit theorems (1980) · Zbl 0431.10030 [6] Fine N. J., Illinois J. Math. 14 pp 88– (1970) [7] Jacobson M. J., Master’s thesis, in: Computational techniques in quadratic fields (1995) [8] Jacobson M. J., Math. Comp. (1999) [9] Joshi P. T., J. Number Theory 2 pp 58– (1970) · Zbl 0208.31103 [10] Lehmer D. H., Math. Comp. 24 pp 433– (1970) [11] Lenstra J. H.W., Journées arithmétiques 1980(Exeter, 1980) pp 123– (1982) [12] ”LiDIA: a C++ library for computational number theory, version 1.3” (1997) [13] Lukes R. F., Nieuw Arch. Wisk. (4) 13 (1) pp 113– (1995) [14] Lukes R. F., Math. Comp. 65 (213) pp 361– (1996) · Zbl 0852.11072 [15] Rosser J. B., Illinois J. Math. 6 pp 64– (1962) [16] Shanks, D. ”Class number, a theory of factorization, and genera”. Proceedings of the 1969 Summer Institutes on Number Theory. 1969, Stony Brook, NY. Edited by: Lewis, D. J. pp.415–440. Providence: Amer. Math. Soc. [Shanks 1971], Proc. Sympos. Pure Math. 20 [17] Shanks, D. ”The infrastructure of a real quadratic field and its applications”. Proceedings of the Number Theory Conference. 1972, Boulder, CO. pp.217–224. Boulder: Univ. Colorado. [Shanks 1972] · Zbl 0334.12005 [18] Zimmer H. G., ”SIMATH: a computer algebra system for number theoretic applications” (1997) · Zbl 0923.11173
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