Teske, Edlyn; Williams, Hugh C. A problem concerning a character sum. (English) Zbl 0938.11061 Exp. Math. 8, No. 1, 63-72 (1999). 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) Cited in 2 Documents MSC: 11Y40 Algebraic number theory computations 11Y99 Computational number theory 11L99 Exponential sums and character sums Keywords:character sum; Legendre symbol Software:SIMATH; LiDIA × Cite Format Result Cite Review PDF Full Text: DOI Euclid EuDML References: [1] Ayoub R., J. London Math. Soc. 42 pp 152– (1967) · Zbl 0146.26804 · doi:10.1112/jlms/s1-42.1.152 [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 · doi:10.1007/BF01418793 [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 · doi:10.1016/0022-314X(70)90006-5 [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 · doi:10.1090/S0025-5718-96-00678-3 [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 · doi:10.1145/258726.258848 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.