Abstract
Let p be a prime congruent to −1 modulo 4, (n/p) the Legendre symbol and \(S(k) = \Sigma _{n = 1}^{p - 1} {\text{ }}n^k \left( {\tfrac{n}{p}} \right)\). 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 Emma Lehmer were ever successful in finding such a p. In this extended abstract we summarize some techniques which were successful in producing, for each k such that 3 ≤ k ≤ 2000, a value for p such that S(k) > 0.
The full version of this paper is to appear in Experimental Mathematics
Research supported by NSERC of Canada grant A7649
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References
R. Ayoub, S. Chowla, and H. Walum. On sums involving quadratic characters. J. London Math. Soc., 42:152–154, 1967.
E. Bach. Improved approximations for Euler products. In Number Theory: CMS Conference Proceedings, volume 15, pages 13–28. AMS, Providence, R.I., 1995.
J. Brillhart, D.H. Lehmer, and J. Selfridge. New primality criteria and factorizations of 2m ± 1. Math. Comp., 29:620–647, 1975.
N.J. Fine. On a question of Ayoub, Chowla and Walum concerning character sums. Illinois J. Math., 14:88–90, 1970.
M.J. Jacobson, Jr. Computational techniques in quadratic fields. Master's thesis, University of Manitoba, 1995. M.Sc. Thesis.
D.H. Lehmer, E. Lehmer, and D. Shanks. Integer sequences having prescribed quadratic character. Math. Comp., 24:433–451, 1970.
LiDIA Group, Technische UniversitÄt Darmstadt, Darmstadt, Germany. LiDIA — A library for computational number theory, Version 1.3, 1997.
R.F. Lukes, C.D. Patterson, and H.C. Williams. Numerical sieving devices: their history and some applications. Nieuw Archief voor Wiskunde, 13(4): 113–139, 1995.
R.F. Lukes, C.D. Patterson, and H.C. Williams. Some results on pseudosquares. Math. Comp., 65:361–372, 1996.
D. Shanks. Class number, a theory of factorization and genera. In Proc. Symp. Pure Math. 20, pages 415–440. AMS, Providence, R.I., 1971.
SIMATH Research Group, Chair of Prof. Dr. H.G. Zimmer, University of Saarland, Saarbrücken, Germany. SIMATH Manual, 1997.
E. Teske and H.C. Williams. A problem concerning a character sum. Experimental Mathematics, to appear.
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Teske, E., Williams, H.C. (1998). A problem concerning a character sum. In: Buhler, J.P. (eds) Algorithmic Number Theory. ANTS 1998. Lecture Notes in Computer Science, vol 1423. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0054874
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DOI: https://doi.org/10.1007/BFb0054874
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