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)


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


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