A Problem Concerning A Character Sum

. Let p be a prime congruent to \Gamma1 modulo 4, i n p j the Legendre symbol and S(k) = P p\Gamma1 n=1 n k i n p j . 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 paper we exhibit some techniques which were successful in producing, for each k such that 3 k 2000, a value for p such that S(k) ? 0. 1. Introduction Shortly after the death of Daniel Shanks, the second author received a collection of correspondence between Shanks and D. H. and Emma Lehmer. This material covers the period between 1968-71 when Shanks was very active in developing ideas that would be of great significance to the development of computational algebraic number theory. Furthermore, it is evident from this correspondence that a rather simple looking problem served as a focus f..