On the distribution of primitive roots and Lehmer numbers

Abstract

In this paper, we study the number of the Lehmer primitive roots solutions of a multivariate linear equation and the number of $ 1\leq x\leq p-1 $ such that for $ f(x)\in {\mathbb{F}}_p[x] $, $ k $ polynomials $ f(x+c_1), f(x+c_2), \ldots, f(x+c_k) $ are Lehmer primitive roots modulo prime $ p $, and obtain asymptotic formulae for these utilizing the properties of Gauss sums and the generalized Kloosterman sums.

Keywords

• primitive roots
• lehmer numbers
• golomb's conjecture
• gauss sums
• the generalized kloosterman sums