On the distribution of primitive roots and Lehmer numbers

Jiafan Zhang; Jiafan Zhang

doi:10.3934/era.2023350

Electronic Research Archive

2023, Volume 31, Issue 11: 6913-6927. doi: 10.3934/era.2023350

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On the distribution of primitive roots and Lehmer numbers

Jiafan Zhang ^,

Research Center for Number Theory and Its Applications, Northwest University, Xi'an 710127, Shaanxi, China

Received: 20 April 2023 Revised: 17 September 2023 Accepted: 19 September 2023 Published: 27 October 2023

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.
- primitive roots,
- Lehmer numbers,
- Golomb's conjecture,
- Gauss sums,
- the generalized Kloosterman sums
Citation: Jiafan Zhang. On the distribution of primitive roots and Lehmer numbers[J]. Electronic Research Archive, 2023, 31(11): 6913-6927. doi: 10.3934/era.2023350

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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.

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