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Research Article | Open Access

On the distribution of primitive roots and Lehmer numbers

Research Center for Number Theory and Its Applications, Northwest University, Xi'an 710127, Shaanxi, China
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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 ≤ x ≤ p − 1 such that for f ( x ) ∈ F p [ x ], k polynomials f ( x + c 1 ) , f ( x + c 2 ) , … , 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 rootsLehmer numbersGolomb's conjectureGauss sumsthe generalized Kloosterman sums

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Electronic Research Archive
Volume 31 Issue 11,
November 2023
Pages 6913-6927
DOI: 10.3934/era.2023350

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Zhang J. On the distribution of primitive roots and Lehmer numbers. Electronic Research Archive, 2023, 31(11): 6913-6927. https://doi.org/10.3934/era.2023350

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Received: 20 April 2023
Revised: 17 September 2023
Accepted: 19 September 2023
Published: 15 November 2023
©2023 the Author(s), licensee AIMS Press.

This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)