The primitive roots and a problem related to the Golomb conjecture

Wenpeng Zhang; Tingting Wang; Wenpeng Zhang; Tingting Wang

doi:10.3934/math.2020252

AIMS Mathematics

2020, Volume 5, Issue 4: 3899-3905. doi: 10.3934/math.2020252

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Research article

The primitive roots and a problem related to the Golomb conjecture

Wenpeng Zhang ¹,
Tingting Wang ^{2
,
,}

1.
School of Mathematics, Northwest University, Xi’an, Shaanxi, P. R. China
2.
College of Science, Northwest A&F University, Yangling, Shaanxi, P. R. China

Received: 10 March 2020 Accepted: 20 April 2020 Published: 26 April 2020
MSC : 11A07, 11D85

In this paper, we use elementary methods, properties of Gauss sums and estimates for character sums to study a problem related to primitive roots, and prove the following result. Let $p$ be a large enough odd prime. Then for any two distinct integers $a, b \in \{1, 2, \cdots, p-1\}$, there exist three primitive roots $\alpha$, $\beta$ and $\gamma$ modulo $p$ such that the congruence equations $\alpha+\gamma\equiv a\bmod p$ and $\beta+\gamma\equiv b\bmod p$ hold.
- primitive roots,
- the Golomb conjecture,
- character sums,
- Gauss sums,
- asymptotic formula
Citation: Wenpeng Zhang, Tingting Wang. The primitive roots and a problem related to the Golomb conjecture[J]. AIMS Mathematics, 2020, 5(4): 3899-3905. doi: 10.3934/math.2020252

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Abstract

In this paper, we use elementary methods, properties of Gauss sums and estimates for character sums to study a problem related to primitive roots, and prove the following result. Let $p$ be a large enough odd prime. Then for any two distinct integers $a, b \in \{1, 2, \cdots, p-1\}$, there exist three primitive roots $\alpha$, $\beta$ and $\gamma$ modulo $p$ such that the congruence equations $\alpha+\gamma\equiv a\bmod p$ and $\beta+\gamma\equiv b\bmod p$ hold.

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