AIMS Mathematics, 2020, 5(4): 3899-3905. doi: 10.3934/math.2020252

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The primitive roots and a problem related to the Golomb conjecture

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

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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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