The primitive roots and a problem related to the Golomb conjecture
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 thre...
| Main Authors: | , |
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| Format: | Article |
| Language: | English |
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AIMS Press
2020-05-01
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| Series: | AIMS Mathematics |
| Subjects: | |
| Online Access: | https://www.aimspress.com/article/10.3934/math.2020252/fulltext.html |
| _version_ | 1818214043827044352 |
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| author | Wenpeng Zhang Tingting Wang |
| author_facet | Wenpeng Zhang Tingting Wang |
| author_sort | Wenpeng Zhang |
| collection | DOAJ |
| description | 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. |
| first_indexed | 2024-12-12T06:13:55Z |
| format | Article |
| id | doaj.art-9639bd35807542748e7f716711bbe32e |
| institution | Directory Open Access Journal |
| issn | 2473-6988 |
| language | English |
| last_indexed | 2024-12-12T06:13:55Z |
| publishDate | 2020-05-01 |
| publisher | AIMS Press |
| record_format | Article |
| series | AIMS Mathematics |
| spelling | doaj.art-9639bd35807542748e7f716711bbe32e2022-12-22T00:35:04ZengAIMS PressAIMS Mathematics2473-69882020-05-01543899390510.3934/math.2020252The primitive roots and a problem related to the Golomb conjectureWenpeng Zhang0Tingting Wang11 School of Mathematics, Northwest University, Xi’an, Shaanxi, P. R. China2 College of Science, Northwest A&F University, Yangling, Shaanxi, P. R. ChinaIn 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.https://www.aimspress.com/article/10.3934/math.2020252/fulltext.htmlprimitive rootsthe golomb conjecturecharacter sumsgauss sumsasymptotic formula |
| spellingShingle | Wenpeng Zhang Tingting Wang The primitive roots and a problem related to the Golomb conjecture AIMS Mathematics primitive roots the golomb conjecture character sums gauss sums asymptotic formula |
| title | The primitive roots and a problem related to the Golomb conjecture |
| title_full | The primitive roots and a problem related to the Golomb conjecture |
| title_fullStr | The primitive roots and a problem related to the Golomb conjecture |
| title_full_unstemmed | The primitive roots and a problem related to the Golomb conjecture |
| title_short | The primitive roots and a problem related to the Golomb conjecture |
| title_sort | primitive roots and a problem related to the golomb conjecture |
| topic | primitive roots the golomb conjecture character sums gauss sums asymptotic formula |
| url | https://www.aimspress.com/article/10.3934/math.2020252/fulltext.html |
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