A Note on the Primitive Roots and the Golomb Conjecture
In this paper, we use the elementary methods and the estimates for character sums to prove the following conclusion. Let p be a prime large enough. Then, for any positive integer n with p1/2+ɛ≤n<p, there must exist two primitive roots α and β modulo p with 1<α,β≤n−1 such that the equation n=α+...
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| Format: | Article |
| Language: | English |
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Wiley
2021-01-01
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| Series: | Journal of Mathematics |
| Online Access: | http://dx.doi.org/10.1155/2021/7639259 |
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| _version_ | 1832566392257249280 |
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| author | Yiwei Hou Hongyan Wang |
| author_facet | Yiwei Hou Hongyan Wang |
| author_sort | Yiwei Hou |
| collection | DOAJ |
| description | In this paper, we use the elementary methods and the estimates for character sums to prove the following conclusion. Let p be a prime large enough. Then, for any positive integer n with p1/2+ɛ≤n<p, there must exist two primitive roots α and β modulo p with 1<α,β≤n−1 such that the equation n=α+β holds, where 0<ɛ<1/2 is a fixed positive number. In other words, n can be expressed as the exact sum of two primitive roots modulo p. |
| format | Article |
| id | doaj-art-22d85f0b31d845e29877dfd900698abf |
| institution | Kabale University |
| issn | 2314-4629 2314-4785 |
| language | English |
| publishDate | 2021-01-01 |
| publisher | Wiley |
| record_format | Article |
| series | Journal of Mathematics |
| spelling | doaj-art-22d85f0b31d845e29877dfd900698abf2025-02-03T01:04:10ZengWileyJournal of Mathematics2314-46292314-47852021-01-01202110.1155/2021/76392597639259A Note on the Primitive Roots and the Golomb ConjectureYiwei Hou0Hongyan Wang1School of Big Data Science, Hebei Science and Technology Finance Key Laboratory, Hebei Finance University, Baoding, Hebei, ChinaSchool of Big Data Science, Hebei Finance University, Baoding, Hebei, ChinaIn this paper, we use the elementary methods and the estimates for character sums to prove the following conclusion. Let p be a prime large enough. Then, for any positive integer n with p1/2+ɛ≤n<p, there must exist two primitive roots α and β modulo p with 1<α,β≤n−1 such that the equation n=α+β holds, where 0<ɛ<1/2 is a fixed positive number. In other words, n can be expressed as the exact sum of two primitive roots modulo p.http://dx.doi.org/10.1155/2021/7639259 |
| spellingShingle | Yiwei Hou Hongyan Wang A Note on the Primitive Roots and the Golomb Conjecture Journal of Mathematics |
| title | A Note on the Primitive Roots and the Golomb Conjecture |
| title_full | A Note on the Primitive Roots and the Golomb Conjecture |
| title_fullStr | A Note on the Primitive Roots and the Golomb Conjecture |
| title_full_unstemmed | A Note on the Primitive Roots and the Golomb Conjecture |
| title_short | A Note on the Primitive Roots and the Golomb Conjecture |
| title_sort | note on the primitive roots and the golomb conjecture |
| url | http://dx.doi.org/10.1155/2021/7639259 |
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