Research article

The primitive roots and a problem related to the Golomb conjecture

  • 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.

    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

    Related Papers:

  • 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.


    加载中


    [1] S. W. Golomb, Algebraic constructions for costas arrays, J. Comb. Theory Ser. A, 37 (1984), 13-21. doi: 10.1016/0097-3165(84)90015-3
    [2] L. Qi, W. P. Zhang, On the generalization of Golomb's conjecture, Journal of Northwest University, Natural Science Edition, 45 (2015), 199-201.
    [3] Q. Sun, On primitive roots in a finite field, Journal of Sichuan University, Natural Science Edition, 25 (1988), 133-139.
    [4] T. Tian, W. Qi, Primitive normal element and its inverse in finite fields, Acta Math. Sin., 49 (2006), 657-668.
    [5] P. Wang, X. Cao, R. Feng, On the existence of some specific elements in finite fields of characteristic 2, Finite Fields Th. App., 18 (2012), 800-813.
    [6] J. P. Wang, On Golomb's conjecture, Sci. China Ser. A, 31 (1988), 152-161.
    [7] T. T. Wang, X. N. Wang, On the Golomb's conjecture and Lehmer's numbers, Open Math., 15 (2017), 1003-1009. doi: 10.1515/math-2017-0083
    [8] W. Q. Wang, W. P. Zhang, A mean aalue related to primitive roots and Golomb's conjectures, Abstr. Appl. Anal., 2014 (2014), 1-5.
    [9] W. P. Zhang, On a problem related to Golomb's conjectures, J. Syst. Sci. Complex., 16 (2003), 13-18.
    [10] S. D. Cohen, W. P. Zhang, Sums of two exact powers, Finite Fields Th. App., 8 (2002), 471-477. doi: 10.1016/S1071-5797(02)90354-0
    [11] S. D. Cohen, Pairs of primitive roots, Mathematica, 32 (1985), 276-285.
    [12] S. D. Cohen, T. Trudgian, Lehmer numbers and primitive roots modulo a prime, J. Number Theory, 203 (2019), 68-79. doi: 10.1016/j.jnt.2019.03.004
    [13] R. K. Guy, Unsolved Problems in Number Theory, Springer-Verlag, 1981.
    [14] W. P. Zhang, H. L. Li, Elementary Number Theory, Shaanxi Normal University Press, Xi'an, 2013.
    [15] T. M. Apostol, Introduction to Analytic Number Theory, Springer-Verlag, New York, 1976.
    [16] W. Narkiewicz, Classical Problems in Number Theory, Polish Scientifc Publishers, Warszawa, 1987.
    [17] J. Bourgain, Z. M. Garaev, V. S. Konyagin, On the hidden shifted power problem, SIAM J. Comput., 41 (2012), 1524-1557. doi: 10.1137/110850414
    [18] K. Gong, C. H. Jia, Shifted character sums with multiplicative coefficients, J. Number Theory, 153 (2015), 364-371. doi: 10.1016/j.jnt.2015.01.015
  • Reader Comments
  • © 2020 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)
通讯作者: 陈斌, bchen63@163.com
  • 1. 

    沈阳化工大学材料科学与工程学院 沈阳 110142

  1. 本站搜索
  2. 百度学术搜索
  3. 万方数据库搜索
  4. CNKI搜索

Metrics

Article views(3259) PDF downloads(762) Cited by(3)

Article outline

Other Articles By Authors

/

DownLoad:  Full-Size Img  PowerPoint
Return
Return

Catalog