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

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


  • 1. S. W. Golomb, Algebraic constructions for costas arrays, J. Comb. Theory Ser. A, 37 (1984), 13-21.    
  • 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.    
  • 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.    
  • 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.    
  • 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.    
  • 18. K. Gong, C. H. Jia, Shifted character sums with multiplicative coefficients, J. Number Theory, 153 (2015), 364-371.    


This article has been cited by

  • 1. Jiafan Zhang, Xingxing Lv, On the primitive roots and the generalized Golomb’s conjecture, AIMS Mathematics, 2020, 5, 6, 5653, 10.3934/math.2020361

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