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On Brizolis' a problem related to primitive roots modulo a prime $ p $

  • Published: 25 November 2025
  • MSC : 11A07, 11L40

  • The main purpose of this paper is to use very simple elementary and analytic methods to study a problem related to the primitive root modulo $ p $ asked by Brizolis and prove a more general and stronger conclusion.

    Citation: Wenpeng Zhang, Xiaoling Xu. On Brizolis' a problem related to primitive roots modulo a prime $ p $[J]. AIMS Mathematics, 2025, 10(11): 27513-27518. doi: 10.3934/math.20251209

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  • The main purpose of this paper is to use very simple elementary and analytic methods to study a problem related to the primitive root modulo $ p $ asked by Brizolis and prove a more general and stronger conclusion.



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    [1] T. M. Apostol, Introduction to analytic number theory, Springer-Verlag, New York, 1976. https://doi.org/10.1007/978-1-4757-5579-4
    [2] R. Ayoub, An introduction to the analytic theory of numbers, American Mathematical Society, Providence, 1963,295.
    [3] K. Ireland, M. Rosen, A classical introduction to modern number theory, Springer-Verlag, New York, 1982.
    [4] W. Narkiewicz, Classical problems in number theory, Polish Scientifc Publishers, WARSZAWA, 1986.
    [5] R. K. Guy, Unsolved problems in number theory, Springer-Vlerlag, Berlin, 1994,244. https://doi.org/10.1007/978-0-387-26677-0
    [6] W. P. Zhang, On a problem of Brizolis, Pure Appl. Math., 11 (1995), 1–3.
    [7] M. Levin, C. Pomerance, K. Soundararajan, Fixed points for discrete logarithms, ANTS-IX 2010, LNCS, 6197 (2010), 6–15. https://doi.org/10.1007/978-3-642-14518-6-5
    [8] H. Hatalová, T. Šalát, Remarks on two results in elementary theory of numbers, Acta Fac. Rer. Natur Univ. Comenian. Math., 20 (1969), 113–117.
    [9] G. Robin, Estimation de la fonction de Tchebychef $\theta$ sur le k-ième nombre premier et grandes valeurs de la fonction $\omega(n)$ nombre de diviseurs premiers de $n$, Acta Arith., 42 (1983), 367–389. http://eudml.org/doc/205883
    [10] G. Robin, Sur la différence $Li(\theta(x))-\pi(x)$, Annales Fac. Sci. Toulouse, 6 (1984), 257–268.
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