Functiones et Approximatio Commentarii Mathematici

On the congruence $\kappa n\equiv a(mod \varphi(n))$

Boqing Xue

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Abstract

Lehmer's totient problem asks whether there exists a composite $n$ such that $\varphi(n)\mid (n-1)$, where $\varphi$ is the Euler's function. This problem is still open. Later, several upper bounds of the derived problem ``$\varphi(n)\mid (n-a)$'' were given. In this paper, we extend it to $n$ with $\varphi(n) \mid (\kappa n-a)$ and obtain some new bounds. As an application, for any integer $\lambda>0$ we have, \[ \#\{n\leq x: \varphi(n)\mid (n-1), n\not \equiv 1 (mod 6^{\lambda})\}\ll x^{1/2}/(\log{x})^{0.559552+\textit{o}(1)},\\ \#\{n\leq x: \varphi(n)\mid (3n-1)\} \ll x^{1/2}/(\log{x})^{0.559552+\textit{o}(1)}. \]

Article information

Source
Funct. Approx. Comment. Math., Volume 51, Number 1 (2014), 189-196.

Dates
First available in Project Euclid: 24 September 2014

Permanent link to this document
https://projecteuclid.org/euclid.facm/1411564623

Digital Object Identifier
doi:10.7169/facm/2014.51.1.11

Mathematical Reviews number (MathSciNet)
MR3263077

Zentralblatt MATH identifier
1352.11009

Subjects
Primary: 11A25: Arithmetic functions; related numbers; inversion formulas

Keywords
Euler function Lehmer's totient problem

Citation

Xue, Boqing. On the congruence $\kappa n\equiv a(mod \varphi(n))$. Funct. Approx. Comment. Math. 51 (2014), no. 1, 189--196. doi:10.7169/facm/2014.51.1.11. https://projecteuclid.org/euclid.facm/1411564623


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References

  • W.D. Banks, A.M. Güloglu and C.W. Nevans, On the congruence $n \equiv a \;(\textup{mod}\,\varphi(n))$, Integers 8 (2008), #A59.
  • W.D. Banks and F. Luca, Composite integers n for which $\varphi(n)\,|\,n-1$, Acta Math. Sinica 23 (2007), 1915–1918.
  • P. Erdős and J.-L. Nicolas, Sur la fonction: nombre de facteurs premiers de N, Enseign. Math. (2) 27 (1981), no. 1–2, 3–27.
  • R.R. Hall and G. Tenenbaum, Divisor, Cambridge University Press.
  • D.H. Lehmer, On Euler's totient function, Bull. Amer. Math. Soc. 38 (1932), 745–757.
  • F. Luca and C. Pomerance, On composite integers n for which $\varphi(n)\,|\,n-1$, Boletin de la Sociedad Matemática Mexicana 17 (2011), 13–21.
  • F. Mertens, Ein Beitrag zur analytischen Zahlentheorie, Crelle's Journal 78 (1874), 46–62.
  • C. Pan and C. Pan, Fundamentals of analytic number theory (in Chinese), Science Press, Beijing 1991.
  • C. Pomerance, On composite n for which $\varphi(n)\,|\,n-1$, Acta Arith. 28 (1976), 387–389.
  • C. Pomerance, On composite n for which $\varphi(n)\,|\,n-1, II$, Pacific J.Math. 69 (1977), 177–186.
  • Z. Shan, `On composite n for which $\varphi(n)\,|\,n-1$, J. China Univ. Sci.Tech. 15 (1985), 109–112.
  • W. Sierpinski, Elementary Theory of Numbers, Warsaw, 1964.