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September 2014 On the congruence $\kappa n\equiv a(mod \varphi(n))$
Boqing Xue
Funct. Approx. Comment. Math. 51(1): 189-196 (September 2014). DOI: 10.7169/facm/2014.51.1.11

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, \[ \#\{n\leq x: \varphi(n)\mid (n-1), n\not \equiv 1 (mod 6^{\lambda})\}\ll x^{1/2}/(\log{x})^{0.559552+\textit{o}(1)},\\ \#\{n\leq x: \varphi(n)\mid (3n-1)\} \ll x^{1/2}/(\log{x})^{0.559552+\textit{o}(1)}. \]

Citation

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Boqing Xue. "On the congruence $\kappa n\equiv a(mod \varphi(n))$." Funct. Approx. Comment. Math. 51 (1) 189 - 196, September 2014. https://doi.org/10.7169/facm/2014.51.1.11

Information

Published: September 2014
First available in Project Euclid: 24 September 2014

zbMATH: 1352.11009
MathSciNet: MR3263077
Digital Object Identifier: 10.7169/facm/2014.51.1.11

Subjects:
Primary: 11A25

Rights: Copyright © 2014 Adam Mickiewicz University

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Vol.51 • No. 1 • September 2014
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