Functiones et Approximatio Commentarii Mathematici

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

Boqing Xue

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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

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

First available in Project Euclid: 24 September 2014

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Zentralblatt MATH identifier

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

Euler function Lehmer's totient problem


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.

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