Generalized Cullen numbers with the Lehmer property
Bull. Korean Math. Soc. 2013 Vol. 50, No. 6, 1981-1988
Printed November 1, 2013
Dae-June Kim and Byeong-Kweon Oh
Seoul National University, Seoul National University
Abstract : We say a positive integer $n$ satisfies the Lehmer property if $\phi(n)$ divides $n-1$, where $\phi(n)$ is the Euler's totient function. Clearly, every prime satisfies the Lehmer property. No composite integer satisfying the Lehmer property is known. In this article, we show that every composite integer of the form $D_{p,n}=np^n+1$, for a prime $p$ and a positive integer $n$, or of the form $\alpha2^{\beta}+1$ for $\alpha \le \beta$ does not satisfy the Lehmer property.
Keywords : Euler's totient function, generalized Cullen number, Lehmer property
MSC numbers : 11A05, 11N25
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