On k-Lehmer numbers

2010/12/10 by Oller-Marcén, Antonio M., Grau, José María
#11A25 #11B99 #FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.1012.2337

Abstract

Lehmer's totient problem consists of determining the set of positive integers n such that φ(n)|n-1 where φ is Euler's totient function. In this paper we introduce the concept of k-Lehmer number. A k-Lehmer number is a composite number such that φ(n)|(n-1)^k. The relation between k-Lehmer numbers and Carmichael numbers leads to a new characterization of Carmichael numbers and to some conjectures related to the distribution of Carmichael numbers which are also k-Lehmer numbers.

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