On k-Lehmer Numbers

Abstract

Lehmer's totient problem consists of determining the set of positive integers n such that '.n/ j .n 1/ where ' 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 '.n/ j .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.

Lehmer's totient problem over Fq[x]

“…where the sets T and T ′′ are defined by the formula (5). Suppose that there exists e i 0 for some 1 ≤ i 0 ≤ s such that e i 0 ∤ n. Hence there is a prime p and r ∈ N * such that p r |e i 0 and p r ∤ n. Thus p r ∈ T ′′ .…”

“…To the best of our knowledge, the current best result is due to Richard G. E. Pinch(see [9]), that the number of prime factors of a Lehmer number n must be at least 15 and there is no Lehmer number less than 10 30 . For further results on this topic we refer the reader to ( [1], [2], [5], [7], [10]). J. Schettler [11] generalizes the divisibilty condition ϕ(n)|(n − 1), constructs reasonable notion of Lehmer numbers and Carmichael numbers in a PID and gets some interesting results.…”

Radically Weakening the Lehmer and Carmichael Conditions

“…Before proving the lemma, we see that using this upper bound in (5) gives us (1) . The theorem then follows immediately from our estimates of K ′ (x) and K ′′ (x).…”

“…Grau and Oller-Marcén [5] define a k-Lehmer number to be an integer n satisfying the condition ϕ(n)|(n−1) k . (Note that they do not require n to be composite, as we have in our definitions.)…”

“…Grau and Oller-Marcén [5] present other possible weakenings of the Lehmer property: looking at the sets of those n such that ϕ(n)|(n − 1) k for a fixed value of k as well as the set of those n for which ϕ(n)|(n − 1) k for some k, that is all of the primes dividing ϕ(n) also divide n − 1. Note that this last set is a weakening of both the Lehmer and Carmichael properties, since λ(n) and ϕ(n) have the same prime divisors.…”