RADICALLY WEAKENING THE LEHMER AND CARMICHAEL CONDITIONS

Carmichael numbers and pseudoprimes

We now establish a pleasantly simple description of Carmichael numbers, due to Korselt. First, we need the following notion. Let a and p be coprime (usually, p will be prime, but this is not essential). The order of a modulo p, denoted by ordp(a), is the smallest positive integer m such that a ≡ 1 mod p. Recall [NT4.5]: If ordp(a) = m and r is any integer such that a ≡ 1 mod p, then r is a mult...

Pseudoprimes and Carmichael Numbers

Compositions with the Euler and Carmichael Functions

Let φ and λ be the Euler and Carmichael functions, respectively. In this paper, we establish lower and upper bounds for the number of positive integers n ≤ x such that φ(λ(n)) = λ(φ(n)). We also study the normal order of the function φ(λ(n))/λ(φ(n)).

Sierpiński and Carmichael numbers

We establish several related results on Carmichael, Sierpiński and Riesel numbers. First, we prove that almost all odd natural numbers k have the property that 2nk + 1 is not a Carmichael number for any n ∈ N; this implies the existence of a set K of positive lower density such that for any k ∈ K the number 2nk + 1 is neither prime nor Carmichael for every n ∈ N. Next, using a recent result of ...

Divisors of the Euler and Carmichael Functions

We study the distribution of divisors of Euler’s totient function and Carmichael’s function. In particular, we estimate how often the values of these functions have ”dense” divisors.