THE CARMICHAEL AND HARLEY PRIZES.

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

News, awards and prizes

In a new study published in Cerebral Cortex, Senior Researcher Jens Hjortkjær together with colleagues from the Danish Research Centre for Magnetic Resonance (DRCMR) demonstrated a method to decode listeners’ brain activity patterns to determine what sound source a person is listening to. The researchers used functional magnetic resonance imaging (fMRI) to record neural activity while participa...

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 ...