Testimonial to Mr. Carmichael

Testimonial to Sir Joseph Lister

Testimonial knowledge through unsafe testimony

Frank is a writer with a strange habit. Every morning, at precisely 7:30 a.m., he wakes up and dumps out whatever is left of the pint of milk he purchased the day before, but places the empty carton back in the fridge until noon. Then, throughout the interval from 7:30 to noon, he always remains in the kitchen, as that is where he writes every morning like clockwork. Finally, at exactly noon, h...

Virchow Testimonial Fund

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...

Higher-order Carmichael numbers

We define a Carmichael number of order m to be a composite integer n such that nth-power raising defines an endomorphism of every Z/nZalgebra that can be generated as a Z/nZ-module by m elements. We give a simple criterion to determine whether a number is a Carmichael number of order m, and we give a heuristic argument (based on an argument of Erdős for the usual Carmichael numbers) that indica...