Counting Carmichael numbers with small seeds

Counting Carmichael numbers with small seeds

Let As be the product of the first s primes, let Ps be the set of primes p for which p−1 divides As but p does not divide As, and let Cs be the set of Carmichael numbers n such that n is composed entirely of the primes in Ps and such that As divides n − 1. Erdős argued that, for any ε > 0 and all sufficiently large x (depending on the choice of ε), the set Cs contains more than x1−ε Carmichael ...

Carmichael Numbers With Three Prime Factors

A Carmichael number (or absolute pseudo-prime) is a composite positive integer n such that n|an − a for every integer a. It is not difficult to prove that such an integer must be square-free, with at least 3 prime factors. Moreover if the numbers p = 6m + 1, q = 12m + 1 and r = 18m + 1 are all prime, then n = pqr will be a Carmichael number. However it is not currently known whether there are i...

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...

Pseudoprimes and Carmichael Numbers