Carmichael numbers in number rings

Carmichael Numbers in Number Rings

for all integers a. This result gives us the rudimentary Fermat Compositeness Test: If a 6≡ a (mod n) for some integer a, then n is composite. While this has the advantage of being computationally simple, it has the distinct disadvantage of failing for some composite n and choice of a. Take, for example, n = 341 = 31 · 11 and a = 2. A quick computation tells us that 2 ≡ 2 (mod 341). Fortunately...

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

Carmichael Numbers in Arithmetic Progressions

We prove that when (a, m) = 1 and a is a quadratic residue mod m, there are infinitely many Carmichael numbers in the arithmetic progression a mod m. Indeed the number of them up to x is at least x1/5 when x is large enough (depending on m). 2010 Mathematics subject classification: primary 11N25; secondary 11A51.