Constructing Carmichael numbers through improved subset-product algorithms

Constructing Carmichael numbers through improved subset-product algorithms

We have constructed a Carmichael number with 10,333,229,505 prime factors, and have also constructed Carmichael numbers with k prime factors for every k between 3 and 19,565,220. These computations are the product of implementations of two new algorithms for the subset product problem that exploit the non-uniform distribution of primes p with the property that p − 1 divides a highly composite Λ.

A new algorithm for constructing large Carmichael numbers

We describe an algorithm for constructing Carmichael numbers N with a large number of prime factors p1, p2, . . . , pk. This algorithm starts with a given number Λ = lcm(p1 − 1, p2 − 1, . . . , pk − 1), representing the value of the Carmichael function λ(N). We found Carmichael numbers with up to 1101518 factors.

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