Constructing Carmichael numbers through improved subset-product algorihms

Abstract. 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 prob-lem that exploit the non-uniform distribution of primes p with the property that p − 1 divides a highly composite Λ. 1