Constructing Carmichael numbers through improved subset-product algorithms

TL;DR

This paper introduces two new algorithms for the subset product problem, enabling the construction of large Carmichael numbers with many prime factors, advancing computational number theory.

Contribution

The paper presents novel algorithms that efficiently construct Carmichael numbers with a vast number of prime factors, improving previous methods.

Findings

Constructed a Carmichael number with over 10 billion prime factors.

Built Carmichael numbers with all sizes between 3 and 19,565,220 prime factors.

Algorithms exploit non-uniform prime distributions for efficient computation.

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 problem that exploit the non-uniform distribution of primes p with the property that p-1 divides a highly composite \Lambda.