Constructing Carmichael numbers through improved subset-product algorithms

Abstract

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 Λ.

Tabulating Carmichael numbers $$n = Pqr$$ with small P

“…The Erdős construction starts with a carefully selected composite number L, and builds Carmichael numbers around L. There exist algorithmic versions of this construction technique and they have produced Carmichael numbers with billions of digits (e.g. [1]). A tabulation algorithm should be able to account for how such a number would be found in a tabulation, even if only in a hypothetical way.…”

Product Subset Problem : Applications to number theory and cryptography

“…Furthermore, we can calculate some Carmichael numbers (the first 16): 561, 1105, 1729, 2465, 2821, 6601, 8911, 10585, 15841, 29341, 41041, 46657, 52633, 62745, 63973, and 75361. In [3] they used an idea of Erdős [10] to find Carmichael numbers with many prime factors. In 1996, Loh and Niebuhr [22] provided a Carmichael with 1, 101, 518 prime factors using Erdős heuristic algorithm (Algorithm 1).…”

“…The following method is based on Erdős idea [10]. It was used in [3,22] to produce Carmichael numbers having large number of prime factors.…”

“…We can implement this without using much memory. Even better, we can use [3,Section 8] where they keep only the exponents of the divisors of Λ. Since the set P contains integers of the form 2 a1 3 a2 • • • p ar r + 1 with 0 ≤ a i ≤ h i , instead of storing 2 a1 3 a2 • • • p ar r + 1 we can store (a 1 , ..., a r ).…”

“…Remark 3.1. In [3] they used another algorithm inspired by the quantum algorithm of Kuperberg and they exploit the distribution of the primes in the set P (which is not uniform).…”