Fast tabulation of challenge pseudoprimes

TL;DR

This paper introduces an efficient algorithm for identifying challenge pseudoprimes, confirming their non-existence up to a large bound and providing asymptotic improvements over prior methods for fixed prime factor counts.

Contribution

The authors develop a new optimized algorithm for tabulating challenge pseudoprimes, with proven asymptotic improvements for fixed prime factors.

Findings

No challenge pseudoprimes with two or three prime factors up to 2^80.

Algorithm provides unconditional asymptotic improvement for fixed prime factors.

Validated the non-existence of certain pseudoprimes within a large computational range.

Abstract

We provide a new algorithm for tabulating composite numbers which are pseudoprimes to both a Fermat test and a Lucas test. Our algorithm is optimized for parameter choices that minimize the occurrence of pseudoprimes, and for pseudoprimes with a fixed number of prime factors. Using this, we have confirmed that there are no PSW challenge pseudoprimes with two or three prime factors up to $2^{80}$. In the case where one is tabulating challenge pseudoprimes with a fixed number of prime factors, we prove our algorithm gives an unconditional asymptotic improvement over previous methods.