Strengthening the Baillie-PSW primality test

TL;DR

This paper introduces a strengthened version of the Baillie-PSW primality test that incorporates Lucas-V pseudoprimes, enhancing its reliability with minimal additional computational effort.

Contribution

The paper presents a novel enhancement to the Baillie-PSW primality test by including Lucas-V pseudoprimes, significantly improving its robustness against composite numbers.

Findings

Only five Lucas-V pseudoprimes less than 10^{15} identified.

Strengthened test maintains low computational cost.

No known odd composite passes the improved test.

Abstract

The Baillie-PSW primality test combines Fermat and Lucas probable prime tests. It reports that a number is either composite or probably prime. No odd composite integer has been reported to pass this combination of primality tests if the parameters are chosen in an appropriate way. Here, we describe a significant strengthening of this test that comes at almost no additional computational cost. This is achieved by including in the test what we call Lucas-V pseudoprimes, of which there are only five less than $10^{15}$.