Strengthening the Baillie-PSW primality 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 $10^{15}$.