Fast tabulation of challenge pseudoprimes

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.

Using Fibonacci factors to create Fibonacci pseudoprimes

“…Sam Wagstaff has checked all of those to determine if there were any $620 winners and there were none (see [Po2]). The search is also described in [BW,CG,CP,MK,Po1,PSW,SW].…”