#### Vol. 2, 2019

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Fast tabulation of challenge pseudoprimes

### Andrew Shallue and Jonathan Webster

Vol. 2 (2019), No. 1, 411–423
##### 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.

pseudoprimes
##### Mathematical Subject Classification 2010
Primary: 11Y11, 11Y16
Secondary: 11A41, 68W40
##### Milestones
Received: 2 March 2018
Revised: 18 June 2018
Accepted: 17 September 2018
Published: 13 February 2019
##### Authors
 Andrew Shallue Department of Mathematics Illinois Wesleyan University Bloomington, IL United States Jonathan Webster Department of Mathematics, Statistics & Actuarial Science Butler University Indianapolis, IN United States