Lots and Lots of Perrin-Type Primality Tests and Their Pseudo-Primes

TL;DR

This paper explores numerous Perrin- and Lucas-style primality tests using computational methods, identifying many tests with infinite families of pseudo-primes, including Perrin pseudo-primes and Carmichael primes.

Contribution

It introduces a large collection of new Perrin- and Lucas-style primality tests and constructs infinite families of pseudo-primes for these tests.

Findings

Discovery of many new primality tests

Construction of infinite pseudo-prime families

Identification of Perrin and Carmichael pseudo-primes

Abstract

We use Experimental Mathematics and Symbolic Computation (with Maple), to search for lots and lots of Perrin- and Lucas- style primality tests, and try to sort the wheat from the chaff. More impressively, we find quite a few such primality tests for which we can explicitly construct infinite families of pseudo-primes, rather, like in the cases of Perrin pseudo-primes and the famous Carmichael primes, only proving the mere existence of infinitely many of them.