Tabulating Absolute Lucas Pseudoprimes

TL;DR

This paper develops algorithms to tabulate absolute Lucas pseudoprimes, finite for fixed prime factors, and provides the first known tabulation for discriminant 5, advancing understanding of these special composite numbers.

Contribution

The paper introduces algorithms for tabulating absolute Lucas pseudoprimes and proves their finiteness for fixed prime factors, including the first tabulation for discriminant 5.

Findings

Finite number of absolute Lucas pseudoprimes for fixed prime factors

Algorithms for efficient tabulation of these pseudoprimes

First known tabulation for discriminant 5

Abstract

In 1977, Hugh Williams studied Lucas pseudoprimes to all Lucas sequences of a fixed discriminant. These are composite numbers analogous to Carmichael numbers and they satisfy a Korselt-like criterion: $n$ must be a product of distinct primes and $p_i - \delta_{p_i} | n - \delta_n $ where $\delta_n$ is a Legendre symbol with the first argument being the discriminant of the Lucas sequence. Motivated by tabulation algorithms for Carmichael numbers, we give algorithms to tabulate these numbers and provide some asymptotic analysis of the algorithms. We show that there are only finitely many absolute Lucas pseudoprimes $n = \prod_{i = 1}^k p_i$ with a given set of $k-2$ prime factors. We also provide the first known tabulation for discriminant $5$.