Average liar count for degree-2 Frobenius pseudoprimes

TL;DR

This paper establishes bounds on the average number of liars in the Quadratic Frobenius Pseudoprime Test, extending previous theoretical work and suggesting the existence of challenging pseudoprimes.

Contribution

It generalizes earlier bounds on liars for Frobenius pseudoprimes, providing new theoretical insights into their distribution and properties.

Findings

Bounds on average number of liars established

Evidence for existence of challenge pseudoprimes

Generalization of previous theoretical results

Abstract

In this paper we obtain lower and upper bounds on the average number of liars for the Quadratic Frobenius Pseudoprime Test of Grantham, generalizing arguments of Erd\H{o}s and Pomerance, and Monier. These bounds are provided for both Jacobi symbol plus and minus cases, providing evidence for the existence of several challenge pseudoprimes.