Counting composites with two strong liars

TL;DR

This paper analyzes the distribution of composite numbers with exactly two strong liars in primality testing, providing asymptotic counts, improved algorithms, and extending previous work on liars in primality tests.

Contribution

It introduces methods to count composites with exactly two strong liars, improves algorithms for exact counts, and extends analysis to cases with two prime factors and Euler liars.

Findings

Asymptotic count of composites with two strong liars

Improved algorithm for exact enumeration

Asymptotic counts for special cases with prime factors and Euler liars

Abstract

The strong probable primality test is an important practical tool for discovering prime numbers. Its effectiveness derives from the following fact: for any odd composite number $n$, if a base $a$ is chosen at random, the algorithm is unlikely to claim that $n$ is prime. If this does happen we call $a$ a liar. In 1986, Erd\H{o}s and Pomerance computed the normal and average number of liars, over all $n \leq x$. We continue this theme and use a variety of techniques to count $n \leq x$ with exactly two strong liars, those being the $n$ for which the strong test is maximally effective. We evaluate this count asymptotically and give an improved algorithm to determine it exactly. We also provide asymptotic counts for the restricted case in which $n$ has two prime factors, and for the $n$ with exactly two Euler liars.