Some thoughts on pseudoprimes

TL;DR

This paper studies the distribution of pseudoprimes, providing statistical data and proving a conjecture about the density of integers that are pseudoprimes to some proper divisor base.

Contribution

It offers new statistical insights into pseudoprimes in various residue classes and proves a conjecture regarding the asymptotic density of certain pseudoprime sets.

Findings

Tabulated all even pseudoprimes up to 10^16

Provided robust statistics for pseudoprimes in different residue classes

Proved the asymptotic density of integers that are pseudoprimes to some proper divisor base

Abstract

We consider several problems about pseudoprimes. First, we look at the issue of their distribution in residue classes. There is a literature on this topic in the case that the residue class is coprime to the modulus. Here we provide some robust statistics in both these cases and the general case. In particular we tabulate all even pseudoprimes to $10^{16}$. Second, we prove a recent conjecture of Ordowski: the set of integers $n$ which are a pseudoprime to some base which is a proper divisor of $n$ has an asymptotic density.