Carmichael Numbers in All Possible Arithmetic Progressions

Abstract

We prove that every arithmetic progression either contains infinitely many Carmichael numbers or none at all. Furthermore, there is a simple criterion for determining which category a given arithmetic progression falls into. In particular, if $m$ is any integer such that $(m,2\ddot{I}•(m))=1$ then there exist infinitely many Carmichael numbers divisible by $m$. As a consequence, we are able to prove that ${\mathrm{liminf}}_{n\text{ Carmichael}}\frac{\ddot{I}•(n)}{n}=0$, resolving a question of Alford, Granville, and Pomerance.

56 pages; minor corrections and clarifications