Advances in Tabulating Carmichael Numbers

Abstract

We report that there are $49679870$ Carmichael numbers less than $10^{22}$ which is an order of magnitude improvement on Richard Pinch's prior work. We find Carmichael numbers of the form $n = Pqr$ using an algorithm bifurcated by the size of $P$ with respect to the tabulation bound $B$. For $P < 7 \cdot 10^7$, we found $35985331$ Carmichael numbers and $1202914$ of them were less than $10^{22}$. When $P > 7 \cdot 10^7$, we found $48476956$ Carmichael numbers less than $10^{22}$. We provide a comprehensive overview of both cases of the algorithm. For the large case, we show and implement asymptotically faster ways to tabulate compared to the prior tabulation. We also provide an asymptotic estimate of the cost of this algorithm. It is interesting that Carmichael numbers are worst case inputs to this algorithm. So, providing a more robust asymptotic analysis of the cost of the algorithm would likely require resolution of long-standing open questions regarding the asymptotic density of Carmichael numbers.