Algorithms for Carmichael numbers

TL;DR

This paper analyzes algorithms for finding Carmichael numbers, showing that the problem is in P conditioned on ERH, introducing a heuristic optimal tabulation method, and presenting extensive computational results up to 10^{24}.

Contribution

It proves CARMICHAELS is in P conditioned on ERH, introduces a new asymptotically optimal tabulation algorithm, and reports the discovery of over 300 million Carmichael numbers below 10^{24}.

Findings

CARMICHAELS is in P conditioned on ERH.

A new heuristic optimal tabulation algorithm is proposed.

Over 300 million Carmichael numbers found below 10^{24}.

Abstract

Our primary concern is the computational complexity of algorithms that find all Carmichael numbers less than some specified bound $B$. We have three related results. First, we show CARMICHAELS is in $\textbf{P}$, where only the run-time is conditioned on the ERH. Second, we state a heuristically optimal tabulation algorithm, which is the first asymptotic improvement to tabulation algorithms in the $50$ years since Swift first described the prime-by-prime approach. Third, we implemented a related algorithm that tabulated $100$ times further while only doing about $5$ times the work of the prior tabulation. We found $308,279,939$ Carmichael numbers less than $10^{24}$ and we provide some statistics on these numbers.