Carmichael number variable relations: three-prime Carmichael numbers up to 10^24

TL;DR

This paper reviews bounds related to Carmichael numbers, introduces a new algorithm for finding three-prime Carmichael numbers up to 10^24, and discusses their distribution in relation to existing conjectures.

Contribution

A new algorithm for identifying three-prime Carmichael numbers and extended bounds and statistics up to 10^24.

Findings

Distribution of three-prime Carmichael numbers analyzed

Algorithm successfully implemented up to 10^24

Insights into conjectures by Granville and Pomerance

Abstract

Bounds and other relations involving variables connected with Carmichael numbers are reviewed and extended. Families of numbers or individual numbers attaining or approaching these bounds are given. A new algorithm for finding three-prime Carmichael numbers is described, with its implementation up to $10^{24}$. Statistics relevant to the distribution of three-prime Carmichael numbers are given, with particular reference to the conjecture of Granville and Pomerance in [A.Granville and C.Pomerance, Two contradictory conjectures concerning Carmichael numbers, Math. Comp. 71 (2001), 883-908].