Progress towards a conjecture of S.W. Graham

Abstract

This article describes progress towards a conjecture of S.W. Graham. He conjectured that the number $C_3\left(X\right)$ of Carmichael numbers up to X with three prime factors is $\le \sqrt{X}$ for all $X\ge 1$. He showed that his conjecture is true for $X\le {10}^{16}$ and $X>{10}^{126}$. In this article, it is shown that the conjecture is true for $X\le {10}^{24}$ and $X>2⁎{10}^{40}$. In both cases, analytical methods establish the conjecture for large X and tables of Carmichael numbers are used for small X.