On primary Carmichael numbers

Abstract

The primary Carmichael numbers were recently introduced as a special subset of the Carmichael numbers. A primary Carmichael number $m$ has the unique property that $s_p(m) = p$ holds for each prime factor $p$, where $s_p(m)$ is the sum of the base-$p$ digits of $m$. The first such number is Ramanujan's famous taxicab number $1729$. Due to Chernick, all Carmichael numbers with three factors can be constructed by certain squarefree polynomials $U_3(t) \in \mathbb{Z}[t]$, the simplest one being $U_3(t) = (6t+1)(12t+1)(18t+1)$. We show that the values of any $U_3(t)$ obey a special decomposition for all $t \geq 2$ and besides certain exceptions also in the case $t=1$. These cases further imply that if all three factors of $U_3(t)$ are simultaneously odd primes, then $U_3(t)$ is not only a Carmichael number, but also a primary Carmichael number. Together with the exceptional cases, all Carmichael numbers with three factors have at least the property that $s_p(m) = p$ holds for the greatest prime factor $p$ of $m$. Subsequently, we show some connections to taxicab and polygonal numbers, involving the number $1729$ as an example again.