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 for each prime factor $p$ the sum of the base-$p$ digits of $m$ equals $p$. The first such number is Ramanujan's famous taxicab number $1729$. Due to Chernick, 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. Subsequently, we show some connections to the taxicab and polygonal numbers.Comment: 39 pages, 12 tables, small revisio