A Korselt Characterization of Carmichael Numbers of the Form $(aC_n+1)(aC_{n+1}+1)(aC_{n+2}+1)$

Abstract

We study products \[ N=(aC_n+1)(aC_{n+1}+1)(aC_{n+2}+1), \] where $(C_n)$ is the Lucas--balancing sequence. Under the assumption that all three factors are prime, we give a necessary and sufficient condition for $N$ to be a Carmichael number: this happens precisely when $n$ is even and $a=kC_{n+1}$ for a positive integer $k$ satisfying \[ 48k\equiv 7\pmod{C_nC_{n+2}}. \] We derive further congruence restrictions, an explicit parametrization of the integer solutions of the resulting congruence, a lower bound for $k$, and a divisibility relation involving the middle prime factor. The characterization is conditional on the primality of the three factors and does not assert that such prime triples exist for infinitely many parameters.