On Narayana numbers which are products of four $b$-repdigits with a consequence

Abstract

In this paper, we focus on Narayana numbers which can be written as a products of four repdigits in base $g$, where $g$ is an integer with $g\ge 2$. We prove that for $g$ between $2$ and $12$, there are finitely many of these numbers. Moreover we have fully determined them.