On repunit generalized Cullen numbers

Let $s$ be an integer with $s\ge 2$ and let $R_s$ be the number of solutions of the equation $ns^{ n}+1=(b^m-1)/(b-1)$ in integers $b\ge 2$, $m\ge 3$ and $n\ge 2$. In 2022, Alahmadi and Luca proved that $R_2=0$. In this paper, we prove that for any prime power $p^\ell $, $R_{p^\ell }=0$.