Linear Combinations of Factorial and $S$-unit in a Ternary recurrence sequence with a double root

Florian Luca, Armand Noubissie

Here, we show that if $u_n=n{2}^n\pm 1$, then the largest prime factor of $u_n\pm m!$ for $n\ge 0,m\ge 2$ tends to infinity with $\max \{m,n\}$. In particular, the largest $n$ partic…