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

Abstract

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 $n$ participating in the equation $u_n\pm m!=2^a{3}^b{5}^c{7}^d$ with $n\ge 1,m\ge 2$ is $n=8$ for which $(8\cdot 2^8+1)-4!=3^4\cdot 5^2$.

18 pages