Special Fibonacci sequence Let $\{F_n\}, n\in \mathbb{N}$ be the sequence of Fibonacci numbers such that $F_1=1$, $F_2=1$ and $F_n=F_{n-1}+F_{n-2}$ $\forall n\geq2$.
Define a new sequence $\{S_n\}$ such that $S_n=F_n+1$ $\forall n\in \mathbb{N}$.
Now the question is: For every prime $p$, does there exist an $N\in  \mathbb{N}$, such that $p|S_N$ ?

Hint. The answer is yes. Show that for any prime $p\not=5$, 
$$p\;\mbox{divides}\;S_{p^2-3}=F_{p^2-3}+1.$$
See for example Jack D'Aurizio's answer here:Fibonacci Sequence problem. Prove that there are infinitely many prime numbers such that $p$ divides $F_{p-1}$
