Fibonacci[n]-1 is always composite for n>6. why? In[11]:= Select[Table[Fibonacci[n], {n, 1, 10000}], PrimeQ[# - 1] &]
Out[11]= {3, 8}
Edit: Fibonacci[n]-1 is always composite for n>6. why?
 $$\sum\limits_{i = 0}^n {{F_i}}  = {F_{n + 2}} - 1$$
In[16]:= Select[Table[Fibonacci[n], {n, 1, 10000}], PrimeQ[# + 1] &]
Out[16]= {1, 1, 2}
Fibonacci[n]+1 is always composite for n>3. why?
 A: mjqxxxx is correct; there is a proof in Honsberger's Mathematical Gems III, which is short enough to reproduce here.  In fact, it suffices to verify the following eight identities:
$$F_{4k} - 1 = F_{2k+1} L_{2k-1}$$
$$F_{4k+1} - 1 = F_{2k} L_{2k+1}$$
$$F_{4k+2} - 1 = F_{2k} L_{2k+2}$$
$$F_{4k+3} - 1 = F_{2k+2} L_{2k+1}$$
$$F_{4k} + 1 = F_{2k-1} L_{2k+1}$$
$$F_{4k+1} + 1 = F_{2k+1} L_{2k}$$
$$F_{4k+2} + 1 = F_{2k+2} L_{2k}$$
$$F_{4k+3} + 1 = F_{2k+1} L_{2k+2}$$
where $L_k$ are the Lucas numbers.  These identities all follow straightforwardly from Binet's formula, although I imagine there are also combinatorial proofs.  

I suspect at least one of these is an open problem, if not both.  Let me record for now the following observation: modular arithmetic is not enough.
Proposition:  For any finite set $p_1, ... p_n$ of primes, there exists a Fibonacci number $F_k$ such that $\prod p_i | F_k$.
Proof.  Since $F_n | F_{mn}$ for all $m, n \ge 1$ it suffices to show this for a single prime.  But the Fibonacci sequence is clearly periodic $\bmod p$ for any $p$ (since the Fibonacci recursion is reversible), and $F_0 = 0 \equiv 0 \bmod p$.  (The computation of the exact period $\bmod p$ is a nice exercise.)
So both $F_n + 1$ and $F_n - 1$ can avoid any finite set of primes.  
A: Wikipedia cites the following source for the assertion that no sufficiently large Fibonacci number is one greater or one less than a prime:
Ross Honsberger, Mathematical Gems III (AMS Dolciani Mathematical Expositions No. 9), 1985, ISBN 0-88385-318-3, p. 133.
Not sure if that reference has a proof.
