TITLE: Fibonacci[n]-1 is always composite for n>6. why?
QUESTION [16 upvotes]: 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?

REPLY [11 votes]: 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.