I noticed that starting with Fibonacci numbers 2 and 3, alternating pairs of sequential Fibonacci numbers, fn and fn+1 switch between having fn2=1 (mod fn+1) and having fn2= -1 (mod fn+1).
22=1 (mod 3)
32= -1 (mod 5)
52=1 (mod 8)
82 = -1 (mod 13),
and so on.
If this holds for all sequential pairs of Fibonacci numbers then one immediate result would be that all Fibonacci primes greater than 3 can only have odd Fibonacci indices. If fn2=1 (mod fn+1) and fn+1 is prime then fn would have to equal 1 or fn+1 -1, which is not possible.
I am sure that if what I am saying is correct then someone has worked it out long ago. I am not looking for detailed proof, just an outline of how it could be done.
I see now how the Casini identity directly relates to what I asked. I will look into the proof of that identity, which apparently is not that difficult if induction is used.
