Fibonacci(n) - prime(n) > 0 for n >= 8. All terms other than 2 and 8 (only two terms producing 2, the only even prime) are divisible by 3 (as Fibonacci(n) is even - and hence |Fibonacci(n) - prime(n)| > 1 and odd - iff n is divisible by 3).
Some of the larger entries may only correspond to probable primes.
9 is a term as Fibonacci(9) - prime(9) = 34 - 23 = 11, a prime.
MATHEMATICA
fQ[n_] := PrimeQ[ Fibonacci[n] - Prime[n]]; Do[ If[ fQ[n], Print[n]], {n, 9, 10^4, 3}] (* Robert G. Wilson v, Nov 18 2004 *)
Select[Range[6000], PrimeQ[Abs[Fibonacci[#]-Prime[#]]]&] (* The program generates the first 15 terms of the sequence. *) (* Harvey P. Dale, Nov 08 2025 *)
