Twin primes and Fibonacci

Question

Let $ n $ be a positive integer such that $ 2n+1 $ is prime and $ n(n+1)\pm 1 $ is a pair of twin primes. Is the Fibonacci number $ F_{2n+1} $ almost surely prime ? Thanks in advance.

Answer 1

The first several $n$'s for which $2n+1$, $n^2+n-1$, and $n^2+n+1$ are prime are $n=2,3,5,6,8$ and $15$. It turns out $F_{2n+1}$ is prime in the first five cases (see OEIS entry A001605), but

$$F_{31}=1346269=557\times2417$$

The next two values of $n$ are $20$ and $21$, with prime triples $(41,419,421)$ and $(43,461,463)$, respectively. $F_{43}$ is prime (see the OEIS entry), but

$$F_{41}=165580141=2789\times59369$$

Added later: OEIS entry A088485 comes in handy here. It consists of $n$'s for which $n^2+n-1$ and $n^2+n+1$ are twin primes. The sequence starts

$$2, 3, 5, 6, 8, 15, 20, 21, 24, 38, 41, 50, 54, 59, 66, 89, 101, 131, 138,\ldots$$

Converting it to a sequence of $2n+1$'s and dropping non-primes, we get

$$5,7,11,13,17,31,41,43,83,101,109,179,263,277,\ldots$$

The primes $5,7,11,13,17,43$ and $83$ appear in A001605; the primes $31,41,101,109,179,263$ and $277$ do not -- i.e., $F_{31},F_{41},F_{101}$, etc. are all composite, not prime.

If I had to hazard a guess, I would speculate that it becomes increasingly rare for $n$'s on the OP's list to give a Fibonacci prime $F_{2n+1}$. Of course it's not known if there are infinitely many twin primes; nor is it known if there are infinitely many Fibonacci primes. It's easy to hazard a guess when little is known.

Answer 2

I have ran a Sage script which has looked at all $n<50000$ for which $2n+1$ and $n(n+1)\pm 1$ are prime and checked for which of them $F_{2n+1}$ is prime as well. Here are the results:

• $F_{2n+1}$ is prime for nine values of $n$, namely: $$3,5,6,8,21,41,215,12780,25416\dots$$
• as opposed to $337$ values of $n$ for which $F_{2n+1}$ is composite.

I guess this is evidence against your conjecture.

(explanation of last edit: initially I had to finish the computation early because it was taking a lot of time. It has turned out the comptuation was aborted around $n=49400$, so I've decided to finish it. Four more composites were found, no more primes.)