Note: These queries had come up during an earlier discussion: [On Fibonacci numbers that are also highly composite](https://mathoverflow.net/questions/408396/on-fibonacci-numbers-that-are-also-highly-composite). Am putting them up as a separate post.


Q: Are there any Fibonacci numbers that are sandwiched between twin primes?
An observation: none of the first 30 odd Fibonacci numbers is sandwiched between twin primes.



Are there any Fibonacci numbers that are sandwiched between twin primes?
(In collaboration with Z. Chase.)
A Fibonacci number $F\_{n}$ is never sandwiched between two twin primes $(p,p+2)$.


This is because this would require $F\_{n}+1$ to be a prime, but that can only happen iff $n=1,2,3$, and one can check that $F\_{n}-1$ is not a prime in these cases.


The fact that $F\_{n}+1$ is a prime iff $n=1,2,3$ is probably quite old. One reference is [this OEIS page](https://oeis.org/A001611), where (the) Richard Guy shows


1. $F\_{4n}+1 = F\_{2n-1} L\_{2n+1}$,
2. $F\_{4n+1}+1 = F\_{2n+1} L\_{2n}$,
3. $F\_{4n+2}+1 = F\_{2n+2} L\_{2n}$,
4. $F\_{4n+3}+1 = F\_{2n+1} L\_{2n+2}$


where $L\_n$ is the $n$th Lucas number.


In fact, we only need the first of these identities for your question, because if $F\_{n}$ is sandwiched between twin primes $(p,p+2)$ then $F\_n \equiv 0 \bmod 6$ implying $n \equiv 0 \bmod 12$.


