(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, where (the) Richard Guy shows
- $F_{4n}+1 = F_{2n-1} L_{2n-1}$,
- $F_{4n}+1 = F_{2n+1} L_{2n}$,
- $F_{4n+2}+1 = F_{2n+2} L_{2n}$,
- $F_{4n+3}+1 = F_{2n+1} L_{2n+2}$
where $L_n$ is the $n$th Lucas number.