Index of Fibonacci primes and Lucas primes.

For an integer $$n\geq 0$$ let $$F_n$$ denote the $$n$$th Fibonacci number and let $$L_n$$ denote the $$n$$th Lucas number.

It is known that $$F_n$$ is prime only if $$n$$ is prime or $$n=4$$.

According to Wikipedia it is known that $$L_n$$ is prime only if $$n$$ is $$0$$, prime or a power of $$2$$.

The reciprocals are not true. Indeed, $$\begin{array}{ccc} F_2 &= &1,\\ F_{19} &= &37\times 113,\\ L_{23} &= &139\times 461,\\ L_{64} &= &1087 \times 4481. \end{array}$$

Let $$S=\{n: F_n\text{ is prime}\}$$ and $$T=\{n: L_n\text{ is prime}\}$$.

Q: Is the set $$S\cap T$$ finite?

Some remarks:

1. Some computations in Mathematica suggest that $$n=4, 5, 7, 11, 13, 17, 47$$ are all the integers in $$S\cap T$$ with $$n\leq 10000$$.

2. It follows by the exposition above that an element of $$S\cap T$$ is either $$4$$ or prime.

3. It is not known whether there are infinitely many Fibonacci prime numbers. So, either my question is an open problem or the answer is yes.

4. My interest in the set $$S\cap T$$ arises from a question related to giving approximations of $$\sqrt{5}$$ as the fraction of two prime numbers. Recall that $$\frac{L_n}{F_n}$$ tends to $$\sqrt 5$$ when $$n$$ tends to infiniy.

5. I have no strong background on Fibonacci numbers. All remarks are welcome.

• (1). I dk...(2). It is also not known whether $T$ is finite or not...(3). Since $F_{2n}=F_nL_n$ we have $n\in S\cap T$ iff $F_{2n}$ is the product of two primes. May 25, 2019 at 17:24
• OEIS A080327 is the related sequence and indicates that the next, and conjectured only other, member of $S\cap T$ is $148091$. This in turn links to Prime Curio which remarks on the precision of the related $L_n/F_n$ approximation to $\sqrt{5}$ Jun 7, 2019 at 19:12

One thing (among many) that we do know is that $$F_{2n} = F_{n}L_{n}$$ with the $$gcd(F_{n}, L_{n})$$ being either 1 or 2.