Proving that there are infinitely-many prime numbers that are not Fibonacci numbers

Question

Fibonacci numbers $\{F_n\}_n\in\mathbb{N}$ are defined by the sequence $F_{n+2}=F_{n+1}+F_{n}$, $F_1=F_2=1$. Now prove that there are infinitely many prime numbers which are not Fibonacci numbers.

I tried very much and tried to get a proof by contradiction, but failed. I assumed the negation, but it takes me nowhere.

I assumed that there is a prime number which is a Fibonacci number and the next prime is also a Fibonacci number. Hence, as there is at least one prime between $n$ and $2n$, then $2n$ will have a Fibonacci number form, which can be hence expressed as a prime. But I am not sure what to do next. Is my assumption correct after all?

Answer 1

Analyzing mod $11$, you can show that no fibonacci-number has the form $11k+4$, but infinite many primes have that form because of Dirichlet's theorem.

Answer 2

Let us assume to the contrary that only finitely many primes are not Fibonacci numbers. Suppose the largest of them is $p_{k}$. Then for every n>k, $p_{n}$ must be some Fibonacci number. Now we know, $F_{2n+1}= F_{2n}+F_{2n-1}> F_{2n-1}+F_{2n-1}=2F_{2n-1}$ or, $F_{2n+1}>2F_{2n-1}$

Now we know there exists at least 1 prime between n and 2n. So, there must exist a prime between $F_{2n-1}$ and $2F_{2n-1}$. Therefore, there should exist a prime between $F_{2n+1}$ and $F_{2n-1}$ .

Now, suppose the prime be $p_{n}$ where n>k. So, according to our assumption $p_{n}$ is a fibonacci number. And $p_{n}$ lies between $F_{2n-1}$ and $F_{2n+1}$. So, $p_{n}$ must be $F_{2n}$. But for any n in N, $F_{2n} = (F_{n-1} + F_{n+1})F_{n}$ . Hence, $F_{2n}$ cannot be a prime. So, we reach a contradiction. [Hence, proved]