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?
Best Answer
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.