Consider the set of prime numbers that are not Fibonacci numbers, $ \mathbb{P}_{\neq \mathcal{F}} = \mathbb{P} \setminus \mathcal{F} $.
The first few numbers in this set are $\mathbb{P}_{\neq \mathcal{F}} = \{7, 11, 17, 19, 23, 29, ...\}$
Can it be shown that $|\mathbb{P}_{\neq \mathcal{F}}| = \aleph_{0}$?
In other words, can it be shown that there are an infinite number of prime numbers that are not Fibonacci numbers?
A comment: note that if the cardinality of $\mathbb{P}_{\neq \mathcal{F}}$ was finite, then the number of Fibonacci primes would necessarily be infinite (which is an open question). However, $|\mathbb{P}_{\neq \mathcal{F}}| = \aleph_{0}$ does not indicate anything about the number of Fibonacci primes.
Yes.
Proof: The sum of the reciprocals of the Fibonacci numbers converges (by comparison with a geometric series or by the ratio test). However the sum of the reciprocals of the primes diverges, see this for instance.