This paper answers this: Primes as sums of Fibonacci numbers, Michael Drmota, Clemens Müllner, Lukas Spiegelhofer https://arxiv.org/abs/2109.04068 Memoirs of the American Mathematical Society

The abstracts includes the following: "One of our main results says that for every sufficiently large integer k there exists a prime number that can be represented as the sum of k different and non-consecutive Fibonacci numbers."

Hence it works for all sufficiently large numbers $k\geq k_0$. On page 4 it is mentioned that an effective bound $k_0$ could in principle be worked out, (but that would be very messy).

Update: as usual in analytic number theory, if one can prove the existence of an object one can often prove that there are many. Theorem 1.2, also on page 4, gives a strong quantitative result, implying that for each sufficiently large $k$ there are infinitely many primes.

This paper almost answers this: Primes as sums of Fibonacci numbers, Michael Drmota, Clemens Müllner, Lukas Spiegelhofer https://arxiv.org/abs/2109.04068 Memoirs of the American Mathematical Society

The abstracts includes the following: "One of our main results says that for every sufficiently large integer k there exists a prime number that can be represented as the sum of k different and non-consecutive Fibonacci numbers."

Hence it works for all sufficiently large numbers $k\geq k_0$. On page 4 it is mentioned that an effective bound $k_0$ could in principle be worked out, (but that would be very messy).

Update: as usual in analytic number theory, if one can prove the existence of an object one can often prove that there are many. Theorem 1.2, also on page 4, gives a strong quantitative result, implying that for each sufficiently large $k$ there are many such primes, (choose $x$ such that $k-\mu \log_{\gamma} x$ is small). The question if there are infinitely many such primes seems open.

This paper answers this: Primes as sums of Fibonacci numbers, Michael Drmota, Clemens Müllner, Lukas Spiegelhofer https://arxiv.org/abs/2109.04068 Memoirs of the American Mathematical Society

The abstracts includes the following: "One of our main results says that for every sufficiently large integer k there exists a prime number that can be represented as the sum of k different and non-consecutive Fibonacci numbers."

Hence it works for all sufficiently large numbers $k\geq k_0$. On page 4 it is mentioned that an effective bound $k_0$ could in principle be worked out, (but that has not yet done).

This paper answers this: Primes as sums of Fibonacci numbers, Michael Drmota, Clemens Müllner, Lukas Spiegelhofer https://arxiv.org/abs/2109.04068 Memoirs of the American Mathematical Society

The abstracts includes the following: "One of our main results says that for every sufficiently large integer k there exists a prime number that can be represented as the sum of k different and non-consecutive Fibonacci numbers."

Hence it works for all sufficiently large numbers $k\geq k_0$. On page 4 it is mentioned that an effective bound $k_0$ could in principle be worked out, (but that would be very messy).

Update: as usual in analytic number theory, if one can prove the existence of an object one can often prove that there are many. Theorem 1.2, also on page 4, gives a strong quantitative result, implying that for each sufficiently large $k$ there are infinitely many primes.