This question is about primes that are sums of two Fibonacci numbers.
$$ \begin{align} 2 &= 1 + 1 \\ 3 &= 1 + 2 \\ 5 &= 2 + 3 \\ 7 &= 2 + 5 \\ 11 &= 3 + 8 \\ 13 &= 5 + 8 \\ &\mathbf{17 = ?} \\ &\mathbf{19 = ?} \\ 23 &= 2 + 21 \\ 29 &= 8 + 21 \\ 31 &= 34 + (-1) \\ &= \text{$F_9 + F_{-4}$} \\ 37 &= 3 + 34 \\ &\mathbf{41 = ?} \\ \cdots \end{align} $$
Decompositions $a+b$ for $17$:
$$ \begin{smallmatrix} a &0 &1 &2 &3 &4 &5 &6 &7 &8 &9 &10 &11 &12 &13 &14 &15 &16 &17 \\ b &17 &16 &15 &14 &13 &12 &11 &10 &9 &8 &7 &6 &5 &4 &3 &2 &1 &0 \\ \hline a &0 &-1 &-2 &-3 &-4 &-5 &-6 &-7 &-8 &-9 &-10 &-11 &-12 &-13 &-14 &-15 &-16 &-17 \\ b &17 &18 &19 &20 &21 &22 &23 &24 &25 &26 &27 &28 &29 &30 &31 &32 &33 &34 \\ \end{smallmatrix} $$
So, obviously there are primes that don't have that representation.
Question: Can we characterize the primes that do have a representation $p = F_a + F_b$ where $a,b \in \mathbb{Z}$ and the Fibonacci series $\{F_k\}$ is extended with $k$ going to negative integers as well.