Let $n \in \mathbb{N}, n \geq 2$. By minimal Goldbach basis $G_{2n}$(if it is nonempty) of $2n$ , I mean the minimal set of primes such that every even number less than or equal to $2n$ can be written as a sum of two primes of that set. Minimal refers to the number of elements of the set. For instance,
$G_{4}= \left \{ 2 \right \}, G_{6}= \left \{ 2,3 \right \} , G_{8}= \left \{ 2,3,5 \right \}, G_{10}= \left \{ 2,3,5 \right \}$
However, $ \left \{ 2,3,5,7 \right \}$ also works for $ 2n = 10$ but it is not minimal.
The minimal set is unique since among many possibilities(if any) we are choosing one that has (1) the smallest number of elements(this is the primary criterion) and (2)say, if we have $\left \{ 2,3,5,11,13\right\}$ and $\left \{ 2,3,5,7,13\right\}$, we would choose the second. That means when ordering the elements in increasing order, the one chosen will have elements that are smaller under pairwise comparison with the other choices.
Question 1:(not yet answered) Obviously, $\left | G_{2p} \right |\leq \pi(p) $, for primes p. Proving that $\left | G_{2n} \right |= 0$ for at least one $n$ is harder than the old Goldbach. Still harder would be to show that $\left | G_{2n}\left | \leq \right |G_{2n+2} \right |$ Also, some experiment shows that $\left | G_{2n} \right |= \pi(n) + 1$ when $n \geq6$ till the numbers I checked (excluding $n = 7$ and $13$). How good is this formula at least for some $n$?
Question 2:(has been answered below) Does $G_{2n}$ always contain the primes in linear order, without jumping any primes? (I asked this because none of the $G_{2n}$'s I computed have missing primes.)
Thanks.
N.B.: I have been making many revisions. So, some comments may not make sense. In that case, you may look up the edit history.
$\left \{ 2,3,5,11,13\right\}$
and$\left \{ 2,3,5,7,13\right\}$
, we would choose the second. That means when ordering the elements in increasing order, the one chosen will have elements that are smaller under pairwise comparison with the other choices. I will add this above. $\endgroup$