<s>[INST] Every sufficiently large positive integer is the average of $n$ distinct primes for certain $n \geq 2$? I want to generalize a stronger Goldbach's conjecture a little bit because that might help solve it.
I was thinking: 

For all $n \geq 2$, every sufficiently large positive integer $x \geq b_n$ is the average of $n$ distinct primes? 

Clearly this implies Goldbach's conjecture.  
So, was wondering if it has already been proven for any $n\gt 2$?
I think Terry Tao proved something different to the above, since it explicitly says at most 5 primes, which is algebraically not the same thing.

Remark. In particular the bound $b_n \geq \dfrac{p_1 + \dots + p_n}{n}$, where $p_i$ is the $i$th prime number: $2, 3, 5, \dots $
Proof of remark.  If $\dfrac{q_1 + \dots +q_n}{n} = x \lt \dfrac{\sum_{i=1}^n p_i}{n}$, then cancel out the $n$ and the result follows.  The $q_i$ are some prime numbers averaging to $x$.  
For example for $n = 2$, the lower bound of $x$, $b_n$ seems to be $b_n = 4$, where $\dfrac{3 + 5}{2} = 4$ shows a solution for $x = 4$.  

Here are some small cases:
$$
n = 2: \\
\dfrac{5 + 3}{2} = 4 \\
\dfrac{7 + 3}{2} = 5 \\
\dfrac{5 + 7}{2} = 6 \\
\dfrac{11 + 3}{2} = 7 \\
\dfrac{11 + 5}{2} = 8 \\
\dfrac{11 + 7}{2} = 9 \\
\ \\
n = 3: \\
\dfrac{3 + 5 + 7}{3} = 5 \\
\dfrac{2 + 5 + 11}{3} = 6 \\
\ \\
n = 4: \\
\dfrac{3 + 5 + 7 + 13}{4} = 7\\
\vdots 
$$
We will need some SymPy Python 3.x code.  I might code it up (testing up to $x = N, n = M$), but not for a few days.  So there is some opportunity to beat me to it.
 [/INST] For $n$ large enough Vinogradov's theorem gives $\sim C \frac{n^2}{\ln^3 n}$ solutions to $2n+1 = p_1+p_2+p_3$ which gives $\sim B \frac{n^3}{\ln^4 n}$ solutions to $2n = p_1+p_2+p_3+p_4$.
And the same method as Vinogradov gives $\sim A \frac{n^2}{\ln^3 n}$ solutions to $2n = 2q_1+q_2+q_3$.  

Thus we have $\sim B \frac{n^3}{\ln^4 n}$ solutions for $2n = p_1+p_2+p_3+p_4$ as the sum of distinct primes.

Vinogradov's theorem follows from a strong form of the PNT in arithmetic progressions. I am quite sure for large enough $k$ there is a three line proof to your $2n = \sum_{j=1}^k p_j$ distinct primes problem just from $\pi(x) \sim \frac{x}{\ln x}$.
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