Prime Mean of Primes
1) Are there infinitely many primes $p_1,p_2$ such that $\frac{p_1+p_2}{2}$ is also prime?
2) What can we say about the more general problem :
Are there infinitely many primes $p_1,p_2,\cdots, p_n$ such that $\frac{p_1+p_2+\cdots+p_n}{n}$ is also a prime for $n\geq 1$?
$\mathbf{Remark}$: Note that for $n=1$, the problem 2 is solved by Euclid's theorem.
Solution 1:
Both cases follow from the fact that there exist progressions in primes of length $k,$ for each $k\ge 2.$ This is Green-Tao Theorem.
Remark: case $n=2$ was proved by K.Roth.