I've stumbled across this playing around and summing primes at random during a boring lecture. Is this a known conjecture? Can it be proven?
My conjecture: There exists at least one non trivial solution such that $2p_n = p_a + p_b$ (the trivial being obviously $a=b=n$) for $n > 2$.
Tested by starting at the trivial $2p_n = p_n + p_n$ then incremented the right as I decremented the left until both were prime again or I've ran below $2$ with the left number. The second condition never occurred though, and I've tested for the first $1000$ primes by writing a simple program.
It fascinated me for the fact that this would mean that $p_b = 2p_n - p_a$ where $b > n > a$ so by knowing primes up to the $n$th you would have enough information to evaluate the $(n + 1)$-th?
This is equivalent to saying that there are infinitely many triples of primes $(p_a,p_n,p_b)$ which form an arithmetic progression. While this is a consequence of the Green-Tao theorem, it was shown in 1939 itself by van der Corput. In the paper "Linear equations in primes" (Mathematika, 39 (1992), pages 367-378), Balog showed that there exists for every $n$, $n$ primes such that the average of every two of them is prime.