Is every composite number the average of two primes? I'm interested in this question because it relates to a bad joke about people in their prime.
It seems to work for the first 20 numbers:
4 is the average of 5 and 3.
6 is the average of 5 and 7.
8 is the average of 5 and 11.
$\vdots$
16 is the average of 13 and 19.
$\vdots$
It would not be hard to write a program that checks more cases, but I suspect the hypothesis is true. If that is the case, then a more mathematical approach is needed.
 A: This is almost the same as the Goldbach conjecture, that every even number from four is the sum of two primes.
It has been checked into the quintillions (no, really!) but not proven.  In 2013, a similar theorem was proven by Harald Helfgott, that every odd number from seven up is the sum of three primes.
A: A generalized equation would be in the form of:
$$x =\frac{P_1+P_2}{2}$$
Where $x \in Z^+$, and $P_1,P_2 \in Z^p$ (Here, I define $Z^p$ to be a prime number).
It can be rewritten as $$2x = P_1+P_2$$
Let $2x=a$ Then the equation is: $$a = P_1+P_2$$

I.e. Your question's generalised form ($x =\frac{P_1+P_2}{2}$)  is akin to the Goldbach Conjecture: http://en.wikipedia.org/wiki/Goldbach%27s_conjecture
A: So you are asking whether for every composite number $n$ there exist primes $p,q$ such that
$$n=\frac{p+q}{2}$$
That is, $2n=p+q$, so you are asking whether $2n$ can be written as the sum of two primes.
The question whether every even integer greater than $2$ is a sum of two primes is a famous open problem known as the Goldbach conjecture.
A: I think that it is all about prime gap.

If all natural numbers larger than 4 can be proved that it is the average of primes, the Goldbach's Conjecture will be solved, immediately
