Prove two subsequent primes cannot be written as a product of two primes Suppose we have two subsequent primes, say $p$ and $p'$. Prove their sum cannot be written as a product of two primes, say $p_1$ and $p_2$.
I wanted to proof by contradiction. I started by thinking about parity of the sum. Suppose $p=2$, then $p'=3$. But this sum cannot be written as a product of two primes $p_1$ and $p_2$. So we know that $p>2$; this implies that both $p$ and $p'$ are odd, so $p+p'$ is even. This means that either $p_1$ or $p_2$ must equal $2$. This results in: 
\begin{equation} p+p'=2p_1.\end{equation}
Now, how can I finish the proof?

For $p=2$ then it's immediate. For odd primes $p,p'$ you would have $p_1p_2=even$ namely $p_1=2$ and $p_2=\frac{p+p'}{2}$. But then $p_2$ is a prime that lies between two consecutive primes $p,p'$. Contradiction.
