If the sum of positive integers $a$ and $b$ is a prime, their gcd is $1$. Proof?
Let $c$ be the gcd. Then $c$ divides $a$ and $b$, hence it divides $a+b$, a prime number.
Let's show the contrapositive, because why not?
So we want to show that if $a,b>0$ and $\gcd(a,b) \neq 1$, then their sum is not prime.
Suppose that $\gcd(a,b) = d > 1$. Then $a = a'd$ and $b = b'd$ for some $a',b'$ natural numbers. But then $a + b = da' + db' = d(a' + b')$, and as each of $d,a',b' \geq 1$, we have that $a + b \geq 2d$, but is divisible by $d$. Thus it is not prime. $\diamondsuit$
Thus if the sum of positive integers is prime, then their gcd is $1$.
Let $d$ be their gcd. Then $d$ divides their sum $p$, so $d$ can be only 1 or $p$.
If $d = p$, then $p$ divides both $a$ and $b$. Since both of these are positive, they are each at least $p$, so their sum is at least $2p$.
I realize that this is a restatement of mixedmath's answer.
This can easily be generalized to show that this holds for the sum of $k$ positive integers for $k \ge 2$.