Sum of prime and non prime number is what?

Question

If $p$ is prime and $q$ is not prime and they are positive, what can you say about the $p+q$, can you get every natural greater then 2?

Answer 1

Let $p=3$. Then $q$ can be any even number greater than $2$, so you get all the odd numbers greater than $5$.

If $n > 6$ is even then (as @lulu comments) one of $q = n-3, n-5, n-7$ is divisible by $3$, so not prime. Then $n = q + $ a prime.

You can check the small cases by hand.

Answer 2

Let $n$ be a natural number greater then $5$. If $n$ is odd, then $n - 3$ is even and since $n >5$ $n-3$ is not a prime. If $n$ is even, $n - 2$ is even and thus not a prime since $n > 4$.

The exceptional cases $3, 4, 5$ can easily be solved by hand: $$3 = 2 + 1$$ $$4 = 3 + 1$$ and $5$ does not have such a decomposition.