Imagine two distinct prime numbers $p$ and $q$. Intuitively, I'd say that there is always a natural number n so that $p+n$ is a prime number, but $q+n$ isn't.
I was given two hints:
- for each natural number $n$ there is a prime $p$ so that $n < p \leq 2n$
- consider the primorials
But I still can't come up with a mathematical proof. My main problem is that I don't understand how I can show that the sum of a prime and another number is a prime. Any help?