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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?

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    What is your motivation and background?2011-11-13
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    Did you really intend to write $n < p \le p$ ?2011-11-13
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    @JoelCohen: Oops, I meant to write n < p <= 2n.2011-11-13
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    @BrandonCarter: I need this to prove the non-regularity of a formal language using the Myhill-Nerode theorem.2011-11-13

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