# difference between 2 prime numbers

We have to prove that if the difference between two prime numbers greater than two is another prime,the prime is $2$. It can be proved in the following way.

1)$Odd -odd =even$.

Therefore the difference will always even.

2)The only even prime number is $2$.Therefore the difference will be $2$ if the difference between primes is another prime.

I am looking for more proofs to this theorem.Any help will be appreciated.

• "it is 2"...who is two: the difference or the other prime?! Jun 11, 2013 at 7:10
• @DonAntonio Both,of course. Jun 11, 2013 at 7:12
• It can't be both, since $4$ is not prime. Still, your proof is adequate, though inelegant. Since you've specified that both primes are greater than $2$, then they are both odd, and so case 2 is non-existent. Jun 11, 2013 at 7:21

The proof you provided is fine way of proving the proposition. Here is an alternate proof, set up as a contradiction,

Suppose that the difference between two odd primes $a=2n+1$ and $b=2m+1$ is an odd prime, $c$.

$$a-b=c \implies (2n+1)-(2m+1)=c \implies 2(n+m)=c$$

Therefore, $2|(a-b)$, a contradiction.

It is therefore impossible for the difference of two odd primes to be an odd. This means that the difference of two odd primes must be even. The only even prime is 2.

So, if the difference of two odd primes is a prime, then it must be two.

• Have you read the greater than 2 part,which I forgot to mention first? Jun 11, 2013 at 7:18
• The question says "the difference of two prime numbers greater than $2$". So you can't have $7-2=5$ as an example.
– user67803
Jun 11, 2013 at 7:19
• @You cannot blame him entirely for that.He probably had not refreshed the page when I edited it. Jun 11, 2013 at 7:23
• @AyushKhaitan His new edit adresses the concern. I will update my answer to reflect his edit. Jun 11, 2013 at 7:27
• Thanks for the formal statement of the proof Jun 11, 2013 at 7:28

Well, since the first prime must be odd, then:

(1) If the "other prime" is odd their difference is even and the only even prime is two

(2) If the "other prime" is even then it is two and we have twin primes, like $\,13-2=11\,$

• Have any of you read the edited greater than two part? Jun 11, 2013 at 7:17
• Well, then it is only part (1) and the claim is trivial... Jun 11, 2013 at 7:18
• Well,I already posted that. Jun 11, 2013 at 7:21
• Great...then what are you asking for? More proofs of a completely trivial fact? Jun 11, 2013 at 7:24
• Pretty much.My homework says to find two different proofs. Jun 11, 2013 at 7:25

Here is a proof by contradiction. Let $p, q, r$ be primes greater than $2$ with $p+q=r$

$p$ and $q$ are odd, so $p=2a+1, q=2b+1$ and $$r=p+q=2a+1+2b+1=2(a+b+1)$$

But this is a factorisation of $r$ and contradicts the fact that $r$ was chosen to be prime.