# Can every even integer be expressed as the difference of two primes?

Can every even integer be expressed as the difference of two primes? If so, is there any elementary proof?

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You could strengthen this to "Every even integer can be expressed as the difference of a pair of consecutive primes" or "Every even integer can be expressed as the difference of an infinite number of pairs of primes", or even to "Every even integer can be expressed as the difference of an infinite number of pairs of consecutive primes". They are all open questions. – Henry Mar 21 '11 at 14:48
– Charles Mar 21 '11 at 18:33
@Charles: seen it. – The Chaz 2.0 Apr 19 '11 at 4:58
Isn't this an implication of Goldbach's conjecture being true? – Anant Saxena Jan 31 at 8:11

## 2 Answers

This is listed as an open question at the Prime Pages: http://primes.utm.edu/notes/conjectures/

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The page you're linking to is a bit too old -- the Odd Golbach Conjecture is already proved (in May $2013$) and the page doesn't say so, but it's still a fair enough source. – user26486 Mar 30 '15 at 15:18

This follows from Schinzel's conjecture H. Consider the polynomials $x$ and $x+2k$. Their product equals $2k+1$ at 1 and $4(k+1)$ at 2, which clearly do not have any common divisors. So if Schinzel's conjecture holds, there are infinitely many numbers $n$ such that the polynomials are both prime at $n$, and so subtracting gives the result.

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This proof is from Sierpinski's Elementary Theory of Numbers (the second edition of which was edited by Schinzel) – Harry Stern Apr 19 '11 at 4:54