# 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?

• 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. Mar 21, 2011 at 14:48
• Mar 21, 2011 at 18:33
• @Charles: seen it. Apr 19, 2011 at 4:58
• Isn't this an implication of Goldbach's conjecture being true? Jan 31, 2016 at 8:11
• @AnantSaxena why?
– YCor
Jun 21, 2017 at 7:55

• 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. Mar 30, 2015 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.