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

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    $\begingroup$ 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. $\endgroup$
    – Henry
    Mar 21, 2011 at 14:48
  • $\begingroup$ See oeis.org/A020483 $\endgroup$
    – Charles
    Mar 21, 2011 at 18:33
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    $\begingroup$ @Charles: seen it. $\endgroup$ Apr 19, 2011 at 4:58
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    $\begingroup$ Isn't this an implication of Goldbach's conjecture being true? $\endgroup$
    – drewdles
    Jan 31, 2016 at 8:11
  • $\begingroup$ @AnantSaxena why? $\endgroup$
    – YCor
    Jun 21, 2017 at 7:55

2 Answers 2

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

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  • $\begingroup$ 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. $\endgroup$
    – user26486
    Mar 30, 2015 at 15:18
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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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    $\begingroup$ This proof is from Sierpinski's Elementary Theory of Numbers (the second edition of which was edited by Schinzel) $\endgroup$ Apr 19, 2011 at 4:54

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