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Every even integer can 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
    
See oeis.org/A020483 –  Charles Mar 21 '11 at 18:33
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@Charles: seen it. –  The Chaz 2.0 Apr 19 '11 at 4:58
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2 Answers

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

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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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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
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