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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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    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.2011-03-21
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    See https://oeis.org/A0204832011-03-21
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    @Charles: seen it.2011-04-19
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    Isn't this an implication of Goldbach's conjecture being true?2016-01-31
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    @AnantSaxena why?2017-06-21

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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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    The page you're linking to is a bit too old -- the Odd Golbach Conjecture is already proved ([in May $2013$](https://plus.google.com/+TerenceTao27/posts/8qpSYNZFbzC)) and the page doesn't say so, but it's still a fair enough source.2015-03-30
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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)2011-04-19
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Further, every odd integer $>3$ can be expressed as the difference of two primes also (subject to proving Goldbach's Conjecture).

In fact this is a restatement of GC. For example: For $3+7$ to equal $10$, it follows that $\frac{3+7}{2}$ must equal $5.$

In other words, every even $n$ is partitioned by the primes for which $\frac{n}{2}$, even or odd, is the average.