Can every even integer be expressed as the difference of two primes? If so, is there any elementary proof?
Can every even integer be expressed as the difference of two primes?
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5You 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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0See https://oeis.org/A020483 – 2011-03-21
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2@Charles: seen it. – 2011-04-19
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2Isn't this an implication of Goldbach's conjecture being true? – 2016-01-31
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0@AnantSaxena why? – 2017-06-21
3 Answers
This is listed as an open question at the Prime Pages: http://primes.utm.edu/notes/conjectures/
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0The 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
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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2This proof is from Sierpinski's Elementary Theory of Numbers (the second edition of which was edited by Schinzel) – 2011-04-19
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.