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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$\begingroup$
number-theory
prime-numbers
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0@AnantSaxena why? – 2017-06-21
2 Answers
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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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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
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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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2This proof is from Sierpinski's Elementary Theory of Numbers (the second edition of which was edited by Schinzel) – 2011-04-19