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I read that the following can be proved using Bertrand's postulate (there's always a prime between $n$ and $2n$): $\forall N\in\mathbb N$, there exists an even integer $k>0$ for which there are at least $N$ prime pairs $p$, $p+k$.

But I have no idea how to prove it. Any help would be much appreciated.

Many thanks.

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    I believe I answered the wrong question. I thought you were asking to show there was a $k$ such that there were $N$ primes between $p$ and $p+k$. I have deleted my answer, and will think about the originally intended question.2011-12-15
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    By the way, where did you read this claim?2011-12-15
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    I read it here, on the very last page: http://www.math.dartmouth.edu/~thompson/Bertrand%27s%20Postulate.pdf2011-12-15
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    @user19012, nice slides!2011-12-15

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