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I understand where I am wrong in my previous post. Also, I am very thankful to all members, who answered and showed my errors in post. Now, I would like to know the proof for the following.

"The difference of any two prime numbers $p$ and $q$ is expressible as $2k$; and the probability of finding primes $p$ at $n + k$ and $q$ at $n - k$ decreases as $n$ increases."

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    @Zev Chonoles! Thank you for proper editing of my post.2011-12-21
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    I understood the intuition behind your question, you are looking for some generalization of the twin-prime conjecture, let me try to answer it2011-12-25
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    I just now re-edited and re-tagged your question asking for reference requests too, so that proficient users can give you some good references @mahi2011-12-25

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