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Given there are infinitely many primes, why does this not then immediately imply there are infinite number of primes of gap 2? Does the infinite nature not imply that there are indeed infinitely many primes for every gap > 1?

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    $\begingroup$ Why should it? There are infinitely many powers of $2$, but among them only $2$ and $4$ have a gap of $2$. $\endgroup$ – Brian M. Scott Jun 1 at 20:19
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    $\begingroup$ Given there are infinitely many primes, why does this not then immediately imply there are infinite number of primes of gap 1? $\endgroup$ – Jair Taylor Jun 1 at 20:21
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    $\begingroup$ This reminds me of the occasional battles I have with my undergraduate level students. If I can't pinpoint immediately where their argument goes wrong, they must be right.. $\endgroup$ – Alvin Lepik Jun 1 at 20:22
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    $\begingroup$ @Alvin: fortunately it is a simple matter to pinpoint the flaw in this particular fallacy. $\endgroup$ – TonyK Jun 1 at 20:25
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    $\begingroup$ Thanks :) Should have thought about this one a little longer.. $\endgroup$ – DMSTA Jun 1 at 20:41
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To repeat what has been said in the comments, the infinite nature of the primes does not trivially resolve the conjecture becaus the number of elements in a set of integers says nothing about their gaps. For instance, there are infinitely many powers of $2$, but only $2$ and $4$ have a gap of $2$. For all we know, it may be the case that by some point far down the sequence of primes, the only prime gaps that occur have length $4,6,$ etc. and there are only finitely many gaps of length $2$. A proof to resolve this question in either direction is the main difficulty of the twin prime conjecture.

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  • $\begingroup$ You mean a gap of 2 (you wrote but only 2 and 4 have a gap of 4). $\endgroup$ – user25406 Jun 3 at 14:47
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    $\begingroup$ @user25406 Thanks for catching that mistake! I've fixed it now. $\endgroup$ – YiFan Jun 3 at 20:37

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