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I have observed something, that either:

  1. Given any natural number $n$, there exists some natural number $k$, such that above $k$, the difference between any two consecutive primes is $> n$ ($\implies$ the prime gap increases steadily, having limit infinity), or

  2. For some $n$, there are infinitely many consecutive prime pairs of the form $(p,p+n)$.

Both (1) and (2) look remarkable results to me and one of them must be true! Is there any information as to which one is true? (can both be true?) And then, why is the classical twin prime conjecture important and why can't the conjecture be put in this way:

There exists infinitely many consecutive prime pairs of the form $(p,p+n)$, for some natural number $n$?

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Since the twin prime conjecture is open, it is unknown which of (1) and (2) is true. However, "there exists infinitely many consecutive prime pairs of the form (p,p+n), for some natural number n" is clearly false - $n$ would have to be even - and even then it is much stronger than the twin prime conjecture so it is not merely a restatement. (Indeed, the TPC would merely be the $n=2$ case.) – anon Jun 13 '12 at 6:23
@anon I don't understand your last remark. "For some natural number $n$" allows that number to be $2$. So if the twin prime conjecture is true, then so is the given statement. If the twin prime conjecture is false, it is conceivable that the statement is still true for some other value of $n$ (though unlikely, unless one allows $n=0$). So if anything, the statement is weaker than the twin prime conjecture. – Marc van Leeuwen Jun 13 '12 at 6:41
@MarcvanLeeuwen You are correct, I had a serious blindspot in reading that sentence! – anon Jun 13 '12 at 6:45
up vote 6 down vote accepted

(2) is universally believed to be true, but we are far from having a proof. It is a recent breakthrough of Goldston, Pintz and Yıldırım (arXiv) that (2) is true if we know more about a certain constant $\theta$ called the level of distribution for primes in arithmetic progressions.

The Bombieri-Vinogradov theorem is equivalent to $\theta \ge 1/2$. Goldston-Pintz-Yıldırım shows that if one can obtain any slight improvement $\theta > 1/2$, then (2) provably holds for some $n$. (This does not contradict @GerryMyerson's answer, since no one has any idea how to get any improvement. Even GRH probably isn't enough to imply this.)

Elliot and Halberstam have conjectured that $\theta = 1$. If this is true, or even if just $\theta > 0.971$, then (2) holds for some $2 \le n \le 16$. So far no one has succeeded in tightening this bound to get twin primes conditionally under Elliot-Halberstam.

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The recent (2013) results by Yitang Zhang and others, esp. James Maynard (Small gaps between primes), establish unconditionally that (2) is true. – hardmath Oct 24 '15 at 5:28

It has in fact been conjectured, and is widely believed to be true, that for any even $n$ there are infinitely many primes $p$ such that the next prime is $p+n$. No one has a clue how to prove anything along these lines.

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