On the gaps between consecutive primes

Question

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 $n$?

Answer 1

(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.

Answer 2

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.