I have observed something, that either:
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
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 $n$?