What this proof shows is you never get to a point where the gap is always bigger than 70 million, forevermore. This is a big step towards the "twin primes conjecture" which claims there are an infinite number of pairs of primes only 2 apart.
since the gap never gets always-bigger than 70 million, and there are an infinite number of gaps between primes, by something resembling the pigeonhole principle it can be shown that there are an infinite number of prime pairs for at least one specific gap size between 2 and 70 million. That is, there might not necessarily be an infinite number of pairs of primes exactly 2 apart, but there are an infinite number of primes exactly N apart for some N<70,000,000.
I don't know whether it has a name.
Assume that 70 million disjoint sets all have finite cardinality. Then the cardinality of the union is the sum of the cardinalities, and is finite by construction.
Contradiction; QED.
Then, sometimes 100's of years later, somebody does prove (or disproves) the theorem and get's some press, and for some very important conjectures, prizes. If the conjecture is disproven, then all the math based on it falls down.
Conjectures interesting because they are puzzles left by people that they can't answer but they want them to be true, so it is up to future generations with their advanced minds, tools and insights to prove them, leading to progress in the field with the advancements of techniques use to prove these conjectures and make them actual theorems.
Part of math school is to learn making and breaking guesses about things mathematical.
As noted elsewhere ITT, a conjecture becomes one when there's a community-acknowledged aura of moral rightness about it. So at the very least, a bunch of people have tried to disprove it and failed.
Of course the problem is you can't show this by exhaustion. And you also can't currently prove it, hence why its a conjecture. But it's interesting because it means you might be missing some integral property that would prove the conjecture.
1. The conjecture has been found to hold for a large number of trial cases, and
2. The conjecture has an intuitive aura of rightness.
A good example is the 3n + 1 conjecture: http://en.wikipedia.org/wiki/Collatz_conjecture
Anyway, obviously point two is extremely subjective, but I think it's the crux of why long-standing conjectures are interesting. Why should something feel correct without necessarily being so? If it does turn out to be correct, what about its nature put people on its scent trail? Mathematics so often seems to be a kind of hermetic universe unto itself, so part of the beauty of conjectures is that they're kind of a contact point between that universe and human congnition.
Often, a conjecture will be invented and then proved or disproved a short time later (I remember my wife disproving one her thesis adviser had proposed in class earlier that day), but occasionally one will remain unproven either way for a long time. With many of the "big" conjectures, this gives people confidence they're true -- they've been tested for millions, billions, or even trillions of cases over the course of decades or centuries, but nobody has found a counterexample. But they also haven't come up with an airtight proof; maybe they'd fail on the ten trillionth try, or maybe they'd hold true forever. That uncertainty is part of what makes them interesting.