It’s a big day for number theory! Harald Helfgott has also proved the odd Goldbach conjecture, that every odd number greater than five is the sum of three primes. http://arxiv.org/abs/1305.2897

If that holds, that's huge! It means, the distance between consecutive prime numbers does not grow unbounded (but is bounded by at least 70e6)!

I think you're making a stronger statement than the article - it's certainly possible for there to be an infinitely many pairs of primes less than 70e6 apart, without there being another prime within 70e6 either side of a given prime number

It is especially obvious given that the goal is to lower the boundary from 70e6 to 3.

There are i such that |prime[i] - prime[i+1]| >= 3 but it doesn't prevent the possibility that there are infinite number of pairs such that |p - q| < 3 where p, q are primes.

You are mistaken - gaps between prime grow unboundedly large - it's a common first year exercise in number theory to show that.

Well, he's mistaken if what he is saying is what you think he is saying, but I think his statement is ambiguous.

I think he might mean when he says the gap does not grow unbounded is that that you never reach a point where the gaps are ALL arbitrarily large. As we run through the primes, we'll always continue to find gaps that are 70 million or less.

You are both correct! ;-)

I wonder how they got to 70e6, I mean, such a round number.

My guess is that they didn't actually get to that exact number, but a slightly lower one, more difficult to remember. I don't know though.