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Edit: Admittedly only skimmed the article, let alone the paper. Obviously there are an infinite number of primes and we've proved that ages ago. Do not read this comment, read the replies if you want a more accurate answer. Leaving it here for posterity.

Yup, at least according to the article:

"His paper shows that there is some number N smaller than 70 million such that there are infinitely many pairs of primes that differ by N. [...] [N]o matter how sparse the primes become — you will keep finding prime pairs that differ by less than 70 million."

So he proved that it there is some number N, not necessarily 70mil but below 70mil, is the "gap" between primes.


This is an amazing development. On the other hand, we are also getting close to nailing the odd Goldbach Conjecture. I think it was just last year that Tao proved, without the Riemann Hypothesis, that every odd number N > 1 can be expressed at by the sum at most 5 primes. Wonderful to witness such great leaps in maths during one's own lifetime.

No, this isn't right.

He showed that there are infinitely many twin primes differing by 70 million from each. However, it could be that the next such twin prime, has its lower prime number more than 70million numbers further along.

What I mean is, there is allowed to be a gap larger than 70million with no primes in it, then all of a sudden, twin primes. But those twin primes with a gap less than 70million between them are guaranteed.

Basically, if you separate all adjacent pairs of primes, with no primes in between, you can separate this into two groups of those adjacent pairs with a gap less or equal to 70mil, or greater than 70mil. The first group has infinitely many members according to the proof. And the second group is probably not empty.

And the second group is probably not empty.

We know for sure it's non-empty by simple factorial arguments ITT or by appeal to the prime number theorem.

No, that’s not what the paper shows. The paper claims that there exists arbitrarily large primes prime pairs with a gap of less than 70 million, not that all primes are part of a prime pair. Your misinterpretation is equivalent to intepreting the twin prime conjecture to mean that all odd numbers are prime.

Harald Helfgott would have a bone to pick with "getting close". :)


Not bad for the first half of 2013!

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