Article: First proof that infinitely many prime numbers come in pairs

[Paulie]

It’s a result only a mathematician could love. Researchers hoping to get ‘2’ as the answer for a long-sought proof involving pairs of prime numbers are celebrating the fact that a mathematician has wrestled the value down from infinity to 70 million.

http://www.nature.com/news/first-pro...-pairs-1.12989

And a related blog: http://golem.ph.utexas.edu/category/...en_primes.html

[firejuggler]

well, http://www.slate.com/articles/health...e_numbers.html

Can't say I have enough knowledge to refute or approve the proof, but it might be of some interest.

What about the gaps between consecutive primes? You might think that, because prime numbers get 
rarer and rarer as numbers get bigger, that they also get farther and farther apart. On average, 
that’s indeed the case. But what Yitang Zhang just proved is that there are infinitely many pairs of 
primes that differ by at most 70,000,000. In other words, that the gap between one prime and 
the next is bounded by 70,000,000 infinitely often—thus, the “bounded gaps” conjecture.

And, a first look at the paper
http://blogs.ethz.ch/kowalski/2013/0...etween-primes/

[only_human]

It's been just over a month since Zhang's paper "Bounded gaps between primes." Since then, the Polymath8 page shows that the bounded gap may have reduced from 70,000,000 to less than 61,000.
http://michaelnielsen.org/polymath1/...between_primes

[MattcAnderson]

It looks like the bound has been reduced to 12,042. So there are an infinite number of prime pairs a distance of 12,042 or less apart. Exciting!

[firejuggler]

Impressive, indeed, and in only 5 week. Now it might become difficult to improve he bound.

[TheMawn]

I thought 70 million was pretty exciting but that seems to be old news.

I should note for those who may be misinterpreting this proof: It does NOT say that there is a prime after 70 million or twelve thousand or whatever numbers. What it is saying is that there are infinitely many primes with at most X in between them. It's actually a pretty weak statement.

The proof does NOT guarantee every prime has a close neighbour. If there is only a single prime number in between 10^100,000,000 and 10^{1,000,000,000} (this is a gap of basically 10^{1,000,000,000} which is a LOT bigger than even 10⁷) the proof still holds. It is just saying that there is ALWAYS a next set of sibling primes.

[LaurV]

Channeling my inner RDS, what you say is a bit o gibberish.. The "statement" is quite strong, and it is a step in proving twin primes conjecture. The other two fragments about what the result "does not say" are "more than a bit" of gibberish, first because we already know that the gap between the primes can be mad made arbitrarily large, and the second because we also already know that there is always a prime between n and 2n for any n. (of course, we understand that you used those powers of ten in a figurative sense, but still..... there is a math forum here...)

[TheMawn]

Alright. I'll give you that one. It's fairly strong in what it has set out to do but there is quite little use outside the twin primes conjecture.

All I meant to say was that it doesn't really affect the actual search for primes.

I overlooked the fact that there is a prime between n and 2n. The fact still remains that, as you said, the gap between primes is absolutely unbounded. I just wanted to point out, before anyone made the mistake, that the 12,042 or whatever thing does not in any way state that there MUST be a prime within 12,042 of another prime.

[retina]

Originally Posted by TheMawn

I just wanted to point out, before anyone made the mistake, that the 12,042 or whatever thing does not in any way state that there MUST be a prime within 12,042 of another prime.

I don't see where anyone suggested such a thing. I think we here all knew what the announcement meant.

[LaurV]

Originally Posted by TheMawn

I just wanted to point out, before anyone made the mistake, that the 12,042 or whatever thing does not in any way state that there MUST be a prime within 12,042 of another prime.

That is indeed very true. To our disappointment, otherwise it should be very easy for us to find primes, and get the EFF's money... , we would only have to test about 12k consecutive numbers of 100M digits, which would be most of them eliminated by as simple Erathostenes sieve, that's life...

[ATH]

Anyone with the knowledge to understand these papers think there will ever be proven a finite bound to consecutive primes?

It does not seem possible if the number of primes below n follows roughly n/ln n which means the average gap increases, but these proofs with infinite pairs of primes below 70,000,000 or even lower also seem counter intuitive.