Why is the Twin Primes Conjecture still relevant to mathematicians today?

[mathnerd15]

Proof towards Twin Primes Conjecture!

http://www.wired.com/wiredscience/2013/05/twin-primes/

 

[jim mcnamara]

Not quite. As I read the article it states that instead of finding that there exist infinitely many twin primes,
Zhang was able to construct a proof that there are infinitely many primes that are "70 million apart". There is some more work to do.

The article also quotes some editors who were very excited about the paper. Zhang spoke at Harvard May 13.

Thanks for the link!

 

[mathnerd15]

so there is no total vacuum in the distribution of prime numbers, the density of twin primes never decreases below a certain value, twin primes are separated by less than 70 million at the most?
I'm excited to see the proof!

 

[mathnerd15]

unfortunately Annals charges a subscription... anyway I'm sure many brilliant minds have already spent a great deal of time and effort and failed to solve the RH with many modern techniques

 

[A. Bahat]

so there is no total vacuum in the distribution of prime numbers, the density of twin primes never decreases below a certain value, twin primes are separated by less than 70 million at the most?
I'm excited to see the proof!

This isn't quite what he proved. Rather than a bound on the distance between twin prime pairs, the new result is that there are infinitely many pairs of primes differing by at most 70 million. If we can get 70 million down to 2, then that's a different story.

 

[Useful nucleus]

Very impressive! Thank you for sharing the news!

 

[mathnerd15]

what would reducing it to 2 mean? that you could make some progress on the Riemann Hypothesis? It's a fascinating problem...

 

[verty]

what would reducing it to 2 mean?

Proofs about whether things happen infinitely many times or only finitely many times are difficult to come by, and many simple conjectures about infinite behaviour are still open. For example, the Collatz conjecture has not been proved or refuted.

So here we have a proof about what happens infinitely many times: it happens infinitely many times that neighbouring primes are within 70 million of each other. This means, for any N, there are neighbouring primes that close > N.

And it would seem odd for there to be infinitely many primes within 70 million of each other but only finitely many primes within 2 of each other. So there is an expectation now that it will eventually be shown that there are infinitely many twin primes. This is a big change with regards to that conjecture, good evidence (but not a proof) that that conjecture is true.

The point is, the status of the twin prime conjecture has changed from being unknown to being likely true. From the Wired article:

The result is “astounding,” said Daniel Goldston, a number theorist at San Jose State University. “It’s one of those problems you weren’t sure people would ever be able to solve.”
...
“This work is a game changer, and sometimes after a new proof, what had previously appeared to be much harder turns out to be just a tiny extension,” he [Granville] said. “For now, we need to study the paper and see what’s what.”

Here are some answers to the question, what affect would proving/disproving the twin prime conjecture have on the Riemann Hypothesis?

 

[mathnerd15]

did Riemann make some kind of educated guess? I think he knew the Lagrange Des Nombres by heart
it's really interesting, but I'm curious why it's important today to mathematicians being a 19th century conjecture?