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n2doc

(47,953 posts)
Tue May 21, 2013, 09:29 AM May 2013

Unheralded Mathematician Bridges the Prime Gap

by: Erica Klarreich

On April 17, a paper arrived in the inbox of Annals of Mathematics, one of the discipline’s preeminent journals. Written by a mathematician virtually unknown to the experts in his field — a 50-something lecturer at the University of New Hampshire named Yitang Zhang — the paper claimed to have taken a huge step forward in understanding one of mathematics’ oldest problems, the twin primes conjecture.

Editors of prominent mathematics journals are used to fielding grandiose claims from obscure authors, but this paper was different. Written with crystalline clarity and a total command of the topic’s current state of the art, it was evidently a serious piece of work, and the Annals editors decided to put it on the fast track.

Just three weeks later — a blink of an eye compared to the usual pace of mathematics journals — Zhang received the referee report on his paper.

“The main results are of the first rank,” one of the referees wrote. The author had proved “a landmark theorem in the distribution of prime numbers.”

Rumors swept through the mathematics community that a great advance had been made by a researcher no one seemed to know — someone whose talents had been so overlooked after he earned his doctorate in 1992 that he had found it difficult to get an academic job, working for several years as an accountant and even in a Subway sandwich shop.

more

http://simonsfoundation.org/features/science-news/unheralded-mathematician-bridges-the-prime-gap/

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Unheralded Mathematician Bridges the Prime Gap (Original Post) n2doc May 2013 OP
Excellent article! CrispyQ May 2013 #1
ah yes, the old relaxed prime seiving trick... phantom power May 2013 #2
That is awesome. TxDemChem May 2013 #3
A summary of the proof...with a hint about the 70 million ... DreamGypsy May 2013 #4
+ struggle4progress May 2013 #5

phantom power

(25,966 posts)
2. ah yes, the old relaxed prime seiving trick...
Tue May 21, 2013, 01:52 PM
May 2013

I love it when people get a result by strategically doing a "worse" job on one of the crucial steps. So many good algorithms in computer science do that.



DreamGypsy

(2,252 posts)
4. A summary of the proof...with a hint about the 70 million ...
Wed May 22, 2013, 04:42 PM
May 2013

...well, apparently the 3.5 million.

A summary by an attendee of a seminar given at Harvard by Yitang Zhang of the University of New Hampshire reporting on his new paper, is available here: Bounded gaps between primes

The strategy

The basic question is whether there exists some constant C so that p n+1?p n<C infinitely often. Now, for the first time, we know that the answer is yes…when C=7×10^7.

Here is the basic proof strategy, supposedly familiar in analytic number theory. A subset H={h 1,…,h k} of distinct natural numbers is admissible if for all primes p the number of distinct residue classes modulo p occupied by these numbers is less than p. (For instance, taking p=2, we see that the gaps between the h j must all be even.) If this condition were not satisfied, then it would not be possible for each element in a collection {n+h 1,…,n+h k} to be prime. Conversely, the Hardy-Littlewood conjecture contains the statement that for every admissible H, there are infinitely many n so that every element of the set {n+h 1,…,n+h k} is prime.


Remaining explanation at the link, including the following by the attendee about her notes:

On the size of the bound

You might be wondering where the number 70 million comes from. This is related to the k in the admissible set. (My notes say k=3.5×10^6 but maybe it should be k=3.5×10^7.) The point is that k needs to be large enough so that the change brought about by the extra condition that d is square free with small prime factors is negligible. But Zhang believes that his techniques have not yet been optimized and that smaller bounds will soon be possible.


A comment to the summary confirms the 3.5×10^6 note.

Been way too long since I worked through a serious number theory proof, so I'm still working on my understanding. Hope others find the discussion useful.

Thanks for the post, N2Doc.
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