Public

There is no publicly available preprint on this yet, and my information is all either second- or third-hand, but my understanding is that Zhang has managed to find a specialised improvement of the Bombieri-Vinogradov theorem (in the spirit of some famous papers of Fouvry-Iwaniec and Bombieri-Friedlander-Iwaniec, see e.g. the introduction to this recent paper http://arxiv.org/abs/1108.0439 for a summary) which, when combined with the method of Goldston, Pintz, and Yildirim http://arxiv.org/abs/math.NT/0508185, establishes bounded gaps between consecutive primes infinitely often. (The original Goldston-Pintz-Yildirim paper already noted that certain types of improvement to the Bombieri-Vinogradov theorem would give such a conclusion; I do not know if Zhang's argument establishes such improvements exactly, or establishes some variant result of this type.)

I hear that some very credible experts have already refereed the paper carefully, but it may still take some time to get "official" confirmation of the correctness of the argument, particularly in the absence of a preprint.

I hear that some very credible experts have already refereed the paper carefully, but it may still take some time to get "official" confirmation of the correctness of the argument, particularly in the absence of a preprint.

Recall that the

http://en.wikipedia.org/wiki/Twin_prime

This has been, to put it mildly,

Note that this is very non-obvious, because it may be that the prime numbers get more and more spaced out as they get larger, in the sense that the

http://en.wikipedia.org/wiki/Bertrand's_postulate

which states that given any prime p there is another prime p' < 2p - 2. In fact, Erdős improved this to a logarithmic bound: for any prime p there is a prime p' < p + (c ln p). The constant c has been creeping down over the decades, and we now know, as of 2005, that c can be chosen to be as small as possible (see the Wikipedia page on twin primes linked above). However, this doesn't help with the Twin Prime Conjecture, which wants to know if we can push N in Conjecture(N) down to 3 (question for the experts: why is this so?).

So, according to Peter Woit, who got a group email from Yau, there is a seminar at Harvard later today (3pm local time) by Yitang (Tom) Zhang of U New Hampshire called "Bounded gaps between primes" (this isn't listed on the Harvard website).

http://www.math.columbia.edu/~woit/wordpress/?p=5865

He claims to have a proof of Conjecture(70 000 000), which would be very big news.

If anyone is there, please report back.

#mathematics #primenumbers #Plus1Experiment

*Twin Prime Conjecture*states that there are infinitely many primes p and q such that | p - q | = 2.http://en.wikipedia.org/wiki/Twin_prime

This has been, to put it mildly,

**EXTREMELY HARD**to prove. An equivalent statement is that there are infinitely many primes p and q such that | p - q | < 3, and so one could try to arrive at a weaker statement, where 3 is replaced by some number N.**Conjecture(N):**There are infinitely many primes p and q such that | p - q | < N.Note that this is very non-obvious, because it may be that the prime numbers get more and more spaced out as they get larger, in the sense that the

*minimum*distance between primes in [M,∞) grows as M grows. We*do*know that this spacing grows at most linearly, by Bertrand's posulate:http://en.wikipedia.org/wiki/Bertrand's_postulate

which states that given any prime p there is another prime p' < 2p - 2. In fact, Erdős improved this to a logarithmic bound: for any prime p there is a prime p' < p + (c ln p). The constant c has been creeping down over the decades, and we now know, as of 2005, that c can be chosen to be as small as possible (see the Wikipedia page on twin primes linked above). However, this doesn't help with the Twin Prime Conjecture, which wants to know if we can push N in Conjecture(N) down to 3 (question for the experts: why is this so?).

So, according to Peter Woit, who got a group email from Yau, there is a seminar at Harvard later today (3pm local time) by Yitang (Tom) Zhang of U New Hampshire called "Bounded gaps between primes" (this isn't listed on the Harvard website).

http://www.math.columbia.edu/~woit/wordpress/?p=5865

He claims to have a proof of Conjecture(70 000 000), which would be very big news.

If anyone is there, please report back.

#mathematics #primenumbers #Plus1Experiment

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- +Terence Tao No, that link doesn't work.May 23, 2013
- +Robert Munafo - you need a subscription to the
*Annals*to access it.May 23, 2013 - Zhang did solve it!!

What do you think Prof. Tao?

The Mathematician Who Could Be a Movie Star

By Stephen L. Carter May 23, 2013 5:00 PM CT

.

In the distraction of the scandal-fever swirling through Washington and the news media, you might have missed the announcement the other day that one of the great puzzles of number theory had been solved.

What makes the news most fascinating is that the solver isn’t on the faculty of a top university and wasn’t known until this month to others who work in the field. He is a Chinese immigrant in his 50s named Yitang Zhang, a onetime accountant and part-time lecturer at the University of New Hampshire who used to make sandwiches in a Subway shop. Said one leading number theorist: “Basically, no one knows him.”

Stephen Carter

About Stephen L Carter»

Stephen L. Carter is a professor of law at Yale, where he teaches courses on contracts, professional responsibility, ... MORE

.

More from Stephen L Carter:

Note to Media: The First Amendment Protects All of Us

Q

To the Class of 2013: Resist Simplicity

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Is Obama Undermining America’s Credibility?

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Cue the agents and film producers.

Because the story gets better. Zhang’s accomplishment tracks our romantic vision of the dedicated genius working alone in his garage. The truth is even more unlikely.

Zhang hit upon the crucial idea not in his garage but at a friend’s house in Colorado, where he had gone to clear his head. He was sitting in the backyard, waiting to leave for a concert. (Imagine it cinematically: Zhang skips the concert, ignoring the entreaties of his skeptical hosts, and refuses to budge from the yard, where he sits all through the frigid Rocky Mountain night, feverishly scratching equations into tree bark.)

Faithful Genius

The achievement that has inspired such awe among mathematicians is Zhang’s proof of the “weak” form of the twin prime conjecture -- a proof so strong that he was recently asked to present it to an audience at Harvard University. This isn’t the place to explain what the twin prime conjecture is, or why it has a strong and weak form, or even why the solution has posed such a challenge. (Here’s a good primer for the mathematically curious.)

The fascinating part is how Zhang succeeded where others had failed. There was no flash of genius, no invention of an entirely new methodology. He saw the promise in an approach that others had abandoned, and -- mirabile dictu! -- had enough faith in his idea to stick with it until everything clicked. (Note to producers: Be sure to write in mocking younger colleagues, who think the old guy is past it. See if Benedict whatshisname -- the “Star Trek” guy -- is available.)

The story’s had a bit of coverage, but not nearly what it deserves. The media by and large aren’t terribly excited about science these days. Technology, sure -- albeit generally on the very personal level. An exciting new smartphone application will get commentators salivating, and smiling news anchors will report the results of the latest clinical study on the efficacy of a popular drug, whether or not they understand the data.

Pure science, however, discovery for discovery’s sake -- in short, using our brains because we have them -- doesn’t get a lot of airtime. (Except when thrillingly dangerous, like the rumors a few years back that the Large Hadron Collider near Geneva might create a black hole that would destroy the world. The world, having survived, immediately lost interest.)

I’ve noted before that we may be losing a generation of pure scientists. It has become a truism that many of the brightest science, technology, engineering and math majors are passing up graduate study for law or business school. I am old enough to remember when young people looked with admiration and even envy on their gifted peers who planned to be scientists. Nowadays, a facility with numbers is a highly valued skill, and the returns on careers in law and finance dwarf what they could earn in the academy or the research laboratory. Whenever I’m asked how the students have changed over my three decades of law teaching, I point to the growing disproportion of science majors.

Scientific Romance

The problem isn’t the public’s lack of scientific literacy. Veteran science writer Daniel S. Greenberg, in his 2001 book “Science, Money, and Politics,” put the point this way: “Science, democracy, and prosperity are said to be at risk, though, mysteriously, all have spread robustly despite the dearth of public understanding.” The problem is the lack of serious public interest.

We need to recover what the late Carl Sagan called “the romance of science.” We can do this in part by coming to appreciate the human side. The media can do their part by paying more attention to stories like that of Yitang Zhang -- “Tom” to his friends and students -- because there is human interest everywhere, if we but choose to look. For example, everybody knows that nobody does important work in mathematics after 40, the age at which one becomes ineligible for the Fields Medal, often referred to as the Nobel Prize of the field. And Zhang is in his 50s, and used to work at Subway, and -- as I said, the story writes itself.

Now, I know you have to get back to the scandals of the moment. Before you do, let’s follow the romance of pure science one act further: Zhang says his great breakthrough during his sojourn in Colorado came on July 3. Hmm. Memo to the producers: Can we push that back a day, and set fireworks behind his head, something like what Baz Luhrmann did for our first sight of Gatsby? Just a thought.

(Stephen L. Carter is a Bloomberg View columnist and a professor of law at Yale University. He is the author of “The Violence of Peace: America’s Wars in the Age of Obama,” and the novel “The Impeachment of Abraham Lincoln.” The opinions expressed are his own.)

To contact the writer of this article: Stephen L. Carter at stephen.carter@yale.edu or @StepCarter on Twitter.

To contact the editor responsible for this article: Michael Newman at mnewman43@bloomberg.net.May 26, 2013 - Perspective is a critical ingredient in problems solving because there are many ways to approach a problem, some more effective then others.Sep 20, 2013
- Why didn't Zhang submit his paper to Terence Tao?Sep 20, 2013
- Both Yitang Zhang's mom and Terence Tao's dad are Shanghaineses.Jul 11, 2014