Counting from Infinity: Yitang Zhang and the Twin Prime Conjecture

Announcer: This program was funded in part by the Simons Foundation-- advancing research in mathematics and in the basic sciences... and by the National Science Foundation, where discoveries begin.

Man: This is sort of a-- a Mount Everest question that you just want to solve because it's... Woman: The discovery is a major breakthrough on a subject that has bedeviled mathematicians since ancient times.

Man: Also, this new result by, uh, Yitang Zhang, which says that there are actually infinite many pairs of primes which is separated by no more than about 70 million.

News started spreading through Twitter that some mathematician had proved that prime gaps were bounded.

So when I saw that, I was immediately interested, and I started calling up some mathematicians and saying, "Is this real?"

You know, "Who is this person?"

Never heard of him.

Absolutely never heard of him.

And when I heard a rumor about that it was possible this was done and roughly what techniques were used, I said, "There's no way somebody I've never heard of has done this."

I mean, really unusual to have someone like, um, you know, sort of outside the math world, or just on the fringe of it, just come up with this kind of result, and-- and he basically seems pretty normal.

Man: I really believe the mathematics to be very pure.

I love math.

No one taught me but myself.

I like to read, to learn, and to think.

Although we say that mathematics, it should be logical-- of course it is logical.

But at the very beginning, you're thinking, you're feeling this seems very unclear, then you try to make it clear and clear again.

That could be, say, intuition.

Because intuition is sometimes harder to describe by words.

A prime number is a number that's not divisible by any smaller numbers except for 1.

So, for example, 2 is prime, and 3 is prime, but 4 isn't prime because 4 is divisible by 2.

Prime numbers are considered the atoms of arithmetic because any number can be written as a product of primes in exactly one way.

For example, 12 is 2 times 2 times 3.

So that's sort of the chemical formula of that number.

Man: Prime numbers are one of the oldest topics in mathematics.

The ancient Greeks, um, studied them.

You could, for example, prove that there's infinitely many prime--prime numbers, called Euclid's theorem.

Um, but despite being one of the oldest, um, objects of study, we still don't understand them nearly as well as-- as we would like.

And it's this combination of them being utterly fundamental and, um, so integral to the whole of mathematics, yet at the same time being so somehow incomprehensible that we're unable to say even some of the most basic questions one might ask about these objects that is just utterly fascinating from my point of view.

How often are there primes?

So, for instance, if you start looking amongst the-- the whole numbers, you get the numbers 2, 3, 5, 7, 11, 13 as prime numbers, and you say, "Well, are there any patterns?"

Well, after the first one, 2, they're odd, right-- 3, 5, 7, 11, 13-- potentially room for infinitely many-- do the first patterns you see persist?

The proportion of primes up to a given point gets smaller and smaller as the number gets larger.

So you might think based on that, that primes keep getting further and further away from each other as we-- as we go out infinitely.

But there's a conjecture which is called the twin prime conjecture which is quite old, where people realize that no matter how far they look, they could always find a pair of primes like 17 and 19, or 29 and 31, which differed only by two.

So this is the smallest possible gap that you could have between prime numbers, and it's believed that this gap appears infinitely often.

With prime numbers, you would never expect to see a pattern of two numbers right next to each other appearing again and again because one of those two numbers would have to be even, and the only even prime number is 2.

But you might expect to see pairs of primes that differ by a spacing of two, like 3 and 5, 5 and 7.

You might expect to see that appearing again and again and again, but no one knows whether that's true, and that's the twin primes conjecture.

It could be, theoretically, that the primes have some weird conspiracy between them, that they have some gentlemen's agreement that every time one number decides to be prime, then the number two spaces down will always agree not to be prime.

Twin prime conjecture is the statement that this doesn't happen, this conspiracy doesn't happen.

But we don't have ways of ruling out these conspiracies because we don't understand the primes well enough to-- to show that they don't have any unexpected patterns like this.

There is this very definite sequence of what the primes are.

But if we zoom out and look at the large scale order of what the primes are, then they seem to behave like just random numbers thrown down with a certain probability.

So you throw down this number and say it's prime with a certain chance, then do the same thing with the next one and so on.

And so the twin primes is kind of a test case of that random model.

More recently, um, prime numbers and number theory have become very important in cryptography.

A lot of our best cryptographic algorithms for encrypting data on the Internet or making sure your ATM card is securely read and so forth.

They use algorithms that involve prime numbers because we believe that various mathematic operations using prime numbers are very good at mixing up data and turning sort of cleartext data into something which is very hard to decipher.

Granville: Internet commerce is based on an understanding of prime numbers, and the understanding of prime numbers doesn't come because people say, "Oh, there'll be Internet commerce with those prime numbers, so figure it out."

It comes because people spent their time having fun with it.

They came up with ideas, and then later, people said, "Oh, you know, we could do something really tricky with this and hide secrets on the Internet."

If the twin primes ever ran out, it'd be a huge shock.

As I said, we'd have to rethink all of our cryptographic assumptions.

We'd have to rethink a lot of-- a lot of the number theory.

It would be very shocking.

Zhang: I was born in Shanghai in 1955.

My father was the college teacher and also an engineer.

And my mother is just-- was secretary in a government agency.

My parents had moved to Beijing, and they leave me in Shanghai.

One reason maybe is my grandmother could take care of me.

My grandmother, grandfather, my father, mother, and then me and my aunts and uncles and the cousins.

When I was 10, I learned something very important in number theory such as the Fermat's last theorem, such as the Goldbach conjecture.

But as I remember, there was no twin primes problem.

He was interested in mathematics, because when he was a little child, he bought a book called "100,000 Whys."

Then I moved to Beijing in-- when I was 13.

Chi: Cultural Revolution, I think his father was being punished.

Everybody has that kind of unlucky experience in China, especially in my age.

Zhang: My father suffered a little more from God.

Before 1949, the year the Communists took over China, he was a secret Communist Party member.

But then they said that maybe you were a spy.

Chi: During the Cultural Revolution, there was a lot of informants.

The situation taught us not to reveal what we're really thinking, even to your parents, or parents to your children.

Zhang: I went to the countryside, the farm, with my mother.

That was because of the Cultural Revolution.

We have to have the meetings together in the dining room.

It is not good.

It was a terrible time.

We did have physical labors, but as a child, I didn't feel too bad.

Still, right now, if I have a chance, I will go back to the countryside.

Chi: All of sudden, the universities closed, and there's, uh, high school and there's school-- no schools, really.

The Communists tried to reeducate us to learn the value of farmers, workers, and also soldiers.

Because of my--my father's-- the political problems deal, there had not been any conclusion.

I couldn't enter the high school.

I worked as a worker for a couple of years.

Then in 1978, after the Cultural Revolution, I got a chance to go to the college, Peking University.

In the exam to the university, I didn't do well on the political exam.

But I passed.

In Peking University, I met many good professors.

Man: I met his advisor from master's when he was a master's at Peking University-- Pan, Professor Pan--and he, uh, made clear that Zhang was, uh, singularly strong and the best student finishing in Beijing.

So it's clear that he set his sights very high from then on, and has always gone for the big thing.

Professor Pan was a very responsible teacher, professor, because that was only the master degree.

So then he--he wished to I could do it very quickly, to get a degree quickly, so I just did it.

It just took about a couple of months.

But unfortunately, in China, the personal relationships should be very careful.

Unfortunately, I didn't realize that.

I had a chance to go to San Diego, UC San Diego, with a good pro-- with Professor Stark.

But Professor Ding prefer me to learn how to break geometry rather than number theory.

Anyway, I change it, but, um, still my interest, uh, had been in number theory.

They didn't think so much about the respect to your personal freedom, your personal choice.

That's the Chinese way, in that time, for something, for the interest of the country, the interest of the society.

I spend around one year in Peking University.

After Ding had blocked the arrangement for Tom to go and work with Harold Stark, the next year, TT Moh came to Beijing to recruit students, and he was an algebraic geometer, so Ding was happy to have Zhang go with him.

Zhang went to Purdue with him and began to work with a problem that Moh assigned him, a famous problem called the Jacobian problem.

Goldston: His thesis was on the Jacoby conjecture, which has just, you know, crushed-- all sorts of mathematicians have made fools out of themselves on that one.

Zhang: I believed that my solution on the Jacoby conjecture should be very close to the final.

Eisenbud: Zhang did some nice work on this problem and actually thought he'd solved it, but the solution was based on a lemma that Moh had proven, except that Moh hadn't proven it-- it turned out to be false.

And so the proof collapsed.

He tried to--to solve another problem, like the resolution of singularities.

Yes.

They worked together on that for about 6 years, and then it became clear that Moh's approach which he wanted to follow was not going to work, and Moh gave up, then never wrong Zhang a letter of recommendation.

That's pretty fatal for a student.

His i--attitude to me was not-- was not good.

He just complained and-- "No Chinese student is good."

Then I couldn't find job.

Chi: He's not only a mathematician, he's good person.

I knew him through a organization called Chinese Alliance for Democracy.

Both of us are members.

I believe still this organization is illegal in mainland China.

The organization actually started with a Chinese, uh, overseas student.

First, it was a magazine to promote the democratic movement in China.

Our slogan is, "Freedom, democracy, rule of law, and pluralism."

They have very, um, special friendship between the two of them.

They both were brave enough to stand up for their ideal and for their, um, belief.

So that's a very special foundation for them to build a friendship.

You have to have this deep trust.

In this movement to end up, the good-- one of the good result is that I have a connection with some good friends.

Chang: They found the common interest in music, in basketball, um, in philosophy.

He feels comfortable with me because I don't have to make him talk about big stuff.

He would help me.

When Julius was little, he would say, "Oh, can I feed him?"

He say, "Can I change the diaper?"

I--I let him change the diaper.

He loves Julius.

Julius: My parents told me that, you know, we're gonna meet a great mathematician.

Because I was pretty confident in myself.

I was 5 years old.

You know, I learned my multiplication tables.

I'm ahead of everyone.

I sort of devised what I thought would be, like, a super-difficult problem that he wouldn't be able to solve.

Chi: Sometime he needs to be just by himself, because he needs time to think about his mathematic things.

And so my--my house is always open for him.

Well, he always seemed to be thinking about something.

You know, he would always zone out, you know?

But then again, me and my dad were-- we're kind of like that, too.

Chang: He would just go on problem-solving in his notebook.

And I said, "Do you have a set of problems?"

"No.

Just in my mind I come up with a problem and I solve them."

Zhang: Unless I was sleeping, always I tried to think about the math problems.

More than 2,000 years ago, the Greek mathematician Eratosthenes developed the first sieve for looking for prime numbers.

It's sort of like a strainer, and you dump all the numbers into it, and then the ones that aren't prime fall through the holes.

You start with 2, and 2 is prime, then you let all the numbers divisible by 2 fall through the holes.

Then you go on to 3, and you let all the numbers divisible by 3 fall through the holes.

Now, 4 has already fallen through, so then you go on to 5 and let all the numbers divisible by 5 fall through the holes, and so on.

So in this particular set, we actually have quite a lot of twin primes.

We have--we have 3 and 5, 5 and 7, 11 and 13, and so forth.

Um, unfortunately, uh, we don't-- our--our knowledge of sieve theory-- um, our sieve theory is not able to, um, guarantee that as you keep going to larger and larger, um, sets of numbers, that you will always keep finding these pairs of-- pairs of, um, these pairs of twins.

So people for a long time have been trying to find, uh, better ways of sieving where you would expend less energy and still be able to come up with the primes.

Maybe you wouldn't find all the primes, but you would hope to find most of the primes.

Maybe you would hope to find the primes plus some junk.

Maybe it's exactly like panning for primes.

If you like, it's like panning for gold.

So you expect to left with the gold, but maybe you get some other things left behind, as well.

Tao: There's a very well-established heuristic, actually, which goes back to Hardy and Littlewood back in the 1920s.

This [indistinct] predicts not only that there's an infinitely many twin primes but actually, um, an almost exact formula for how many twin primes there should be up to any large number, like up to a trillion, up to--up to a quadrillion.

How many twin primes there should be, you can check numerically by computer, um, but this prediction is accurate to really high accuracy.

If you count how many twin primes there are to a trillion-- think up to, like, 4 or 5 decimal places-- it's the same as these predictions.

So we have these heuristic models for the primes coming from--from probability, and they're extremely accurate, as far as we can tell.

Uh, we just can't prove that they're accurate.

We can just observe numerically that they're accurate.

So even up to about 20 years back, the only thing that we knew was that every once in a while we could get two primes whose average--whose gap is, uh, about a quarter of the average size of a gap.

Uh, so this is still something which is getting larger and larger, and it seems very far away from getting a gap of two, which is what the twin prime conjecture would predict.

Man: My mentor managed to improve the distribution of prime gaps, and mentioned the problem to me.

And around '85, I work with, uh, Friedlander and Iwaniec on these type of problems.

We tried to--very hard to go beyond this group of buggers for the case of prime numbers.

We obtain some partial results.

I had read the Bombieri, Friedlander, and Iwaniec, the papers, and they provide us very good ideas about that.

I worked on it only because I wanted to figure out if this method from 1965, um, whether the result you got was the best you could do or whether you could do better.

Slowly what happened was we moved over to this other method.

In trying to answer that question, we eventually came up with what's now called a GPY sieve.

It was a beautiful theorem that it did, in fact, prove that there were gaps between primes which were less than a millionth of the average size of the gap.

And the people also had another remarkable idea which nobody had guessed before, which was that if, uh, you could understand something about the distribution of prime numbers in arithmetic progression, which is an area that people had spent a lot of time thinking about and which is connected to problems in mathematics like the Riemann hypothesis, then you would, in fact, get bounded gaps between primes.

Klarreich: The Goldston-Pintz- Yildirim, or GPY sieve, makes the connection between prime gaps and something called an arithmetic progression.

An arithmetic progression is just a sequence of numbers that have the same spacing from one number to the next, like 5, 9, 13, 17, 21, where the numbers all differ by four.

Or 7, 11, 15, 19, where again you have a spacing of four.

But if you look at the two odd progressions, you start to notice that the primes flow evenly into the two progressions.

There's kind of a balance between how many primes are in one progression and in the other.

And in the GPY paper, they made a connection between this balance and the gaps between prime numbers.

Goldston: That phenomena that the primes sort of flow into the different arithmetic progressions evenly should continue to happen even when the difference between the primes gets pretty large.

So when you look at primes up to "X," that you want to look at progressions that have jumps of size square root of "X," and they should still evenly distribute among them.

The Goldston-Pintz-Yildirim, uh, technique is a very ingenious version of the sieve which works in this very special situation where you have a small set of numbers from "M" to "M" plus some big number.

It gives a mechanism by which you could somehow find every once in a while there are two numbers inside this-- inside this interval which are primes.

So if a technique like that works, then it shows because there are two numbers in the short interval that are primes, that there are small gaps between consecutive prime numbers.

When we first came up with our result in 2005, we were able to do all these things that no one had done before.

So at that point, I--I actually remember claiming that maybe we'd be able to prove the twin prime conjecture in a--you know, and I was working on it.

But that was before we, um, sort of found the limitations in our method.

In other words, there was no cheap trick which would allow you to get this, uh, this result.

So they--they had gotten to a barrier, and they managed to squeeze the maximum they could without going past that barrier, but you really had to cross this barrier in order to get the next big result.

We know that if you look at all the arithmetic progressions with the given spacing, the prime numbers flow evenly into the different progressions.

But how far out in the progressions do you have to go in order to see this balance in the number of primes they have?

Mathematicians showed that you'll usually start seeing this balance by the time you've gotten out to about the square of the spacing in between the-- the numbers.

So, for example, if you have progressions with the spacing of 10, you'll start seeing this balance by the time you get out to about 100.

But this wasn't quite enough to prove that prime gaps are bounded.

To do that, mathematicians needed to show that this balance appeared a little bit before the square, but no one could figure out how to do that, and mathematicians called this the square barrier.

Granville: So Zhang came along and he addressed the hardest and most difficult approach that had ever been considered, and somehow he-- his head was hard enough and he broke through that wall, and he actually made progress on the technique proposed by Goldston, Pintz, and Yildirim.

Chi: I learned that he-- sometimes he live in his car.

He never talk about that.

That's when he left Purdue and he's looking for job.

Chang: After Purdue, he did some fellowship work at Princeton, and then that was a temporary arrangement, one year or two years.

And then, um, this other pro-democracy person bought some Subways and he needed some help.

And Tom was between jobs, so he came over and he helped with him, accounting work and all that.

Zhang: Between 1992 to 1999, sometimes I worked in the Subway store.

Even though people talk about it's a hard time, I think he-- he treats it as a money-earning position.

And because most of the time, if he wants to shut off, he can.

If he wants to just go into his world, he can.

Zhang: Then after several years, eventually I got just a lecture job at University of New Hampshire.

Tom Zhang first came to UNH, uh, in late 1999 and, uh, was hired as lecturer and taught for the first time in the spring semester of 2000.

Students value his classes because of the way that he organizes both course and the material.

They, uh, feel that that organization of material, uh, reduces the barriers to passing the course.

Zhang: ...also a zero.

He always really cared about the math more than, you know, worrying about the grades or anything.

It was more that you understand it.

That was the important thing.

For clarity and effectiveness, I would say that Professor Tom, uh, was one of the top professors that I had at the University of New Hampshire.

When he taught at University of New Hampshire, he was-- the best teacher award was given to him.

Hinson: I can read these evaluations here that give some sort of sense of their overall, uh, view of him.

Uh, this one says, "Tom is the Best.

"He's a great teacher and he makes it easy to understand "potentially difficult concepts in calculus.

"He's funny, too.

Everybody loves Tom."

And then a footnote: "He should stop smoking-- it's not good for him."

Zhang: So then a "H" is contained in "P." Student: I think "P" should be solved with all of "H." Hinson: He could be seen at all hours of the day or night in this building, wandering, thinking.

This was recognized, uh, by members of the department, by his own students who would see him at times in the building when they would see no other professors.

Could be elementary.

I'm shy.

Hinson: He says he's shy.

He proclaims this with regularity, but he says put him in front of a mathematical audience and give him mathematics to talk about, and he's not shy anymore.

Also, we can verify a tease... Brown: You could just see that he was happy.

And just by the way he started class and the way he went over, you could see that he really enjoyed what he was doing.

He just loved the mathematics behind it, and he just loved being able to write it on the board and describe it to the students, and I could really feel that energy.

To teach a course, of course you have to understand, right?

It--otherwise, you--you couldn't do nothing.

You have to reorganize the materials.

You have to think about what should be the best way to express it, to present it.

That's my philosophy.

Chi: It's interesting, right?

He's a professor.

You know that he was an adjunct.

He should have his own apartment, but for some reason that he chooses to live with the Chinese students.

And every weekend, he will cook for them.

He cooks so well.

He... he cooks for us.

And he cooks, um, mostly Chinese food.

And he's a very nice cook.

He made the wonton.

He'll make them.

He arranged them like an army.

He's such an organized person.

Zhang, voice-over: Because my lifestyle is very simple, I don't worry about the houses, the cars... [laughs] the enjoyments.

Enjoy-- there's lots of things.

So I--so I--still I have lots of time.

After getting the job in New Hampshire, I went to New York to see my friends.

Then a friend took me to the-- a Chinese buffet in Long Island.

She was the waitress there, mm, my friends had just introduced.

"Oh, this is the doctor.

He came from Peking University," then ask her, "Do you like him?"

Then--and that was--that-- that day, I--I might drink s--so much.

Too much.

Zhang, voice-over: But then after a couple of days, my friend brought me and her together in another restaurant.

And we got to know--know each other.

All of a sudden, he said, "I'm married."

Bubby person, happy.

And they just get along.

I don't know how she--get Tom to talk to her.

[Helen speaking English in voice-over] Zhang: She is, uh, good at dance.

I--I couldn't dance.

Ha ha!

I used to be very quiet.

It's a miracle how they get along and still being a couple.

Zhang, voice-over: That's OK if you have, uh, very different personalities.

Still, you can find the way to get used to each other.

No problem.

Hinson: On the morning of April 17th, a Wednesday, I was sitting in my office.

And Tom came in.

And across the desk passed, uh, a, uh, manuscript to me.

And it was, uh, the, uh, the paper that he was, uh, telling me he was going to submit to the "Annals," uh, later that morning.

That was the first time that I was aware that he was even working on that problem.

Man: I am deluged with manuscripts in elementary number theory by people who are not professional mathematicians who claim, typically, to have solved simultaneously Fermat's last theorem, the twin prime conjecture, Goldbach's conjecture, and usually, uh, some unified theory of physics.

I consider my duty as a scientist to answer once.

So to protect myself, I-- I answer, give-- say why it's wrong, and say, "I've received many requests like yours.

"I cannot devote all my time.

"You've got my report.

And I don't look at revisions.

"And you don't send anything because I will not answer.

Find somebody else."

A lot of these people, you can just tell from reading a few words.

Usually they're very defensive, um, how everybody else is wrong.

Um, that--that's often a giveaway that you're probably not going to enjoy this experience.

I'm the expert on rejecting twin prime proofs.

You would think they would be better proofs.

They're just garbage.

And so our default presumption is one of skepticism, when--when we hear about a--a--big result, especially if it's from someone who hasn't been that active.

Man: The paper came in.

It was assigned to me.

It's not a subject I'm particularly expert in, but I knew who was expert in it.

So I--I sent it to such an expert and got basically an immediate reply-- certainly within 24 hours-- Um, "If this paper is correct, it's a fantastic breakthrough."

Bombieri: I was having lunch here at the institute.

And one of the editors of "Annals" was there and said, "We received a paper on something.

It is not quite twin prime.

It's on prime gaps and..." uh, "by some unknown Chinese mathematician."

And, you know, "We receive so many of these things," um... "How should we handle it?"

I can only imagine, uh, what went through the, uh, minds of the, uh--of the staff at the "Annals" when in through the electronic transom comes this, uh, perfectly crafted paper.

At some point, one has to decide, uh, what to do with this.

There was a line-by-line checking of the paper.

No mistakes, two misprints, one item to add to the bibliography.

Some parts could be simplified, but the result was, uh, significant, important.

The way the--Zhang managed to break the square root barrier was new.

Then just after three weeks, it--it was accepted, it was approved, correct.

Bombieri: One month later, it was already published in electronic form.

And now it's appeared in printed form.

And so Zhang became a star.

Zhang: We tried to find a certain up bound for this one as good as possible, but still it is--lots of things remaining.

Before doing that, let me mention something.

If--the--the classical treatment of such a problem... Bombieri: My reaction was, "Wow!

Who is this guy?"

And he showed a mastery of the--the subject, of, uh, understanding.

Somebody working at not a big research institution, on their own without the right sort of mentors, who we've basically never heard of, I mean, it doesn't happen.

It's-- it's absolutely incredible.

Then that happen in just one thing.

[applause] Thank you.

Chi, voice-over: The next year, when I call him-- I said, "Congratulations, Tom," and, uh, "You made it.

I'm so happy for you."

He said, "You know what?

I solved this problem in your backyard."

I said, "When?!"

2012, I called Yitang.

I said, "Well, now Julius is ready for calculus.

"Can you come?

And, also, of course, I have great wines in the house."

And I know he--he loves wine.

So he came with one bag, one baseball cap.

He spent one hour, teach Julius in the morning and then eat with us, talk to Jacob about music, drink.

Zhang: I tried to get a break.

So I didn't bring any paper, any pen, nothing.

After that, I found my mind was more clear, more productive or something.

["Stars and Stripes Forever" playing on soundtrack] Chi: Every July 4th that since I'm in Pueblo, I give a public this free concert at Riverwalk.

On the July 3rd, I have a dress rehearsal.

So before I get ready, he said, "Well, let me see where the deers are coming to the yard."

Zhang: I tried to find some deers.

Lots of time, the deers went to their house.

Ha ha!

But I didn't see any deer.

That time just very suddenly, I got a idea: Oh, we can go this way.

I just think, first from main term, Oh, we can do something like this.

Then to the error term, also, we can do something for that.

Again, main, error.

Many errors.

Then, eventually, after maybe-- maybe half-hour or maybe 20 minutes, I--I was sure there is a certain way to--to do it.

["Stars and Stripes Forever" playing on soundtrack] And he didn't say anything.

And after he went to the rehearsal, you know, of the concert, he was really excited, t--to-- to hear "Stars and Stripes Forever."

He loved that!

You know, he--after the concert, he was, you know, like, uh, hum this tune all the time.

I didn't think anything more.

All the things that coming to-- the fame I hadn't expected.

Chi: He told his wife, "Look at what YouTube says."

He said, "Well, I have good news for you."

And then he says, "Well, what news?," you know.

I tried to make a trick to her.

I didn't tell her the story.

I just, uh, told her, "Pay attention to the media, to see the newspapers.

Should have something to do with me."

Then she asked, "What do you mean?

What do you mean?"

I say, "You will find."

Zhang: She said, "What?

What do you mean?

Are you drunk?"

"Oh, my God..." "Comb your hair."

I don't want his hair to look like that.

Ha ha!

It was very timely because the Cole Prize for Number Theory, which is one of the most prestigious prizes in the world, was due to be handed out in January 2014.

And, um, the committee was meeting in April 2013 when the news of Zhang came up and was very quickly verified.

And so they chose to give a medal to Goldston, Pintz, Yildirim, and Zhang for the fantastic breakthrough.

[indistinct chatter] Zhang, voice-over: Then my life--ooh--became complicated.

Lots of interviews, maybe too many emails and invitations.

They treat me just like a hero, like a star.

It was difficult to me.

Maybe not to her.

Tom told his wife, "I prom... "When we were married, when we were getting married, I told you that I would give you a lot."

"At that time, I couldn't give you much.

Now what do you want?"

Zhang, voice-over: She has been happier with the change in the-- the life with us.

And, uh, she's very proud of me, of me and herself.

She went to many places, like Tehran, to Beijing-- mainland China-- and maybe, like, at, you know, at Berkeley, at-- at--Columbia University, she now has many chances to-- to travel with me.

And, of course... Of course, we're getting just a little richer in money.

You know, he should enjoy life a little bit more rather than everything is mathematics and philosophy.

Benson: You know, the twin prime conjecture, it's one of the longest standing unsolved problems in mathematics.

And the fact that he would make such a groundbreaking discovery about it, just a sense of awe.

Just a sense of awe.

Where Zhang, I think, is--was exceptional was to go into the proof of the work of the paper of Bombieri, Friedlander, and Iwaniec, understand the exact mechanism that makes it tick, and tinker with that, combined with Goldston, Yildirim... to change it enough to make these things meet with an unconditional proof.

When his paper came out, he said that he could find pairs of primes whose distance apart was less than 70 million, which is a nice round number.

But almost immediately, people started looking at his paper and noticing that if you were a little bit more careful in certain steps of the proof, you could improve this bound of 70 million a little bit.

Klarreich: Zhang's work raised a question: Why 70 million?

There's nothing magical about that number.

It served Zhang's purposes and simplified his proof.

Other mathematicians quickly realized that it should be possible to push this separation down quite a bit lower, although not all the way down to 2.

By the end of May, mathematicians had uncovered simple tweaks to Zhang's argument that brought the bound below 60 million.

A May 30th blog post by Scott Morrison of the Australian National University in Canberra ignited a firestorm of activity as mathematicians vied to bring this bound lower and lower.

By June 4th, Terence Tao of the University of California, Los Angeles, the winner of the Fields Medal, mathematics' highest honor, had created a Polymath project, an online collaboration that attracted dozens of participants.

And lots of people started paying attention to--to this sort of game, if you wish, of improving the numbers.

For, like, two months, that bound just kept going down and down.

It was actually quite, uh, quite exciting to watch.

Klarreich: By July 27th, the team had succeeded in reducing the proven bound on prime gaps from 70 million to 4,680.

Granville: The Polymath project has been very valuable.

It's really changed the landscape.

So I started working on an article about the Polymath project.

And just around the time when I was finishing it up and getting ready to turn in my first draft, I suddenly heard that James Maynard, who I knew nothing about, had produced an alternate proof that brought the prime gap down to 600.

Bombieri: Here comes another proof.

Completely different.

This was by Maynard, the young guy from Oxford.

And, uh, the paper's just about a sieve.

However, here was this, uh, multidimensional sieve with the dimension going to infinity or, at least, extremely large.

Goldston, Pintz, and Yildirim had introduced weights that depended fundamentally on one variable.

And a natural generalization that I was looking at was ways in which, um, the weights could somehow incorporate some slightly more arithmetic structure.

And one way of doing this was to incorporate many variables.

Then the key modification that I had was that I found the right way to introduce a multivariable generalization of the Goldston, Pintz, and Yildirim weights.

And by introducing more variables, I was able to, um, have some weights that were slightly more concentrated in prime numbers.

And this is what enabled me to get, um, certain improvements to the results about bounded gaps between primes.

The fact that there is an absolute bound, an absolute number so that there are infinitely many pairs of primes that differ by no more than that number is an extraordinary breakthrough.

It's got nothing to do with the average gap.

He's shown that there's just some finite number.

And now the number's down to 200 or 300.

So what do we know?

We know that no matter how far you go out, you can find prime numbers such that the next prime is no more than a couple of hundred away.

You can take many, many theorems in the theory of prime numbers and sort of just add now a sentence thanks to Maynard.

And there will be, uh-- you can produce the prime numbers we're looking for in such a way that there will be bounded gaps between them.

These gaps could be very big, and you don't know what the gaps are, but they are bounded.

Now one can combine Zhang's idea, then my ideas to get a, um, stronger result in some questions to do with bounded gaps between prime numbers precisely because these ideas are somewhat separate, and so they can be quite naturally combined with each other.

In fact, a second form of the Polymath project, Polymath8b, was set up to try and optimize the new ideas that I'd introduced and combine it with the ideas of Zhang to try and get the strongest possible gaps between primes.

And so I've been an active participant.

The current world record, at least as of the moment, is that there's infinitely many pairs of primes that differ by at most 246.

We're still nowhere near 2.

And the question of, um, How long will it be until we prove there are infinitely many twin primes-- in other words, that the gap 2 occurs infinitely often-- your guess is as good as mine.

It could be a year.

It could be a century.

Maybe we're ready to start working at that level, and maybe other problems can get solved.

For example, um, there's the Goldbach conjecture.

Uh, the Goldbach conjecture is the statement, every even number, say bigger than 4, should be bi--written as the sum of two primes.

So, like, like, 8 can be written as 3+5 and so on and so forth.

So this is the conjecture.

It's very similar to the twin prime conjecture.

Actually, the twin prime conjecture is a conjecture about differences of primes.

Goldbach's is a conjecture about sums of primes.

They're very similar.

And it's quite likely that a lot of the methods that, um, that have given these partial results towards the twin prime conjecture should also give some partial results towards things like the Goldbach conjecture.

I think a lot of people now are revisiting questions which were thought to be quite difficult.

The great thing about great breakthroughs is that they not only allow you to address old questions, but they inspire new questions.

So what else can we now do with primes that we never could do before?

Zhang: After you--you solve the bigger problem, then, to me, I was not interested to do this problem again.

I tried to change--to turn my int--interests quickly to some other problems.

I think doing math, you can go everywhere.

You just need the one place to do the math yourself.

Doing proofs, and maybe my personality is-- is so quiet, I didn't contact so much with, uh, other people.

Of course, I love this area.

So I've got to know him a little bit.

One day I saw him, and I said, "Hi."

And he said, "Hello."

And he said, "That's the first word he'd said to anybody in the last 10 days," 'cause he'd been so isolated.

So I was a little concerned that that's a little too much.

I mean, he clearly likes to just work, uh, at his own pace.

So, I--we agree to have lunch once a week, and which we have done since.

And during that time, he's opened up every now and then.

And he's discussed a little bit this very, uh, pressing problem he's working on.

Zhang: Next is the so-called Landau-Siegel zero.

I have been working on it, uh, for many years.

The L-function might have a zero very, very close to 1.

That's called a Siegel zero.

The--the problem is to prove such a zero never exists.

I have, uh, lots of partial results.

But I'm not going to publish them.

It's correct.

I did it.

Ha ha!

Emily Corwin: It's been quite a year for Zhang, who goes by "Tom" at UNH.

As a recipient of the prestigious MacArthur grant, Zhang will receive a stipend of $625,000 paid over 5 years with no strings attached.

Zhang: I had heard something about this prize.

But, uh, I had never expected that it would come to me.

They informed me about, uh, th--this one.

And the one question is, "Do not tell other person, with one exception."

Now, this exception is Helen, my wife Helen.

I just told her.

On behalf of the--eh, the--Rolf Schock Foundation and the three academies, I wish to convey our warmest congratulations.

[reading speech in French] [applause] Thank you so much.

Really good finally to meet you...

Thank you.

Hinson: He had every intention of staying at the place where he had accomplished these things and continuing to work.

Zhang: Doing mathematics isn't the most important, uh, thing.

I may not care so much where I can do it.

Here, this is very quiet place.

I can concentrate on what I like.

So I just prefer to be here.

Other universities, they make you--me better offer, much more money, something, but I say no.

"It's OK. We'll just stay here."

[Zhang's lecture continues, indistinct] My belief is that what he has found here, he still values here.

And, uh, our goal is to make sure that those conditions are still here for him.

Chi: He doesn't really talk about, you know, his achievement.

He's, uh, always the same.

He said, "Well, I just want a situation that I still can do my thinking."

He chooses the loneliness.

He chooses solitude.

He proved a very beautiful theorem that was believed by the experts to be out of reach.

He went to the deepest of the deep, and he fully understood.

And his paper establishes him just in-- in the depth of thinking as one of the top half-dozen people in the world in the field, from nowhere.

Goldston: It's gratifying that you're alive and you're seeing this.

The human race actually has proved this result.

Chang: I was so happy that he gets to solve this problem for the world, but I feel like, "Yeah, if anybody, "he probably should be the one.

He's great!"

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