从张益唐身上总能学点什么？

  刚刚看了网上对张益唐的详细报道，感觉有点感动，那么从张的身上总能学点什么吧？

  1、55岁发表论文轰动，而这个年龄一般认为作为数学家，不会有突破性的成果出现。

  2、到了快60了，还是一名讲师，并且没有终身教职，没有按照一般的社会树立的成长阶梯爬副教授、教授，那样，每年必须发表一定数量论文。

  3、能够利用10年多的时间，专攻自己喜欢的研究问题，一般人早就放弃了。

  结论：张的成功基本是游离于在体制外的一种成功，他能耐住寂寞和清贫，围绕兴趣保持志向，能够坚持数十年围绕一个问题锲而不舍，并且是依赖一个人做研究，因此能够在大龄保持活跃的思维大脑。

       另外一个非常重要的一点也许是他应该保持着少有的一种自信---因为他整日与问题所伴--就如攀登心中的珠穆朗玛一样，他一定是坚信终有一天能够登顶，即使不是这样，也会坚信终结在临近珠穆朗玛的路上而一生无憾！

  也许大家不会进行模仿，不会完全在社会成长阶梯之外，但没有人能够随随便便成功，因此从中总能得到一点启示和激励吧？

附：

The Pursuit of Beauty

Yitang Zhang solves a pure-math mystery.

BY ALEC WILKINSON

  CREDI

I don’t see what difference it can make now to reveal that I passed high-school math only because I cheated. I could add and subtract and multiply and divide, but I entered the wilderness when words became equations and x’s and y’s. On test days, I sat next to Bob Isner or Bruce Gelfand or Ted Chapman or DonnyChamberlain—smart boys whose handwriting I could read—and divided my attentionbetween his desk and the teacher’s eyes. Having skipped me, the talent for math concentrated extravagantly in one of my nieces, Amie Wilkinson, a professor at the University of Chicago. From Amie I first heard about Yitang Zhang, asolitary, part-time calculus teacher at the University of New Hampshire whoreceived several prizes, including a MacArthur award in September, for solvinga problem that had been open for more than a hundred and fifty years.

The problem thatZhang chose, in 2010, is from number theory, a branch of pure mathematics. Puremathematics, as opposed to applied mathematics, is done with no practical purposesin mind. It is as close to art and philosophy as it is to engineering. “Myresult is useless for industry,” Zhang said. The British mathematician G. H.Hardy wrote in 1940 that mathematics is, of “all the arts and sciences, themost austere and the most remote.” Bertrand Russell called it a refuge from“the dreary exile of the actual world.” Hardy believed emphatically in theprecise aesthetics of math. A mathematical proof, such as Zhang produced,“should resemble a simple and clear-cut constellation,” he wrote, “not a scattered cluster in the Milky Way.” Edward Frenkel, a math professor at the University of California, Berkeley, says Zhang’s proof has “a renaissancebeauty,” meaning that though it is deeply complex, its outlines are easilyapprehended. The pursuit of beauty in pure mathematics is a tenet. Last year,neuroscientists in Great Britain discovered that the same part of the brainthat is activated by art and music was activated in the brains ofmathematicians when they looked at math they regarded as beautiful.

Zhang’s problem isoften called “bound gaps.” It concerns prime numbers—those which can be dividedcleanly only by one and by themselves: two, three, five, seven, and so on—andthe question of whether there is a boundary within which, on an infinite numberof occasions, two consecutive prime numbers can be found, especially out in theregion where the numbers are so large that it would take a book to print asingle one of them. Daniel Goldston, a professor at San Jose State University;János Pintz, a fellow at the Alfréd Rényi Institute of Mathematics, inBudapest; and Cem Yıldırım, of Boğaziçi University, in Istanbul, workingtogether in 2005, had come closer than anyone else to establishing whetherthere might be a boundary, and what it might be. Goldston didn’t think he’d seethe answer in his lifetime. “I thought it was impossible,” he told me.

Zhang, who alsocalls himself Tom, had published only one paper, to quiet acclaim, in 2001. In2010, he was fifty-five. “No mathematician should ever allow himself to forgetthat mathematics, more than any other art or science, is a young man’s game,”Hardy wrote. He also wrote, “I do not know of an instance of a majormathematical advance initiated by a man past fifty.” Zhang had received a Ph.D.in algebraic geometry from Purdue in 1991. His adviser, T. T. Moh, with whom heparted unhappily, recently wrote a description on his Web site of Zhang as agraduate student: “When I looked into his eyes, I found a disturbing soul, aburning bush, an explorer who wanted to reach the North Pole.” Zhang leftPurdue without Moh’s support, and, having published no papers, was unable tofind an academic job. He lived, sometimes with friends, in Lexington, Kentucky,where he had occasional work, and in New York City, where he also had friendsand occasional work. In Kentucky, he became involved with a group interested inChinese democracy. Its slogan was “Freedom, Democracy, Rule of Law, andPluralism.” A member of the group, a chemist in a lab, opened a Subwayfranchise as a means of raising money. “Since Tom was a genius at numbers,”another member of the group told me, “he was invited to help him.” Zhang keptthe books. “Sometimes, if it was busy at the store, I helped with the cashregister,” Zhang told me recently. “Even I knew how to make the sandwiches, butI didn’t do it so much.” When Zhang wasn’t working, he would go to the libraryat the University of Kentucky and read journals in algebraic geometry andnumber theory. “For years, I didn’t really keep up my dream in mathematics,” hesaid.

“Youmust have been unhappy.”

He shrugged. “Mylife is not always easy,” he said.

With a friend’shelp, Zhang eventually got his position in New Hampshire, in 1999. Havingchosen bound gaps in 2010, he was uncertain of how to find a way into theproblem. “I am thinking, Where is the door?” Zhang said. “In the history ofthis problem, many mathematicians believed that there should be a door, butthey couldn’t find it. I tried several doors. Then I start to worry a littlethat there is no door.”

“Were you everfrustrated?”

“I was tired,” hesaid. “But many times I just feel peaceful. I like to walk and think. This ismy way. My wife would see me and say, ‘What are you doing?’ I said, ‘I’mworking, I’m thinking.’ She didn’t understand. She said, ‘What do youmean?’ ” The problem was so complicated, he said, that “I had no way totell her.”

According to DeaneYang, a professor of mathematics at the New York University Polytechnic Schoolof Engineering, a mathematician at the beginning of a difficult problem is“trying to maneuver his way into a maze. When you try to prove a theorem, youcan almost be totally lost to knowing exactly where you want to go. Often, whenyou find your way, it happens in a moment, then you live to do it again.”

Zhang is deeply reticent, and his manneris formal and elaborately polite. Recently, when we were walking, he said, “MayI use these?” He meant a pair of clip-on shades, which he held toward me as ifI might want to examine them first. His enthusiasm for answering questionsabout himself and his work is slight. About half an hour after I had met himfor the first time, he said, “I have a question.” We had been talking about hischildhood. He said, “How many more questions you going to have?” He dependsheavily on three responses: “Maybe,” “Not so much,” and “Maybe not so much.”From diffidence, he often says “we” instead of “I,” as in, “We may not thinkthis approach is so important.” Occasionally, preparing to speak, he hums.After he published his result, he was invited to spend six months at theInstitute for Advanced Study, in Princeton. The filmmaker George Csicsery hasmade a documentary about Zhang, called “Countingfrom Infinity,” forthe Mathematical Sciences Research Institute, in Berkeley, California. In it,Peter Sarnak, a member of the Institute for Advanced Study, says that one dayhe ran into Zhang and said hello, and Zhang said hello, then Zhang said that itwas the first word he’d spoken to anyone in ten days. Sarnak thought that wasexcessive, even for a mathematician, and he invited Zhang to have lunch once aweek.

Matthew Emerton, aprofessor of math at the University of Chicago, also met Zhang at Princeton. “Iwouldn’t say he was a standard person,” Emerton told me. “He wasn’t gregarious.I got the impression of him being reasonably internal. He had received anotherprize, so the people around him were talking about that. Probably mostmathematicians are very low-key about getting a prize, because you’re not in itfor the prize, but he seemed particularly low-key. It didn’t seem to affect himat all.” Deane Yang attended three lectures that Zhang gave at Columbia in2013. “You expect a guy like that to want to show off or explain how smart heis,” Yang said. “He gave beautiful lectures, where he wasn’t trying to show offat all.” The first talk that Zhang gave on his result was at Harvard, beforethe result was published. A professor there, Shing-Tung Yau, heard aboutZhang’s paper, and invited him. About fifty people showed up. One of them, aHarvard math professor, thought Zhang’s talk was “pretty incomprehensible.” Headded, “The problem is that this stuff is hard to talk about, becauseeverything hinges on some delicate technical understandings.” Another Harvardprofessor, Barry Mazur, told me that he was “moved by his intensity and howbrave and independent he seemed to be.”

In New Hampshire,Zhang works in an office on the third floor of the math and computer-sciencebuilding. His office has a desk, a computer, two chairs, a whiteboard, and somebookshelves. Through a window he looks into the branches of an oak tree. Thebooks on his shelves have titles such as “An Introduction to Hilbert Space” and“Elliptic Curves, Modular Forms, and Fermat’s Last Theorem.” There are alsobooks on modern history and on Napoleon, who fascinates him, and copies ofShakespeare, which he reads in Chinese, because it’s easier than ElizabethanEnglish.

Eric Grinberg, thechairman of the math department at the University of Massachusetts Boston, wasa colleague of Zhang’s in New Hampshire from 2003 to 2010. “Tom was verymodest, very unassuming, never asked for anything,” Grinberg told me. “We knewhe was working on something important. He uses paper and a pencil, but the onlycopy was on his computer, and about once a month I would go in and ask, ‘Do youmind if I make a backup?’ Of course, it’s all in his head anyway. He’s aboveaverage in that.”

Zhang’s memory isabnormally retentive. A friend of his named Jacob Chi said, “I take him to aparty sometimes. He doesn’t talk, he’s absorbing everybody. I say, ‘There’s ahuman decency; you must talk to people, please.’ He says, ‘I enjoy yourconversation.’ Six months later, he can say who sat where and who started aconversation, and he can repeat what they said.”

“I may think socializingis a way to waste time,” Zhang says. “Also, maybe I’m a little shy.”

A few years ago,Zhang sold his car, because he didn’t really use it. He rents an apartmentabout four miles from campus and rides to and from his office with students ona school shuttle. He says that he sits on the bus and thinks. Seven days aweek, he arrives at his office around eight or nine and stays until six orseven. The longest he has taken off from thinking is two weeks. Sometimes hewakes in the morning thinking of a math problem he had been considering when hefell asleep. Outside his office is a long corridor that he likes to walk up anddown. Otherwise, he walks outside.

Zhang met hiswife, to whom he has been married for twelve years, at a Chinese restaurant onLong Island, where she was a waitress. Her name is Yaling, but she callsherself Helen. A friend who knew them both took Zhang to the restaurant andpointed her out. “He asked, ‘What do you think of this girl?’ ” Zhangsaid. Meanwhile, she was considering him. To court her, Zhang went to New Yorkevery weekend for several months. The following summer, she came to NewHampshire. She didn’t like the winters, though, and moved to California, whereshe works at a beauty salon. She and Zhang have a house in San Jose, and hespends school vacations there.

Until Zhang waspromoted to professor, last year, as a consequence of his proof, hisappointment had been tenuous. “I was chair of the math department, and I had togo to him from time to time and remind him this was not a permanent position,”Eric Grinberg said. “We were grateful to him, but it’s not guaranteed. Healways said that he very much appreciated the time he had spent in NewHampshire.”

Zhang devotedhimself to bound gaps for a couple of years without finding a door. “Wecouldn’t see any hope,” he said. Then, on July 3, 2012, in the middle of theafternoon, “within five or ten minutes, the way is open.”

Zhang was inPueblo, Colorado, visiting his friend Jacob Chi, who is a music professor atColorado State University-Pueblo. A few months earlier, Chi had reminded Zhangthat he had promised one day to teach his son, Julius, calculus, and sinceJulius was about to be a senior in high school Chi had called and asked, “Doyou keep your promise?” Zhang spent a month at the Chis’. Each morning, he andJulius worked for about an hour. “He didn’t have a set curriculum,” Julius toldme. “It all just flowed from his memory. He mentioned once that he didn’t haveany numbers in his phone book. He memorized them all.”

Zhang had planneda break from work in Colorado, and hadn’t brought any notes with him. On July3rd, he was walking around the Chis’ back yard. “We live in the mountains, andthe deer come out, and he was smoking a cigarette and watching for the deer,” Chisaid. “No deer came,” Zhang said. “Just walking and thinking, this is my way.”For about half an hour, he walked around at a loss.

In “The Psychologyof Invention in the Mathematical Field,” published in 1945, Jacques Hadamardquotes a mathematician who says, “It often seems to me, especially when I amalone, that I find myself in another world. Ideas of numbers seem to live.Suddenly, questions of any kind rise before my eyes with their answers.” In theback yard, Zhang had a similar experience. “I see numbers, equations, andsomething even—it’s hard to say what it is,” Zhang said. “Something veryspecial. Maybe numbers, maybe equations—a mystery, maybe a vision. I knew that,even though there were many details to fill in, we should have a proof. Then I wentback to the house.”

Zhang didn’t sayanything to Chi about his breakthrough. That evening, Chi was conducting adress rehearsal for a Fourth of July concert in Pueblo, and Zhang went withhim. “After the concert, he couldn’t stop humming ‘The Stars and StripesForever,’ ” Chi said. “All he would say was ‘What a great song.’ ”

I asked Zhang,“Are you very smart?” and he said, “Maybe, a little.” He was born in Shanghaiin 1955. His mother was a secretary in a government office, and his father wasa college professor whose field was electrical engineering. As a small boy, hebegan “trying to know everything in mathematics,” he said. “I became verythirsty for math.” His parents moved to Beijing for work, and Zhang remained inShanghai with his grandmother. The revolution had closed the schools. He spentmost of his time reading math books that he ordered from a bookstore for lessthan a dollar. He was fond of a series whose title he translates as “A HundredThousand Questions Why.” There were volumes for physics, chemistry, biology,and math. When he didn’t understand something, he said, “I tried to solve theproblem myself, because no one could help me.”

Zhang moved toBeijing when he was thirteen, and when he was fifteen he was sent with hismother to the countryside, to a farm, where they grew vegetables. His fatherwas sent to a farm in another part of the country. If Zhang was seen readingbooks on the farm, he was told to stop. “People did not think that math wasimportant to the class struggle,” he said. After a few years, he returned toBeijing, where he got a job in a factory making locks. He began studying totake the entrance exam for Peking University, China’s most respected school: “Ispent several months to learn all the high-school physics and chemistry, andseveral to learn history. It was a little hurried.” He was admitted when he wastwenty-three. “The first year, we studied calculus and linear algebra—it wasvery exciting,” Zhang said. “In the last year, I selected number theory as myspecialty.” Zhang’s professor insisted, though, that he change his major toalgebraic geometry, his own field. “I studied it, but I didn’t really like it,”Zhang said. “That time in China, still the idea was like this: the individualhas to follow the interest of the whole group, the country. He thoughtalgebraic geometry was more important than number theory. He forced me. He wasthe university president, so he had the authority.”

Duringthe summer of 1984, T. T. Moh visited Peking University from Purdue and invitedZhang and several other students, recommended to him by Chinese professors, todo graduate work in his department. One of Moh’s specialties is the Jacobianconjecture, and Zhang was eager to work on it. The Jacobian conjecture, aproblem in algebraic geometry that was introduced in 1939 and is stillunsolved, stipulates certain simple conditions that, if satisfied, enablesomeone to solve a series of complicated equations. It is acknowledged as beingbeyond the capacities of a graduate student and approachable by only the mostaccomplished algebraic geometrists. A mathematician described it to me as a“disaster problem,” for the trouble it has caused. For his thesis, Zhangsubmitted a weak form of the conjecture, meaning that he attempted to provesomething implied by the conjecture, rather than to prove the conjectureitself.

After Zhangreceived his doctorate, he told Moh that he was returning to number theory. “Iwas not the happiest,” Moh wrote me. “However, I was for the student’s right tochange fields, so I kept my smile and said bye to him. For the past 22 years, Iknew nothing about him.”

After graduating,most of the Chinese students went into either computer science or finance. Oneof them, Perry Tang, who had known Zhang in China, took a job at Intel. In1999, he called Zhang. “I thought it was unfair for him not to have aprofessional job,” Tang said. He and Zhang had a classmate at Peking Universitywho had become a professor of math at the University of New Hampshire, and whenthe friend said that he was looking for someone to teach calculus Tangrecommended Zhang. “He decided to try him at a temporary position,” Tang said.

Zhang finished “Bounded Gaps BetweenPrimes” in late 2012; then he spent a few months methodically checking eachstep, which he said was “very boring.” On April 17, 2013, without tellinganyone, he sent the paper to Annals of Mathematics, widely regarded as the profession’s mostprestigious journal. In the Annals archives are unpublished papersclaiming to have solved practically every math problem that anyone has everthought of, and others that don’t really exist. Some are from people who “knowa lot of math, then they go insane,” a mathematician told me. Such people oftenclaim that everyone else who has attacked the problem is wrong. Or theyannounce that they have solved several problems at once, or “they say they havesolved a famous problem along with some unified-field theory in physics,” themathematician said. Journals such as Annals are always skeptical of work from someonethey have never heard of.

In 2013, Annals received nine hundred and fifteenpapers and accepted thirty-seven. The wait between acceptance and publicationis typically around a year. When a paper arrives, “it is read quickly, forworthiness,” Nicholas Katz, the Princeton professor who is the journal’seditor, told me, and then there is a deep reading that can take months. “Thepaper I can’t evaluate off the top of my head, my role is to know whom to ask,”Katz said. “In this case, the person wrote back pretty quickly to say, ‘If thisis correct, it’s really fantastic. But you should be careful. This guy posted apaper once, and it was wrong. He never published it, but he didn’t take itdown, either.’ ” The reader meant a paper that Zhang posted on the Website arxiv.org, where mathematicians often post results before submitting themto a journal, in order to have them seen quickly. Zhang posted a paper in 2007that fell short of a proof. It concerned another famous problem, theLandau-Siegel zeros conjecture, and he left it up because he hopes to correctit.

Katz sent “BoundedGaps Between Primes” to a pair of readers, who are called referees. One of themwas Henryk Iwaniec, a professor at Rutgers, whose work was among that whichZhang had drawn on. “I glanced for a few minutes,” Iwaniec told me. “My firstimpression was: So many claims have become wrong. And I thought, I have otherwork to do. Maybe I’ll postpone it. Remember that he was an unknown guy. Then Igot a phone call from a friend, and it happened he was also reading the paper.We were going to be together for a week at the Institute for Advanced Study,and the intention was to do other work, but we were interrupted with this paperto read.”

Iwaniec and hisfriend, John Friedlander, a professor at the University of Toronto, read withincreasing attention. “In these cases, you don’t read A to Z,” Iwaniec said.“You look first at where is the idea. There had been nothing written on thesubject since 2005. The problem was too difficult to solve. As we read more andmore, the chance that the work was correct was becoming really great. Maybe twodays later, we started looking for completeness, for connections. A few dayspassed, we’re checking line by line. The job is no longer to say the work isfine. We are looking to see if the paper is truly correct.”

After a few weeks, Iwaniec and Friedlanderwrote to Katz, “We have completed our study of the paper ‘Bounded Gaps BetweenPrimes’ by Yitang Zhang.” They went on, “The main results are of the first rank.The author has succeeded to prove a landmark theorem in the distribution ofprime numbers.” And, “Although we studied the arguments very thoroughly, wefound it very difficult to spot even the smallest slip. . . . Weare very happy to strongly recommend acceptance of the paper for publication inthe Annals.”

Once Zhang heardfrom Annals, he called his wife in San Jose. “I say,‘Pay attention to the media and newspapers,’ ” he said. “ ‘You maysee my name,’ and she said, ‘Are you drunk?’ ”

No formulapredicts the occurrence of primes—they behave as if they appear randomly.Euclid proved, in 300 B.C., that there is an infinite number of primes. If youimagine a line of all the numbers there are, with ordinary numbers in green andprime numbers in red, there are many red numbers at the beginning of the line:2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, and 47 are the primes belowfifty. There are twenty-five primes between one and a hundred; 168 between oneand a thousand; and 78,498 between one and a million. As the primes get larger,they grow scarcer and the distances between them, the gaps, grow wider.

Prime numbers haveso many novel qualities, and are so enigmatic, that mathematicians have grownfetishistic about them. Twin primes are two apart. Cousin primes are fourapart, sexy primes are six apart, and neighbor primes are adjacent at somegreater remove. From “Prime Curios!,” by Chris Caldwell and G. L. Honaker, Jr.,I know that an absolute prime is prime regardless of how its digits arearranged: 199; 919; 991. A beastly prime has 666 in the center. The number700666007 is a beastly palindromic prime, since it reads the same forward andbackward. A circular prime is prime through all its cycles or formulations:1193, 1931, 9311, 3119. There are Cuban primes, Cullen primes, and curved-digitprimes, which have only curved numerals—0, 6, 8, and 9. A prime from which youcan remove numbers and still have a prime is a deletable prime, such as 1987.An emirp is prime even when you reverse it: 389, 983. Gigantic primes have morethan ten thousand digits, and holey primes have only digits with holes (0, 4, 6,8, and 9). There are Mersenne primes; minimal primes; naughty primes, which aremade mostly from zeros (naughts); ordinary primes; Pierpont primes; plateauprimes, which have the same interior numbers and smaller numbers on the ends,such as 1777771; snowball primes, which are prime even if you haven’t finishedwriting all the digits, like 73939133; Titanic primes; Wagstaff primes;Wall-Sun-Sun primes; Wolstenholme primes; Woodall primes; and Yarboroughprimes, which have neither a 0 nor a 1.

“Bounded GapsBetween Primes” is a back-door attack on the twin-prime conjecture, which wasproposed in the nineteenth century and says that, no matter how far you travelon the number line, even as the gap widens between primes you will alwaysencounter a pair of primes that are separated by two. The twin-prime conjectureis still unsolved. Euclid’s proof established that there will always be primes,but it says nothing about how far apart any two might be. Zhang establishedthat there is a distance within which, on an infinite number of occasions,there will always be two primes.

“You have toimagine this coming from nothing,” Eric Grinberg said. “We simply didn’t know.It is like thinking that the universe is infinite, unbounded, and finding ithas an end somewhere.” Picture it as a ruler that might be applied to the lineof green and red numbers. Zhang chose a ruler of a length of seventy million,because a number that large made it easier to prove his conjecture. (If he hadbeen able to prove the twin-prime conjecture, the number for the ruler wouldhave been two.) This ruler can be moved along the line of numbers and enclosetwo primes an infinite number of times. Something that holds for infinitelymany numbers does not necessarily hold for all. For example, an infinite numberof numbers are even, but an infinite number of numbers are not even, becausethey are odd. Similarly, this ruler can also be moved along the line of numbersan infinite number of times and not enclose two primes.

From Zhang’s result,a deduction can be made, which is that there is a number smaller than seventymillion which precisely defines a gap separating an infinite number of pairs ofprimes. You deduce this, a mathematician told me, by means of the pigeonholeprinciple. You have an infinite number of pigeons, which are pairs of primes,and you have seventy million holes. There is a hole for primes separated bytwo, by three, and so on. Each pigeon goes in a hole. Eventually, one hole willhave an infinite number of pigeons. It isn’t possible to know which one. Theremay even be many, there may be seventy million, but at least one hole will havean infinite number of pigeons.

Having discoveredthat there is a gap, Zhang wasn’t interested in finding the smallest numberdefining the gap. This was work that he regarded as a mere technical problem, atype of manual labor—“ambulance chasing” is what a prominent mathematiciancalled it. Nevertheless, within a week of Zhang’s announcement mathematiciansaround the world began competing to find the lowest number. One of theobservers of their activity was Terence Tao, a professor at U.C.L.A. Tao hadthe idea for a coöperative project in which mathematicians would work to lowerthe number rather than “fighting to snatch the lead,” he told me.

The project,called Polymath8, started in March of 2013 and continued for about a year.Incrementally, relying also on work by a young British mathematician namedJames Maynard, the participants reduced the bound to two hundred and forty-six.“There are several problems with going lower,” Tao said. “More and morecomputer power is required—someone had a high-powered computer running for twoweeks to get that calculation. There were also theoretical problems. Withcurrent methods, we can never get better than six, because of something calledthe parity problem, which no one knows how to get past.” The parity problemsays that primes with certain behaviors can’t be detected with current methods.“We never strongly believed we would get to two and prove the twin-primeconjecture, but it was a fun journey,” Tao said.

“Is there a talent a mathematician should have?”

“Concentration,”Zhang said. We were walking across the campus in a light rain. “Also, youshould never give up in your personality,” he continued. “Maybe something infront of you is very complicated, it’s lengthy, but you should be able to pickup the major points by your intuition.”

When we reachedZhang’s office, I asked how he had found the door into the problem. On awhiteboard, he wrote, “Goldston-Pintz-Yıldırım”and“Bombieri-Friedlander-Iwaniec.” He said, “The first paper is on bound gaps, andthe second is on the distribution of primes in arithmetic progressions. Icompare these two together, plus my own innovations, based on the years of readingin the library.”

When I asked PeterSarnak how Zhang had arrived at his result, he said, “What he did was look wayout of reach. Maybe forty years ago the problem appeared hopeless, but in 2005Goldston-Pintz-Yıldırım put it at the doorstep. Everybody thought, Now we’revery close, but by 2011 no one was making any progress. Bombieri, Friedlander,and Iwaniec had the other important work, but it looked like you couldn’tcombine their ideas with Goldston. Their work was just not flexible enough tojive—it insisted on some side conditions. Then Zhang comes along. A lot ofpeople use theorems like a computer. They think, If it is correct, then good,I’ll use it. You couldn’t use the Bombieri-Friedlander-Iwaniec, though, becauseit wasn’t flexible enough. You have to take my word, because even to a seriousmathematician this would be difficult to explain. Zhang understood thetechniques deeply enough so as to be able to modifyBombieri-Friedlander-Iwaniec and cross this bridge. This is the mostsignificant thing about what he has done mathematically. He’s made theBombieri-Friedlander-Iwaniec technique about the distribution of prime numbersa tool for any kind of study of primes. A development that began in theeighteen-hundreds continued through him.”

“Our conditionsneeded to be relaxed,” Iwaniec told me. “We tried, but we couldn’t remove them.We didn’t try long, because after failing you just start thinking there is somekind of natural barrier, so we gave up.”

I asked if he wassurprised by Zhang’s result. “What Zhang did was sensational,” he said. “Hiswork is a masterpiece. When you talk of number theory, a lot of the beauty isthe machinery. Zhang somehow completely understood the situation, even thoughhe was working alone. That’s how he surprised. He just amazingly pushed furthersome of the arguments in these papers.”

Zhang used a verycomplicated form of a simple device for finding primes called a sieve, inventedby a Greek named Eratosthenes, a contemporary of Archimedes. To use a simplesieve to find the primes less than a thousand, say, you write down all thenumbers, then cross out the multiples of two, which can’t be prime, since theyare even. Then you cross out the multiples of three, then five, and so on. Youhave to go only as far as the multiples of thirty-one. Zhang used a differentsieve from the one that others had used. The previous sieve excluded numbersonce they grew too far apart. With it, Goldston, Pintz, and Yıldırım had provedthat there were always two primes separated by something less than the averagedistance between primes that large. What they couldn’t identify was a precisegap. Zhang succeeded partly by making the sieve less selective.

I asked Zhang ifhe was working on something new. “Maybe two or three problems I would like tosolve,” he said. “Bound gaps is successful, but still I have something else.”

“Will it be asimportant?”

“Yes.”

According to othermathematicians, Zhang is working on his incomplete result for the Landau-Siegelzeros conjecture. “If he succeeds, it would be much more dramatic,” PeterSarnak said. “We don’t know how close he is, but he’s proved that he’s agenius. There’s no question about that. He’s also proved that he can speak withsomething over many years. Based on that, his chances are not zero. They’repositive.”

“Many people havetried that problem,” Iwaniec said. “He’s a private guy. Nothing is rushed. Ifit takes him another ten years, that’s fine with him. Unless you tackle aproblem that’s already solved, which is boring, or one whose solution is clearfrom the beginning, mostly you are stuck. But Zhang is willing to be stuck muchlonger.”

Zhang’spreference for undertaking only ambitious problems is rare. The pursuit oftenure requires an academic to publish frequently, which often means refiningone’s work within a field, a task that Zhang has no inclination for. He doesnot appear to be competitive with other mathematicians, or resentful abouthaving been simply a teacher for years while everyone else was a professor. Noone who knows him thinks that he is suited to a tenure-track position. “I thinkwhat he did was brilliant,” Deane Yang told me. “If you become a good calculusteacher, a school can become very dependent on you. You’re cheap and reliable,and there’s no reason to fire you. After you’ve done that a couple of years,you can do it on autopilot; you have a lot of free time to think, so long asyou’re willing to live modestly. There are people who try to work nontenurejobs, of course, but usually they’re nuts and have very dysfunctionalpersonalities and lives, and are unpleasant to deal with, because they feeldisrespected. Clearly, Zhang never felt that.”

Oneday, I arrived at Zhang’s office as he was making tea. There was a piece ofpaper on his desk with equations on it and a pen on top of the paper. Zhang hadan envelope in one hand. “I had a letter from an old friend,” he said. “We havebeen separated for many years, and now he found me.”

He took a pair ofscissors from a drawer and cut open the envelope so slowly that he seemed to beperforming a ritual. The letter was written in Chinese characters. He sat onthe edge of his chair and read slowly. He put the letter down and took from theenvelope a photograph of a man and a woman and a child on a sofa with a curtainin the background. He returned to reading the letter and then he put it back inthe envelope and in the drawer and closed the drawer. “His new address is inQueens,” he said. Then he picked up his tea and blew on it and faced me,looking at me over the top of the cup like someone peering over a wall.

I asked aboutHardy’s observations regarding age—Hardy also wrote, “A mathematician may stillbe competent enough at sixty, but it is useless to expect him to have originalideas.”

“This may not apply to me,” Zhang said. Heput his tea on the desk and looked out the window. “Still I think I haveintuition,” he said. “Still I am confident of myself. Still I have some othervisions.”