接受《纽约客》专访时，张益唐59岁。仅仅两年前，他不过是个美国非一流大学的普通讲师，只发表过两篇论文，没有研究经费，曾有近十年的时间找不到学术职位，“流浪”美国各州，不时借住朋友家安身。
2013年5月，他因出色地证明了一个关于素数分布的“里程碑式的定理”而蜚声全球。英国著名数学家哈代说，数学比起其他技艺和科学来，更像是“年轻人的游戏”，没有哪一个重大成就是50岁之后提出来的。然而张益唐用天才般的工作证明：年龄、职位、论文统统不是登顶的“标配”。
2月2日，《纽约客》杂志正式刊发特约撰稿人亚历克•威尔金森（AlecWilkinson）专访张益唐的长文。《赛先生》求教一流数论专家，补正部分内容，力求准确编译，以飨国内读者。

张益唐证明了什么
张益唐所做的工作通常被称作“素数间的有界距离”，是“孪生素数”猜想证明的弱形式。
所谓“素数”，又称“质数”，是指只能被1和它本身整除的数字，例如：2、3、5、7等等。但随着数字增大，素数在数轴上的分布越来越稀疏。想像一条数轴，普通数字是绿色的，素数是红色的。轴线开始时有许多红色的数字：2、3、5、7、11、13、17、19、23、29、31、41、43和47，它们都是小于50的素数。在1-100之间有25个素数，1到1000之间有168个素数，1到100万之间有78498个素数。素数越来越大时，它们变得越来越稀少，素数与素数间的平均距离越来越大。那么，相邻两个素数之间的距离是否是有限的呢？特别是当数字趋于无穷大时，一个数字的位数之多需要一本书的厚度才能写下，此时是否还能找到相邻的两个素数呢？
没有一个方程式可以预言素数的分布特征——它们看起来非常随机。欧几里得在公元前300年证明存在无穷多个素数，但并没有证明两个素数之间的距离可能是多远。他曾大胆猜想：存在无穷多对之差为2的素数。由于人们把这种素数对称为“孪生素数”，如（3,5），（11,13），因此这一猜想被称作“孪生素数猜想”。
1849年，法国数学家阿尔方•波利尼亚克提出了更一般的猜想（即“波利尼亚克猜想”）：对所有正整数k，存在无穷多个素数对（p，p+2k）。k=1时就是孪生素数猜想，而k等于其他正整数时就称为弱孪生素数猜想。
1900年，德国数学家大卫•希尔伯特在巴黎举行的第2届国际数学家大会上发表题为《数学问题》的著名讲演。他根据过去特别是19世纪数学的研究成果和发展趋势，提出了23个最重要的数学问题（通称“希尔伯特问题”）；孪生素数猜想是希尔伯特问题的第8个的一部分（和“孪生素数猜想”一起被提出的，是著名的“哥德巴赫猜想”和“黎曼猜想”）。
张益唐的论文《素数间的有界距离》就是“孪生素数猜想”的弱化版，他证明了在数字趋于无穷大的过程中，存在无穷多个之差小于7000万的素数对。
此前最接近证明孪生素数猜想的一次努力，是圣何塞州立大学的教授丹尼尔•戈德斯通（Daniel Goldston）、布达佩斯阿尔弗雷德•莱利（Alfréd Rényi）数学研究所研究员平兹（János Pintz）和伊斯坦布尔海峡大学的伊尔迪里姆（CemYildirim）教授于2005年共同开展的一项工作。不过，一直到2011年，关于孪生素数猜想的研究仍没有取得任何进展。Goldston认为，他在有生之年可能都看不到答案，“我曾以为解开这个难题是不可能的了。”
尽管张益唐得到的7000万这个结果看起来与2还有很大差距，但国际数学界公认这是一项伟大的成就。英国《自然》杂志称张益唐的工作为一个“重要的里程碑”。美国数学家丹尼尔•戈德斯坦说：“从7000万到2的距离相比从无穷大到7000万的距离来说是微不足道的。”他认为，每缩小一段范围，都是在获得终极答案（k=1）道路上的一个脚印。
“你必须想像这完全是从无到有，”麻省大学波士顿分校的数学系主任埃里克•格林贝格（EricGrinberg）说。“我们确实不知道。这就像我们以为宇宙无限大，没有界限，却发现它在某个地方存在终点。”想象有一把度量绿色与红色数字的尺子。张益唐选择了一把长度为7000万的尺子，因为这么大的数字更容易证明他的猜想。（如果他已能证明孪生素数猜想，这把尺子的长度就是2。）我们可以拿这把尺子沿数轴移动，无数次地将两个素数圈起来。但圈住无穷多个数不一定就是圈住了所有的数，因为有一些情况，比如有无穷多个数是偶数，但还有无穷多个数是奇数。同样道理，这把尺子也能沿着数轴移动无数次时，但圈不到两个素数。
从张益唐的结果来看，他的推导是成立的，存在无穷多个之差小于7000万的素数对。接受《纽约客》采访的一位数学家解释说，这是根据鸽巢原理推出的。假设有7000万个鸽巢和无穷多只鸽子，每只鸽子代表一个素数对。把之差为2的素数对（鸽子）放进一个鸽巢，之差为3的放进另一个鸽巢，以此类推，把所有间隔不同的素数对（鸽子）都放进一个鸽巢。最后，会有放了无穷多只鸽子的鸽巢，但无法知道具体是哪一个鸽巢有无穷多只鸽子，不过至少有一个鸽巢里有无穷多只鸽子。

引来全球数学家开展竞赛
发现存在无穷多个素数对的那个最大的素数间隔后，张益唐对找到间隔的最小数并不感兴趣。他觉得这种工作纯粹只是个技术活，一种体力劳动——一位杰出的数学家把这种行为叫做“追赶救护车”。
不过，张益唐研究成果面世不到一周，就引来全世界数学家的围观，他们竞相刷新这个最小距离数。围观者当中就有31岁即获得“菲尔茨”奖（数学界的最高荣誉）的著名数学家陶哲轩（TerenceTao，生于澳大利亚的华人家庭），他现在是加州大学洛杉矶分校的教授。他希望建立一个合作项目，让数学家一起工作去寻找更小的数字，而不是“抢夺领先的位置”。
他建立的这个项目名为Polymath-8（博学者8号难题），于2013年6月正式启动，持续了大约一年时间。凭借英国一位年轻数学家JamesMaynard的贡献，项目参与者逐渐将无穷多个素数的差缩减到246。但“数字减小的同时也发现一些问题，”陶哲轩说，“需要越来越多的计算机资源——有人为了做一个计算要让一台高性能的计算机运行两周。此外也有些理论上的问题。用现在的方法，我们不可能得到比6（即k=3）更好的数字。因为存在奇偶校正问题，没有人知道如何绕过这个槛。”陶哲轩说：“我们并没有强烈地认为，我们可以把数值减小到2，从而证出孪生素数猜想，但这是段有趣的旅程。”

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 Donny Chamberlain—smart boys whose handwriting I could read—and divided my attention between 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, a solitary, part-time calculus teacher at the University of New Hampshire who received several prizes, including a MacArthur award in September, for solving a problem that had been open for more than a hundred and fifty years.
The problem that Zhang chose, in 2010, is from number theory, a branch of pure mathematics. Pure mathematics, as opposed to applied mathematics, is done with no practical purposes in mind. It is as close to art and philosophy as it is to engineering. “My result is useless for industry,” Zhang said. The British mathematician G. H. Hardy wrote in 1940 that mathematics is, of “all the arts and sciences, the most austere and the most remote.” Bertrand Russell called it a refuge from “the dreary exile of the actual world.” Hardy believed emphatically in the precise 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 renaissance beauty,” meaning that though it is deeply complex, its outlines are easily apprehended. The pursuit of beauty in pure mathematics is a tenet. Last year, neuroscientists in Great Britain discovered that the same part of the brain that is activated by art and music was activated in the brains of mathematicians when they looked at math they regarded as beautiful.

Zhang’s problem is often called “bound gaps.” It concerns prime numbers—those which can be divided cleanly only by one and by themselves: two, three, five, seven, and so on—and the question of whether there is a boundary within which, on an infinite number of occasions, two consecutive prime numbers can be found, especially out in the region where the numbers are so large that it would take a book to print a single one of them. Daniel Goldston, a professor at San Jose State University; János Pintz, a fellow at the Alfréd Rényi Institute of Mathematics, in Budapest; and Cem Yıldırım, of Boğaziçi University, in Istanbul, working together in 2005, had come closer than anyone else to establishing whether there might be a boundary, and what it might be. Goldston didn’t think he’d see the answer in his lifetime. “I thought it was impossible,” he told me.
Zhang, who also calls himself Tom, had published only one paper, to quiet acclaim, in 2001. In 2010, he was fifty-five. “No mathematician should ever allow himself to forget that 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 major mathematical 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 he parted unhappily, recently wrote a description on his Web site of Zhang as a graduate student: “When I looked into his eyes, I found a disturbing soul, a burning bush, an explorer who wanted to reach the North Pole.” Zhang left Purdue without Moh’s support, and, having published no papers, was unable to find 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 friends and occasional work. In Kentucky, he became involved with a group interested in Chinese democracy. Its slogan was “Freedom, Democracy, Rule of Law, and Pluralism.” A member of the group, a chemist in a lab, opened a Subway franchise 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 kept the books. “Sometimes, if it was busy at the store, I helped with the cash register,” Zhang told me recently. “Even I knew how to make the sandwiches, but I didn’t do it so much.” When Zhang wasn’t working, he would go to the library at the University of Kentucky and read journals in algebraic geometry and number theory. “For years, I didn’t really keep up my dream in mathematics,” he said.

Matthew Emerton, a professor of math at the University of Chicago, also met Zhang at Princeton. “I wouldn’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 another prize, so the people around him were talking about that. Probably most mathematicians are very low-key about getting a prize, because you’re not in it for the prize, but he seemed particularly low-key. It didn’t seem to affect him at all.” Deane Yang attended three lectures that Zhang gave at Columbia in 2013. “You expect a guy like that to want to show off or explain how smart he is,” Yang said. “He gave beautiful lectures, where he wasn’t trying to show off at all.” The first talk that Zhang gave on his result was at Harvard, before the result was published. A professor there, Shing-Tung Yau, heard about Zhang’s paper, and invited him. About fifty people showed up. One of them, a Harvard math professor, thought Zhang’s talk was “pretty incomprehensible.” He added, “The problem is that this stuff is hard to talk about, because everything hinges on some delicate technical understandings.” Another Harvard professor, Barry Mazur, told me that he was “moved by his intensity and how brave and independent he seemed to be.”
In New Hampshire, Zhang works in an office on the third floor of the math and computer-science building. His office has a desk, a computer, two chairs, a whiteboard, and some bookshelves. Through a window he looks into the branches of an oak tree. The books on his shelves have titles such as “An Introduction to Hilbert Space” and “Elliptic Curves, Modular Forms, and Fermat’s Last Theorem.” There are also books on modern history and on Napoleon, who fascinates him, and copies of Shakespeare, which he reads in Chinese, because it’s easier than Elizabethan English.

Zhang met his wife, to whom he has been married for twelve years, at a Chinese restaurant on Long Island, where she was a waitress. Her name is Yaling, but she calls herself Helen. A friend who knew them both took Zhang to the restaurant and pointed her out. “He asked, ‘What do you think of this girl?’ ” Zhang said. Meanwhile, she was considering him. To court her, Zhang went to New York every weekend for several months. The following summer, she came to New Hampshire. She didn’t like the winters, though, and moved to California, where she works at a beauty salon. She and Zhang have a house in San Jose, and he spends school vacations there.
Until Zhang was promoted to professor, last year, as a consequence of his proof, his appointment had been tenuous. “I was chair of the math department, and I had to go 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. He always said that he very much appreciated the time he had spent in New Hampshire.”

asked Zhang, “Are you very smart?” and he said, “Maybe, a little.” He was born in Shanghai in 1955. His mother was a secretary in a government office, and his father was a college professor whose field was electrical engineering. As a small boy, he began “trying to know everything in mathematics,” he said. “I became very thirsty for math.” His parents moved to Beijing for work, and Zhang remained in Shanghai with his grandmother. The revolution had closed the schools. He spent most of his time reading math books that he ordered from a bookstore for less than a dollar. He was fond of a series whose title he translates as “A Hundred Thousand Questions Why.” There were volumes for physics, chemistry, biology, and math. When he didn’t understand something, he said, “I tried to solve the problem myself, because no one could help me.”
Zhang moved to Beijing when he was thirteen, and when he was fifteen he was sent with his mother to the countryside, to a farm, where they grew vegetables. His father was sent to a farm in another part of the country. If Zhang was seen reading books on the farm, he was told to stop. “People did not think that math was important to the class struggle,” he said. After a few years, he returned to Beijing, where he got a job in a factory making locks. He began studying to take the entrance exam for Peking University, China’s most respected school: “I spent several months to learn all the high-school physics and chemistry, and several to learn history. It was a little hurried.” He was admitted when he was twenty-three. “The first year, we studied calculus and linear algebra—it was very exciting,” Zhang said. “In the last year, I selected number theory as my specialty.” Zhang’s professor insisted, though, that he change his major to algebraic 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 individual has to follow the interest of the whole group, the country. He thought algebraic geometry was more important than number theory. He forced me. He was the university president, so he had the authority.”
During the summer of 1984, T. T. Moh visited Peking University from Purdue and invited Zhang and several other students, recommended to him by Chinese professors, to do graduate work in his department. One of Moh’s specialties is the Jacobian conjecture, and Zhang was eager to work on it. The Jacobian conjecture, a problem in algebraic geometry that was introduced in 1939 and is still unsolved, stipulates certain simple conditions that, if satisfied, enable someone to solve a series of complicated equations. It is acknowledged as being beyond the capacities of a graduate student and approachable by only the most accomplished algebraic geometrists. A mathematician described it to me as a “disaster problem,” for the trouble it has caused. For his thesis, Zhang submitted a weak form of the conjecture, meaning that he attempted to prove something implied by the conjecture, rather than to prove the conjecture itself.

Katz sent “Bounded Gaps Between Primes” to a pair of readers, who are called referees. One of them was Henryk Iwaniec, a professor at Rutgers, whose work was among that which Zhang had drawn on. “I glanced for a few minutes,” Iwaniec told me. “My first impression was: So many claims have become wrong. And I thought, I have other work to do. Maybe I’ll postpone it. Remember that he was an unknown guy. Then I got 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 paper to read.”
Iwaniec and his friend, John Friedlander, a professor at the University of Toronto, read with increasing 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 the subject since 2005. The problem was too difficult to solve. As we read more and more, the chance that the work was correct was becoming really great. Maybe two days later, we started looking for completeness, for connections. A few days passed, we’re checking line by line. The job is no longer to say the work is fine. We are looking to see if the paper is truly correct.”
After a few weeks, Iwaniec and Friedlander wrote to Katz, “We have completed our study of the paper ‘Bounded Gaps Between Primes’ 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 of prime numbers.” And, “Although we studied the arguments very thoroughly, we found it very difficult to spot even the smallest slip. . . . We are very happy to strongly recommend acceptance of the paper for publication in the Annals.”
Once Zhang heard from Annals, he called his wife in San Jose. “I say, ‘Pay attention to the media and newspapers,’ ” he said. “ ‘You may see my name,’ and she said, ‘Are you drunk?’ ”

Bounded Gaps Between Primes” is a back-door attack on the twin-prime conjecture, which was proposed in the nineteenth century and says that, no matter how far you travel on the number line, even as the gap widens between primes you will always encounter a pair of primes that are separated by two. The twin-prime conjecture is 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 established that there is a distance within which, on an infinite number of occasions, there will always be two primes.
“You have to imagine this coming from nothing,” Eric Grinberg said. “We simply didn’t know. It is like thinking that the universe is infinite, unbounded, and finding it has an end somewhere.” Picture it as a ruler that might be applied to the line of 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 had been able to prove the twin-prime conjecture, the number for the ruler would have been two.) This ruler can be moved along the line of numbers and enclose two primes an infinite number of times. Something that holds for infinitely many numbers does not necessarily hold for all. For example, an infinite number of numbers are even, but an infinite number of numbers are not even, because they are odd. Similarly, this ruler can also be moved along the line of numbers an 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 seventy million which precisely defines a gap separating an infinite number of pairs of primes. You deduce this, a mathematician told me, by means of the pigeonhole principle. 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 by two, by three, and so on. Each pigeon goes in a hole. Eventually, one hole will have an infinite number of pigeons. It isn’t possible to know which one. There may even be many, there may be seventy million, but at least one hole will have an infinite number of pigeons.

“Our conditions needed 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 some kind of natural barrier, so we gave up.”
I asked if he was surprised by Zhang’s result. “What Zhang did was sensational,” he said. “His work is a masterpiece. When you talk of number theory, a lot of the beauty is the machinery. Zhang somehow completely understood the situation, even though he was working alone. That’s how he surprised. He just amazingly pushed further some of the arguments in these papers.”
Zhang used a very complicated form of a simple device for finding primes called a sieve, invented by a Greek named Eratosthenes, a contemporary of Archimedes. To use a simple sieve to find the primes less than a thousand, say, you write down all the numbers, then cross out the multiples of two, which can’t be prime, since they are even. Then you cross out the multiples of three, then five, and so on. You have to go only as far as the multiples of thirty-one. Zhang used a different sieve from the one that others had used. The previous sieve excluded numbers once they grew too far apart. With it, Goldston, Pintz, and Yıldırım had proved that there were always two primes separated by something less than the average distance between primes that large. What they couldn’t identify was a precise gap. Zhang succeeded partly by making the sieve less selective.
I asked Zhang if he was working on something new. “Maybe two or three problems I would like to solve,” he said. “Bound gaps is successful, but still I have something else.”
“Will it be as important?”
“Yes.”
According to other mathematicians, Zhang is working on his incomplete result for the Landau-Siegel zeros conjecture. “If he succeeds, it would be much more dramatic,” Peter Sarnak said. “We don’t know how close he is, but he’s proved that he’s a genius. There’s no question about that. He’s also proved that he can speak with something over many years. Based on that, his chances are not zero. They’re positive.”
“Many people have tried that problem,” Iwaniec said. “He’s a private guy. Nothing is rushed. If it takes him another ten years, that’s fine with him. Unless you tackle a problem that’s already solved, which is boring, or one whose solution is clear from the beginning, mostly you are stuck. But Zhang is willing to be stuck much longer.”

很牛啊，还有其他重要问题研究，神人啊。

涨姿势了