12年级高中生解决世界数论难题,被MIT麻省理工录取

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咱村有些家长还对数论问题不屑一顾,切得晓伐? :buttrock:


高中生有志向,刻苦专研,才能荡舟查尔斯河呀。:zhichi:


只用300小时,17岁高中生解开困扰数学家27年难题​

文章来源: 量子位QbitAI 于 2022-10-15 19:42:15 - 新闻取自各大新闻媒体,新闻内容并不代表本网立场!
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只因在电视上多看了一眼数学家张益唐的纪录片,中学生开始沉迷数论,还独立发表了一篇“博士级别”数学论文。

解决的数学问题,还是曾难住3位正经数学家整整27年的那种。

当这3位数学家中的卡尔·波梅兰斯(Carl Pomerance)本人,看到这篇出自17岁少年之手的论文时,也不禁感慨:


这是一篇会让任何一位数学家都为之自豪的论文。

e611a8a2221cb457b7c440d71e4acda9.jpg


△图源:Quantamagazine

少年名叫丹尼尔·拉森(Daniel Larsen)。

就在今年,他这篇有关卡迈克尔数的论文,已经正式发表在《国际数学研究通告》上,还为他赢得了10万美元(约合人民币72万元)奖学金。


他本人也告别高中生活,成为了麻省理工学院数学专业的大一新生。

aa3f9d791326905fdcaeefb63bc44161.jpg


事实上,丹尼尔本人在家乡早已是小有名气的“神童”:

一家子都是数学家,他本人13岁就在《纽约时报》上发表过填字游戏,是这个项目史上最年轻作者。

看上去一路顺风顺水,但在与外界交流时,他却说,自己做啥都“像是在挣扎”。

还自曝:喜欢走捷径。

因张益唐“入坑”数论,屡挫屡战

正如开头所说,丹尼尔和数论的缘分,始于一部有关张益唐的纪录片。

张益唐是传奇美籍华裔数学家,因“孪生素数猜想”一举成名。但在功成名就之前,张益唐的经历可谓半生潦倒:

博士毕业后因未拿到导师推荐信,学术道路坎坷,甚至不得不靠快餐店收银员等工作糊口。

但即便如此,张益唐也并未放弃对数论的钻研,直到58岁终于大器晚成。

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△张益唐,图源:北京大学招生网

或许正是这种对数论的执着触动了丹尼尔。他开始抑制不住地在脑海中不断思考数论。

一开始,他同样把目光瞄向了“孪生素数猜想”:张益唐的成果首次证明了存在无穷多对间隔有限的质数,但他证明的间隔是7000万,这个数字仍可以进一步缩小。

陶哲轩和今年的新晋菲尔兹奖得主詹姆斯·梅纳德,就都做过这方面的工作。

虽然只是一名中学生,丹尼尔还是试图通过阅读张益唐、陶哲轩和梅纳德在这一问题上发表的论文,搞清楚背后的数学原理。

但最终他不得不承认:

这对我来说几乎是不可能的。他们的论文太复杂了。

尽管如此,丹尼尔并没有被当场劝退。相反,他一头扎进了数论论文的海洋,坚持寻找能激发他灵感的那一个“巨人的肩膀”。

终于在2021年2月,17岁的丹尼尔·拉森和卡迈克尔数邂逅了。

72a4349fa5ca58ccf4c43a270357fc1f.jpg


△图源:THE SOCIETY FOR SCIENCE

300小时攻克数论难题

卡迈克尔数的定义是:

一个正合数n,对于所有跟n互质的整数b,b^n-b都是n的倍数,那么n就是一个卡迈克尔数。

根据费马小定理,所有质数都具备这种特质,因此卡迈克尔数又被称为“伪质数”。

在1899年,数学家Alwin Korselt还提出了一种卡迈克尔数的等效定义,当正合数n满足以下三个性质时:

必须包含不止一个质因数;

质因数均不重复;

对于每一个能被n整除的质数p,p-1也可以被n-1整除

它就是一个卡迈克尔数。

举个例子,最小的卡迈克尔数是561,561=3×11×17,而2、10和16均能被560整除。

1994年,雷德·阿尔福德(Red Alford)、安德鲁·格兰维尔(Andrew Granville),以及前文提到的卡尔·波梅兰斯三位数学家,在《数学年刊》上发表论文,证明了卡迈克尔数有无穷多个。

0738d5468d71db1ee63e2e4e029084b9.jpg


但当他们试图证明这无穷多个卡迈克尔数之间的间隔时,新的困难出现了。

三位数学家认为,这个问题可以转化为这样一种证明:给定一个足够大的数字X,在X和2X之间一定存在一个卡迈克尔数。

遗憾的是,从1994年到2021年的27年之间,并没有人完成这个证明。

难度可想而知。因此当丹尼尔的爸爸——印第安纳大学路明顿分校数学教授迈克尔·拉森(Michael Larsen)得知儿子想要攻克这个问题时,他的第一反应是“这可能会变成一段负面经历”。

但丹尼尔的反应却是:

你的意思是我仍有10%的机会!

于是,他坚定地投身其中。并且在约300个小时(12.5天)的努力之后,他的论文出炉了。

前面说到,一开始接触数论,丹尼尔就研究过陶哲轩和梅纳德的论文。而在这个有关卡迈克尔数的证明上,他巧妙地站在了前辈的肩膀上。

他修改了梅纳德在证明孪生素数间隔时的用到的方法,将之与阿尔福德、格兰维尔和波梅兰斯的方法相结合。如此一来,就能够确保他最终得到足以产生卡迈克尔数的素数区间。

7f172b7fdb52ba2f374c5263f7b08a5d.jpg


实际上,这篇论文不仅证明了卡迈克尔数一定会出现在X和2X之间,其证明方法还适用于更小的间隔。

另一位致力于伪质数研究的数学家、沃福德学院的Thomas Wright就表示,“这篇论文改变了研究卡迈克尔数的许多事情”。

值得一提的是,卡迈克尔数与密码学和通信安全息息相关。

最典型的非对称加密算法RSA中,生成公钥的第一步就是选取一对很大的随机质数。

而当数字比较大时,想要判断其是否为质数就很麻烦,也很容易与其它数字混淆。这时候,卡迈克尔数的相关研究就能派上用场了。

出身数学世家

如果说与数论的机缘是从张益唐的纪录片开始,那么丹尼尔与数学的缘分在他更小的时候就已经显现。

这与他的家庭氛围息息相关。

丹尼尔出身数学世家,父母都是印第安纳大学的数学教授,他在浓厚的数学氛围下长大。

他的父亲迈克尔·拉森是1977的IMO(国际数学奥林匹克竞赛)金牌得主,本科毕业于哈佛大学,后于普林斯顿大学取得博士学位。

2013年,迈克尔·拉森因“对群论、数论、拓扑学和代数几何的贡献”而成为美国数学会会员。

5cd68e0da4ac566aed014a5dfa14dba5.jpg


△图源:印第安纳大学

丹尼尔4岁的时候,父亲组织了一个“数学圈”,周六下午为当地孩子开设免费小组,谈论一些能让孩子们对数学产生兴趣的古怪话题,丹尼尔也参与其中。

在这样的培养之下,丹尼尔从小就对解谜感兴趣,并且虽然不太喜欢打游戏,但却很喜欢鼓捣电脑,去钻研游戏背后的工作机制。

12岁时,丹尼尔就写出了填字游戏生成软件,并且13岁就在《纽约时报》发表了自己的作品,到现在还保持着“最年轻作者”的记录。

8c6921c7ed6949e5c7987dacd3713524.jpg


△右一丹尼尔,图源:indianapolismonthly

实际上,在丹尼尔研究卡迈克尔数时,他的父亲就是他的第一任导师。

虽然父亲并不对儿子搞定这么难的数学问题抱太大希望,但他还是给予了儿子情感上的支持,并且以对待博士生的态度来指导儿子。

值得一提的是,丹尼尔的姐姐安妮也是“数学神童”,她在高中时就已经开始学习印第安纳大学研究生水平的数学课程。现在,安妮正在MIT攻读数学博士学位。

从小在数学的熏陶下长大,丹尼尔自己也逐渐形成了对数学的一套理解。

他认为现如今互联网时代削弱了人们的社区意识以及共同目标,人们与外界的联系越来越少,这形成一个“元问题”,阻碍了其他问题的解决。

而丹尼尔将这视作数学的另外一面,称数学可以建立共识,它充满了和谐与统一。他还说:“上帝是个数学家。”

偶尔躺平的“小数学家”

数学之外,丹尼尔可谓兴趣广泛,小提琴、钢琴、魔方、国际象棋等均有涉猎。

980f0dbe8ee1924e892f886621902223.jpg


△图源:Quantamagazine

他小时候就能够45秒内解出一个魔方,并且还设计了一个乐高机器人,可以把铜便士和锌便士分开。

不过“少年天才”的光环之外,丹尼尔也像其他年轻人一样,喜欢偶尔躺平,喜欢走“捷径”。

就比如说,丹尼尔不太喜欢夏天,一到夏天,他的心情就开始低落,即使在研究卡迈克尔数也会这样,那时他暂时“撂下挑子”,去看了夏季奥运会。

甚至还自述有过这种情况:

感觉冷但是就是懒得拿外套。

想看黑板上的字,但离得有点远就没过去看。

并且当被问到“不擅长什么时”,丹尼尔表示自己“做什么都像是在挣扎”。

我经常选择阻力最小的道路。如果我对某种情况感到不快,我可能并不会积极地去处理它。

目前,丹尼尔已经步入了麻省理工的校园,他同样面临着和大多数人一样的迷茫,不确定下一步要解决什么:

我只是在上课…… 并试图保持开放的心态。




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  1. MIT News
  2. In the Media
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In the Media​

Media Outlet:
Quanta Magazine
Publication Date:
October 13, 2022
Description:
During his senior year of high school, MIT first-year student Daniel Larsen successfully proved a key theorem about Carmichael numbers, entities that mimic prime numbers, writes Jordana Cepelewicz for Quanta Magazine. “His proof is really quite advanced,” says Dartmouth Prof. Carl Pomerance. “It would be a paper that any mathematician would be really proud to have written.”
 

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Teenager Solves Stubborn Riddle About Prime Number Look-Alikes​

In his senior year of high school, Daniel Larsen proved a key theorem about Carmichael numbers — strange entities that mimic the primes. “It would be a paper that any mathematician would be really proud to have written,” said one mathematician.
1

A teenager wearing a blue polo shirt and glasses.

After he posted his proof, Daniel Larsen enrolled at the Massachusetts Institute of Technology as a math major.
Katherine Taylor for Quanta Magazine

Jordana Cepelewicz​

Senior Writer


October 13, 2022

VIEW PDF/PRINT MODE
mathematicsnumber theoryprime numberstwin primes conjectureAll topics

When Daniel Larsen was in middle school, he started designing crossword puzzles. He had to layer the hobby on top of his other interests: chess, programming, piano, violin. He twice qualified for the Scripps National Spelling Bee near Washington, D.C., after winning his regional competition. “He gets focused on something, and it’s just bang, bang, bang, until he succeeds,” said Larsen’s mother, Ayelet Lindenstrauss. His first crossword puzzles were rejected by major newspapers, but he kept at it and ultimately broke in. To date, he holds the record for youngest person to publish a crossword in The New York Times, at age 13. “He’s very persistent,” Lindenstrauss said.
Still, Larsen’s most recent obsession felt different, “longer and more intense than most of his other projects,” she said. For more than a year and a half, Larsen couldn’t stop thinking about a certain math problem.
It had roots in a broader question, one that the mathematician Carl Friedrich Gauss considered to be among the most important in mathematics: how to distinguish a prime number (a number that is divisible only by 1 and itself) from a composite number. For hundreds of years, mathematicians have sought an efficient way to do so. The problem has also become relevant in the context of modern cryptography, as some of today’s most widely used cryptosystems involve doing arithmetic with enormous primes.
Over a century ago, in that quest for a fast, powerful primality test, mathematicians stumbled on a group of troublemakers — numbers that fool tests into thinking they’re prime, even though they’re not. These pseudoprimes, known as Carmichael numbers, have been particularly difficult to grasp. It was only in the mid-1990s, for instance, that mathematicians proved there are infinitely many of them. Being able to say something more about how they’re distributed along the number line has posed an even greater challenge.
Then along came Larsen with a new proof about just that, one inspired by recent epochal work in a different area of number theory. At the time, he was just 17.

The Spark

Growing up in Bloomington, Indiana, Larsen was always drawn to mathematics. His parents, both mathematicians, introduced him and his older sister to the subject when they were young. (She’s now pursuing a doctorate in math.) When Larsen was 3 years old, Lindenstrauss recalls, he started asking her philosophical questions about the nature of infinity. “I thought, this kid has a mathematical mind,” said Lindenstrauss, a professor at Indiana University.
Then a few years ago — around the time that he was immersed in his spelling and crossword projects — he came across a documentary about Yitang Zhang, an unknown mathematician who rose from obscurity in 2013 after proving a landmark result that put an upper bound on the gaps between consecutive prime numbers. Something clicked in Larsen. He couldn’t stop thinking about number theory, and about the related problem that Zhang and other mathematicians still hoped to solve: the twin primes conjecture, which states that there are infinitely many pairs of primes that differ by only 2.

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Video: Daniel Larsen wouldn’t let go of an old question about Carmichael numbers. “It was just stubbornness on my part,” he said.
Photo by Katherine Taylor for Quanta Magazine; Video by Emily Buder, Noah Hutton, Taylor Hess and Rui Braz for Quanta Magazine
After Zhang’s work, which showed that there are infinitely many pairs of primes that differ by less than 70 million, others jumped in to lower this bound even further. Within months, the mathematicians James Maynard and Terence Tao independently proved an even stronger statement about the gaps between primes. That gap has since shrunk to 246.
Larsen wanted to understand some of the mathematics underlying Maynard and Tao’s work, “but it was pretty much impossible for me,” he said. Their papers were far too complicated. Larsen tried to read related work, only to find it impenetrable as well. He kept at it, jumping from one result to another, until finally, in February 2021, he came across a paper he found both beautiful and comprehensible. Its subject: Carmichael numbers, those strange composite numbers that could sometimes pass themselves off as prime.

All but Prime

In the mid-17th century, the French mathematician Pierre de Fermat wrote a letter to his friend and confidant Frénicle de Bessy, in which he stated what would later be known as his “little theorem.” If N is a prime number, then bNb is always a multiple of N, no matter what b is. For instance, 7 is a prime number, and as a result, 27 – 2 (which equals 126) is a multiple of 7. Similarly, 37 – 3 is a multiple of 7, and so on.
Mathematicians saw the potential for a perfect test of whether a given number is prime or composite. They knew that if N is prime, bNb is always a multiple of N. What if the reverse was also true? That is, if bNb is a multiple of N for all values of b, must N be prime?
Alas, it turned out that in very rare cases, N can satisfy this condition and still be composite. The smallest such number is 561: For any integer b, b561 – b is always a multiple of 561, even though 561 is not prime. Numbers like these were named after the mathematician Robert Carmichael, who is often credited with publishing the first example in 1910 (though the Czech mathematician Václav Šimerka independently discovered examples in 1885).
A teenager sitting and reading a notebook.

When Larsen finished his proof, he sent a draft to some of the top people in number theory. To his surprise, they read it and replied.
Katherine Taylor for Quanta Magazine
Mathematicians wanted to better understand these numbers that so closely resemble the most fundamental objects in number theory, the primes. It turned out that in 1899 — a decade before Carmichael’s result — another mathematician, Alwin Korselt, had come up with an equivalent definition. He simply hadn’t known if there were any numbers that fit the bill.
According to Korselt’s criterion, a number N is a Carmichael number if and only if it satisfies three properties. First, it must have more than one prime factor. Second, no prime factor can repeat. And third, for every prime p that divides N, p – 1 also divides N – 1. Consider again the number 561. It’s equal to 3 × 11 × 17, so it clearly satisfies the first two properties in Korselt’s list. To show the last property, subtract 1 from each prime factor to get 2, 10 and 16. In addition, subtract 1 from 561. All three of the smaller numbers are divisors of 560. The number 561 is therefore a Carmichael number.
Though mathematicians suspected that there are infinitely many Carmichael numbers, there are relatively few compared to the primes, which made them difficult to pin down. Then in 1994, Red Alford, Andrew Granville and Carl Pomerance published a breakthrough paper in which they finally proved that there are indeed infinitely many of these pseudoprimes.
Unfortunately, the techniques they developed didn’t allow them to say anything about what those Carmichael numbers looked like. Did they appear in clusters along the number line, with large gaps in between? Or could you always find a Carmichael number in a short interval? “You’d think if you can prove there’s infinitely many of them,” Granville said, “surely you should be able to prove that there are no big gaps between them, that they should be relatively well spaced out.”
In particular, he and his co-authors hoped to prove a statement that reflected this idea — that given a sufficiently large number X, there will always be a Carmichael number between X and 2X. “It’s another way of expressing how ubiquitous they are,” said Jon Grantham, a mathematician at the Institute for Defense Analyses who has done related work.
But for decades, no one could prove it. The techniques developed by Alford, Granville and Pomerance “allowed us to show that there were going to be many Carmichael numbers,” Pomerance said, “but didn’t really allow us to have a whole lot of control about where they’d be.”
Then, in November 2021, Granville opened up an email from Larsen, then 17 years old and in his senior year of high school. A paper was attached — and to Granville’s surprise, it looked correct. “It wasn’t the easiest read ever,” he said. “But when I read it, it was quite clear that he wasn’t messing around. He had brilliant ideas.”
Pomerance, who read a later version of the work, agreed. “His proof is really quite advanced,” he said. “It would be a paper that any mathematician would be really proud to have written. And here’s a high school kid writing it.”
The key to Larsen’s proof was the work that had drawn him to Carmichael numbers in the first place: the results by Maynard and Tao on prime gaps.

Unlikely — Not Impossible

When Larsen first set out to show that you can always find a Carmichael number in a short interval, “it seemed that it was so obviously true, how hard can it be to prove?” he said. He quickly realized it could be very hard indeed. “This is a problem which tests the technology of our time,” he said.
A teenager with a blue shirt outside.

Larsen established a tighter constraint on Carmichael numbers than what he set out to prove.
Katherine Taylor for Quanta Magazine
In their 1994 paper, Alford, Granville and Pomerance had shown how to create infinitely many Carmichael numbers. But they hadn’t been able to control the size of the primes they used to construct them. That’s what Larsen would need to do to build Carmichael numbers that were relatively close in size. The difficulty of the problem worried his father, Michael Larsen. “I didn’t think it was impossible, but I thought it was unlikely he’d succeed,” he said. “I saw how much time he was spending on it … and I felt it would be devastating for him to give so much of himself to this and not get it.”
Still, he knew better than to try to dissuade his son. “When Daniel commits to something that really interests him, he sticks with it through thick and thin,” he said.
So Larsen returned to Maynard’s papers — in particular, to work showing that if you take certain sequences of enough numbers, some subset of those numbers must be prime. Larsen modified Maynard’s techniques to combine them with the methods used by Alford, Granville and Pomerance. This allowed him to ensure that the primes he ended up with would vary in size — enough to produce Carmichael numbers that would fall within the intervals he wanted.
“He has more control over things than we’ve ever had,” Granville said. And he achieved this through a particularly clever use of Maynard’s work. “It’s not easy … to use this progress on short gaps between primes,” said Kaisa Matomäki, a mathematician at the University of Turku in Finland. “It’s quite nice that he’s able to combine it with this question about the Carmichael numbers.”

RELATED:​


  1. Graduate Student Solves Decades-Old Conway Knot Problem

  2. Major Quantum Computing Advance Made Obsolete by Teenager

  3. Unheralded Mathematician Bridges the Prime Gap

In fact, Larsen’s argument didn’t just allow him to show that a Carmichael number must always appear between X and 2X. His proof works for much smaller intervals as well. Mathematicians now hope it will also help reveal other aspects of the behavior of these strange numbers. “It’s a different idea,” said Thomas Wright, a mathematician at Wofford College in South Carolina who works on pseudoprimes. “It changes a lot of things about how we might prove things about Carmichael numbers.”
Grantham agreed. “Now you can do things you never thought of,” he said.
Larsen, meanwhile, just started his freshman year at the Massachusetts Institute of Technology. He’s not sure what problem he might work on next, but he’s eager to learn what’s out there. “I’m just taking courses … and trying to be open-minded,” he said.
“He did all this without an undergraduate education,” Grantham said. “I can only imagine what he’s going to be coming up with in graduate school.”
 

茶马盐铁

我想看看自定义头衔到底能有多少字。继续加,看系统什么时候把这个字符串截断。呃,居然还有?那就继续吧。
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还以为又是瑞鸡山庄子弟,EOM学子呢。

跟这俩都不沾边,说个*8
 

CFC250

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对数论不屑一顾的是哈代。。。咱村有对袋鼠不屑一顾的
 

Hainiu

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数论长期以来被认为是一门纯数学,没有什么实际应用,所以一般大学理工科的数学课都不包括这方面内容。不过现在在通信和信息工程中得到越来越多的应用,譬如现代的纠错编码和加密编码等理论和技术都有赖于数论的理论,所以有些相关的研究生会被要求学习数论的课程。
 

Hainiu

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说到 卡迈克尔数 (Carmichael Numbers,以下简称 卡数),前些年媒体上报道过,国内河南一位痴迷于数学的打工者,余建春,提出了一个构建卡数的新公式。经过多年奔走,最终引起了浙江大学蔡天新教授的注意,邀请他赴浙大交流。余建春的新公式得到蔡教授的认可,并被蔡教授收入即将出版的一本数论专著。
 

ottawa_tj

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老外要是爱数学,那可是真爱,绝对不是父母推的。。。
 

CFC250

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对数论不屑一顾的是哈代。。。咱村有对袋鼠不屑一顾
说到 卡迈克尔数 (Carmichael Numbers,以下简称 卡数),前些年媒体上报道过,国内河南一位痴迷于数学的打工者,余建春,提出了一个构建卡数的新公式。经过多年奔走,最终引起了浙江大学蔡天新教授的注意,邀请他赴浙大交流。余建春的新公式得到蔡教授的认可,并被蔡教授收入即将出版的一本数论专著。
有些哈代-拉马努金的意味。。
 

billwanhua

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老外要是爱数学,那可是真爱,绝对不是父母推的。。。
只有爱还不行,这种人需要上帝给的gift, 普通人别想这个。

我年轻时候对足球的爱肯定可以跟梅西比,可是后来我知道我不仅成不了梅西,还不可能成为郝海东
 
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