2018年04月16日 18:36 爱语吧


  On March 20, American-Canadian mathematician Robert Langlands received the Abel Prize, celebrating lifetime achievement in mathematics. Langlands’ research demonstrated how concepts from geometry, algebra and analysis could be brought together by a common link to prime numbers.


  When the King of Norway presents the award to Langlands in May, he will honor the latest in a 2,300-year effort to understand prime numbers, arguably the biggest and oldest data set in mathematics. As a mathematician devoted to this “Langlands program,” I’m fascinated by the history of prime numbers and how recent advances tease out their secrets. Why they have captivated mathematicians for millennia?


  How to find primes


  To study primes, mathematicians strain whole numbers through one virtual mesh after another until only primes remain. This sieving process produced tables of millions of primes in the 1800s. It allows today’s computers to find billions of primes in less than a second. But the core idea of the sieve has not changed in over 2,000 years.


  “A prime number is that which is measured by the unit alone,” mathematician Euclid wrote in 300 B.C. This means that prime numbers can’t be evenly divided by any smaller number except 1. By convention, mathematicians don’t count 1 itself as a prime number.

  数学家欧几里德(Euclid)在公元前300年写道:“只能为一个单位量测尽的数是质数。” 这意味着质数不能被除了1之外的任何数字整除。根据惯例,数学家不将1计为质数。

  Euclid proved the infinitude of primes – they go on forever – but history suggests it was Eratosthenes who gave us the sieve to quickly list the primes.


  Here’s the idea of the sieve. First, filter out multiples of 2, then 3, then 5, then 7 – the first four primes. If you do this with all numbers from 2 to 100, only prime numbers will remain.


  With eight filtering steps, one can isolate the primes up to 400. With 168 filtering steps, one can isolate the primes up to 1 million. That’s the power of the sieve of Eratosthenes.


  Tables and tables


  An early figure in tabulating primes is John Pell, an English mathematician who dedicated himself to creating tables of useful numbers. He was motivated to solve ancient arithmetic problems of Diophantos, but also by a personal quest to organize mathematical truths. Thanks to his efforts, the primes up to 100,000 were widely circulated by the early 1700s. By 1800, independent projects had tabulated the primes up to 1 million.

  为质数制表的早期人物代表是 John Pell,一位致力于创建有用数字的表格的英国数学家。他的动力来源于想要解决古老的丢番图算术问题,同时也有着整理数学真理的个人追求。在他的努力之下,10万以内的质数得以在18世纪早期广泛传播。到了1800年,各种独立项目已列出了100万以内的质数。

  To automate the tedious sieving steps, a German mathematician named Carl Friedrich Hindenburg used adjustable sliders to stamp out multiples across a whole page of a table at once. Another low-tech but effective approach used stencils to locate the multiples. By the mid-1800s, mathematician Jakob Kulik had embarked on an ambitious project to find all the primes up to 100 million.

  为了自动化冗长乏味的筛分步骤,德国数学家 Carl Friedrich Hindenburg 用可调节的滑动条在整页表格上一次排除所有倍数。另一种技术含量低但非常有效的方法是用漏字板来查找倍数的位置。到了19世纪中叶,数学家 Jakob Kulik 开始了一项雄心勃勃的计划,他要找出1亿以内的所有质数。

  This “big data” of the 1800s might have only served as reference table, if Carl Friedrich Gauss hadn’t decided to analyze the primes for their own sake. Armed with a list of primes up to 3 million, Gauss began counting them, one “chiliad,” or group of 1000 units, at a time. He counted the primes up to 1,000, then the primes between 1,000 and 2,000, then between 2,000 and 3,000 and so on.


  Gauss discovered that, as he counted higher, the primes gradually become less frequent according to an “inverse-log” law. Gauss’s law doesn’t show exactly how many primes there are, but it gives a pretty good estimate. For example, his law predicts 72 primes between 1,000,000 and 1,001,000. The correct count is 75 primes, about a 4 percent error.


  A century after Gauss’ first explorations, his law was proved in the “prime number theorem.” The percent error approaches zero at bigger and bigger ranges of primes. The Riemann hypothesis, a million-dollar prize problem today, also describes how accurate Gauss’ estimate really is.


  The prime number theorem and Riemann hypothesis get the attention and the money, but both followed up on earlier, less glamorous data analysis.


  Modern prime mysteries


  Today, our data sets come from computer programs rather than hand-cut stencils, but mathematicians are still finding new patterns in primes.


  Except for 2 and 5, all prime numbers end in the digit 1, 3, 7 or 9. In the 1800s, it was proven that these possible last digits are equally frequent. In other words, if you look at the primes up to a million, about 25 percent end in 1, 25 percent end in 3, 25 percent end in 7, and 25 percent end in 9.

  除了2和5之外,所有质数都以数字1、3、7、9结尾。在19世纪,数学家证明了这些可能的结尾数字有着同样的出现频率。 换句话说,如果数100万以内的质数,会发现大约25%的质数以1结尾,25%以3结尾,25%以7结尾,以及25%以9结尾。

  A few years ago, Stanford number theorists Robert Lemke Oliver and Kannan Soundararajan were caught off guard by quirks in the final digits of primes. An experiment looked at the last digit of a prime, as well as the last digit of the very next prime. For example, the next prime after 23 is 29: One sees a 3 and then a 9 in their last digits. Does one see 3 then 9 more often than 3 then 7, among the last digits of primes?

  几年前,斯坦福大学的数论学家 Robert Lemke Oliver 和 Kannan Soundararajan 在一个观察质数和下一个质数的最后一位数字的实验中,发现了质数的结尾数的奇异之处。例如质数23之后的下一个质数是29,它们的结尾数字分别是3和9。那么是否在质数的结尾数中,3和9的出现要多过于3和7吗?

  Number theorists expected some variation, but what they found far exceeded expectations. Primes are separated by different gaps; for example, 23 is six numbers away from 29. But 3-then-9 primes like 23 and 29 are far more common than 7-then-3 primes, even though both come from a gap of six.


  Mathematicians soon found a plausible explanation. But, when it comes to the study of successive primes, mathematicians are (mostly) limited to data analysis and persuasion. Proofs – mathematicians’ gold standard for explaining why things are true – seem decades away.