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.

　　3月20日，数学界的最高荣誉之一—阿贝尔奖颁发给了数学家罗伯特·朗兰兹，以表彰他对数学作出的终生成就。朗兰兹提出的纲领探讨了数论和调和分析之间的深层联系，这种联系被数学家用来解答与质数性质相关的问题。

　　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?

　　当挪威国王5月给朗兰兹颁奖的时候，这一研究已经进行了2300多年，数学家一直都在试图更好的理解质数。可以说，相关的研究构成了数学史上最大最古老的数据集。朗兰兹说他着迷于质数的历史和最近的进展，并热衷于如何揭示他们的秘密。我们不免好奇，质数如何能让数学家为之着迷上千年?

　　How to find primes

　　如何寻找质数?

　　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.

　　为了研究质数，数学家将整数一个个通过他们的虚拟网格，将质数“筛选”出来。这种筛分过程在19世纪就产生了含有数百万个质数的表格。现代计算机可以用这种方法在不到一秒的时间内找到数十亿个质数。但筛分的核心思想却在2000多年间从没改变过。

　　“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.

　　欧几里德证明了质数的无限性，但历史表明是埃拉托色尼(Eratosthenes)为我们提供了快速列出质数的筛分方法。

　　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.

　　筛分的想法是这样的：首先依次过滤出2、3、5、7这四个质数的倍数。如果对2到100之间的所有数字执行这一操作，很快就会只剩下质数。

　　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.

　　通过8个过滤步骤，就可以分离出400以内的全部质数。通过168个过滤步骤，可以分离出100万以内的所有质数。这就是埃拉托色尼筛法的力量。

　　Tables and tables

　　表格×表格

　　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。