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双语:两千年来数学家为何痴迷于质数研究

质数的研究已经进行了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?



  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。

  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.

  若没有高斯等人对质数的研究,这个19世纪的“大数据”或许只能作为一张参考表。在有了这张包含300万以内所有质数的列表之后,高斯开始着手数它们,每次以1000为分界点分组。他找出1000以内的质数,然后再找出1000到2000之间的质数,然后是2000到3000之间,以此类推。






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  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.

  高斯发现,随着数值的增高,质数出现的频率会遵循“反对数”定律逐渐下降。虽然高斯定律没确切地给出质数的数量,但它给出了一个非常好的估计。例如他预测了从1,000,000至1,001,000之间大约有72个质数;而正确的计数是75个,误差值约为4%。

  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.

  在高斯的第一次探索之后的一个世纪里,他的定律在“质数定理”中得到了证明。在数值越大的质数范围内,它的误差百分比接近于零。作为世界七大数学难题之一的黎曼假设,也描述了高斯估算的准确程度。

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

  质数定理和黎曼假设都得到了应有的关注和资金,但这两者都是在早期不那么迷人的数据分析中得到的。

  Modern prime mysteries

  现代质数之谜

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

  现在,我们的数据集来自计算机程序而非手工切割的漏字模板,但数学家仍在努力寻找质数中的新模式。

  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?

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  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.

  数论学家预计会有一些变化,但他们的发现远远超出预期。质数与质数之间被不同大小的间隔分开;例如,23与29之间相差6。但是像23和29那样的先以3再以9结尾的质数比先以7再以3结尾的质数要普遍得多,尽管这两种质数组合的间隔都是6。

  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.

  虽然数学家很快找到了合理的解释。但是,在研究连续质数时,数学家大多能做的仅限于数据分析和尽力说服。而数学家用以解释某事物为何为真的黄金标准——证明,似乎仍距我们数十年之远。

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