質(zhì)數(shù)的研究已經(jīng)進(jìn)行了2300多年，數(shù)學(xué)家一直都在試圖更好的理解質(zhì)數(shù)�？梢哉f，相關(guān)的研究構(gòu)成了數(shù)學(xué)史上最大最古老的數(shù)據(jù)集。朗蘭茲說他著迷于質(zhì)數(shù)的歷史和最近的進(jìn)展，并熱衷于如何揭示他們的秘密。我們不免好奇，質(zhì)數(shù)如何能讓數(shù)學(xué)家為之著迷上千年?

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，數(shù)學(xué)界的最高榮譽(yù)之一—阿貝爾獎(jiǎng)?lì)C發(fā)給了數(shù)學(xué)家羅伯特·朗蘭茲，以表彰他對數(shù)學(xué)作出的終生成就。朗蘭茲提出的綱領(lǐng)探討了數(shù)論和調(diào)和分析之間的深層聯(lián)系，這種聯(lián)系被數(shù)學(xué)家用來解答與質(zhì)數(shù)性質(zhì)相關(guān)的問題。

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?

當(dāng)挪威國王5月給朗蘭茲頒獎(jiǎng)的時(shí)候，這一研究已經(jīng)進(jìn)行了2300多年，數(shù)學(xué)家一直都在試圖更好的理解質(zhì)數(shù)�？梢哉f，相關(guān)的研究構(gòu)成了數(shù)學(xué)史上最大最古老的數(shù)據(jù)集。朗蘭茲說他著迷于質(zhì)數(shù)的歷史和最近的進(jìn)展，并熱衷于如何揭示他們的秘密。我們不免好奇，質(zhì)數(shù)如何能讓數(shù)學(xué)家為之著迷上千年?

How to find primes

如何尋找質(zhì)數(shù)?

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.

為了研究質(zhì)數(shù)，數(shù)學(xué)家將整數(shù)一個(gè)個(gè)通過他們的虛擬網(wǎng)格，將質(zhì)數(shù)“篩選”出來。這種篩分過程在19世紀(jì)就產(chǎn)生了含有數(shù)百萬個(gè)質(zhì)數(shù)的表格。現(xiàn)代計(jì)算機(jī)可以用這種方法在不到一秒的時(shí)間內(nèi)找到數(shù)十億個(gè)質(zhì)數(shù)。但篩分的核心思想?yún)s在2000多年間從沒改變過。

“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.

數(shù)學(xué)家歐幾里德(Euclid)在公元前300年寫道：“只能為一個(gè)單位量測盡的數(shù)是質(zhì)數(shù)。” 這意味著質(zhì)數(shù)不能被除了1之外的任何數(shù)字整除。根據(jù)慣例，數(shù)學(xué)家不將1計(jì)為質(zhì)數(shù)。

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.

歐幾里德證明了質(zhì)數(shù)的無限性，但歷史表明是埃拉托色尼(Eratosthenes)為我們提供了快速列出質(zhì)數(shù)的篩分方法。

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.

篩分的想法是這樣的：首先依次過濾出2、3、5、7這四個(gè)質(zhì)數(shù)的倍數(shù)。如果對2到100之間的所有數(shù)字執(zhí)行這一操作，很快就會(huì)只剩下質(zhì)數(shù)。

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.

通過8個(gè)過濾步驟，就可以分離出400以內(nèi)的全部質(zhì)數(shù)。通過168個(gè)過濾步驟，可以分離出100萬以內(nèi)的所有質(zhì)數(shù)。這就是埃拉托色尼篩法的力量。

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.

為質(zhì)數(shù)制表的早期人物代表是 John Pell，一位致力于創(chuàng)建有用數(shù)字的表格的英國數(shù)學(xué)家。他的動(dòng)力來源于想要解決古老的丟番圖算術(shù)問題，同時(shí)也有著整理數(shù)學(xué)真理的個(gè)人追求。在他的努力之下，10萬以內(nèi)的質(zhì)數(shù)得以在18世紀(jì)早期廣泛傳播。到了1800年，各種獨(dú)立項(xiàng)目已列出了100萬以內(nèi)的質(zhì)數(shù)。

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.

為了自動(dòng)化冗長乏味的篩分步驟，德國數(shù)學(xué)家 Carl Friedrich Hindenburg 用可調(diào)節(jié)的滑動(dòng)條在整頁表格上一次排除所有倍數(shù)。另一種技術(shù)含量低但非常有效的方法是用漏字板來查找倍數(shù)的位置。到了19世紀(jì)中葉，數(shù)學(xué)家 Jakob Kulik 開始了一項(xiàng)雄心勃勃的計(jì)劃，他要找出1億以內(nèi)的所有質(zhì)數(shù)。

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.

若沒有高斯等人對質(zhì)數(shù)的研究，這個(gè)19世紀(jì)的“大數(shù)據(jù)”或許只能作為一張參考表。在有了這張包含300萬以內(nèi)所有質(zhì)數(shù)的列表之后，高斯開始著手?jǐn)?shù)它們，每次以1000為分界點(diǎn)分組。他找出1000以內(nèi)的質(zhì)數(shù)，然后再找出1000到2000之間的質(zhì)數(shù)，然后是2000到3000之間，以此類推。

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.

高斯發(fā)現(xiàn)，隨著數(shù)值的增高，質(zhì)數(shù)出現(xiàn)的頻率會(huì)遵循“反對數(shù)”定律逐漸下降。雖然高斯定律沒確切地給出質(zhì)數(shù)的數(shù)量，但它給出了一個(gè)非常好的估計(jì)。例如他預(yù)測了從1,000,000至1,001,000之間大約有72個(gè)質(zhì)數(shù);而正確的計(jì)數(shù)是75個(gè)，誤差值約為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個(gè)世紀(jì)里，他的定律在“質(zhì)數(shù)定理”中得到了證明。在數(shù)值越大的質(zhì)數(shù)范圍內(nèi)，它的誤差百分比接近于零。作為世界七大數(shù)學(xué)難題之一的黎曼假設(shè)，也描述了高斯估算的準(zhǔn)確程度。

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

質(zhì)數(shù)定理和黎曼假設(shè)都得到了應(yīng)有的關(guān)注和資金，但這兩者都是在早期不那么迷人的數(shù)據(jù)分析中得到的。

Modern prime mysteries

現(xiàn)代質(zhì)數(shù)之謎

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

現(xiàn)在，我們的數(shù)據(jù)集來自計(jì)算機(jī)程序而非手工切割的漏字模板，但數(shù)學(xué)家仍在努力尋找質(zhì)數(shù)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，所有質(zhì)數(shù)都以數(shù)字1、3、7、9結(jié)尾。在19世紀(jì)，數(shù)學(xué)家證明了這些可能的結(jié)尾數(shù)字有著同樣的出現(xiàn)頻率。 換句話說，如果數(shù)100萬以內(nèi)的質(zhì)數(shù)，會(huì)發(fā)現(xiàn)大約25%的質(zhì)數(shù)以1結(jié)尾，25%以3結(jié)尾，25%以7結(jié)尾，以及25%以9結(jié)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?

幾年前，斯坦福大學(xué)的數(shù)論學(xué)家 Robert Lemke Oliver 和 Kannan Soundararajan 在一個(gè)觀察質(zhì)數(shù)和下一個(gè)質(zhì)數(shù)的最后一位數(shù)字的實(shí)驗(yàn)中，發(fā)現(xiàn)了質(zhì)數(shù)的結(jié)尾數(shù)的奇異之處。例如質(zhì)數(shù)23之后的下一個(gè)質(zhì)數(shù)是29，它們的結(jié)尾數(shù)字分別是3和9。那么是否在質(zhì)數(shù)的結(jié)尾數(shù)中，3和9的出現(xiàn)要多過于3和7嗎?

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.

數(shù)論學(xué)家預(yù)計(jì)會(huì)有一些變化，但他們的發(fā)現(xiàn)遠(yuǎn)遠(yuǎn)超出預(yù)期。質(zhì)數(shù)與質(zhì)數(shù)之間被不同大小的間隔分開;例如，23與29之間相差6。但是像23和29那樣的先以3再以9結(jié)尾的質(zhì)數(shù)比先以7再以3結(jié)尾的質(zhì)數(shù)要普遍得多，盡管這兩種質(zhì)數(shù)組合的間隔都是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.

雖然數(shù)學(xué)家很快找到了合理的解釋。但是，在研究連續(xù)質(zhì)數(shù)時(shí)，數(shù)學(xué)家大多能做的僅限于數(shù)據(jù)分析和盡力說服。而數(shù)學(xué)家用以解釋某事物為何為真的黃金標(biāo)準(zhǔn)——證明，似乎仍距我們數(shù)十年之遠(yuǎn)。