質數的研究已經進行了2300多年，數學家一直都在試圖更好的理解質數�？梢哉f，相關的研究構成了數學史上最大最古老的數據集。朗蘭茲說他著迷于質數的歷史和最近的進展，并熱衷于如何揭示他們的秘密。我們不免好奇，質數如何能讓數學家為之著迷上千年?

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，數學界的最高榮譽之一—阿貝爾獎頒發(fā)給了數學家羅伯特·朗蘭茲，以表彰他對數學作出的終生成就。朗蘭茲提出的綱領探討了數論和調和分析之間的深層聯系，這種聯系被數學家用來解答與質數性質相關的問題。

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?

當挪威國王5月給朗蘭茲頒獎的時候，這一研究已經進行了2300多年，數學家一直都在試圖更好的理解質數�？梢哉f，相關的研究構成了數學史上最大最古老的數據集。朗蘭茲說他著迷于質數的歷史和最近的進展，并熱衷于如何揭示他們的秘密。我們不免好奇，質數如何能讓數學家為之著迷上千年?

How to find primes

如何尋找質數?

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.

為了研究質數，數學家將整數一個個通過他們的虛擬網格，將質數“篩選”出來。這種篩分過程在19世紀就產生了含有數百萬個質數的表格�，F代計算機可以用這種方法在不到一秒的時間內找到數十億個質數。但篩分的核心思想卻在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.

數學家歐幾里德(Euclid)在公元前300年寫道：“只能為一個單位量測盡的數是質數。” 這意味著質數不能被除了1之外的任何數字整除。根據慣例，數學家不將1計為質數。

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.

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

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這四個質數的倍數。如果對2到100之間的所有數字執(zhí)行這一操作，很快就會只剩下質數。

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個過濾步驟，就可以分離出400以內的全部質數。通過168個過濾步驟，可以分離出100萬以內的所有質數。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.

為質數制表的早期人物代表是 John Pell，一位致力于創(chuàng)建有用數字的表格的英國數學家。他的動力來源于想要解決古老的丟番圖算術問題，同時也有著整理數學真理的個人追求。在他的努力之下，10萬以內的質數得以在18世紀早期廣泛傳播。到了1800年，各種獨立項目已列出了100萬以內的質數。

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.

為了自動化冗長乏味的篩分步驟，德國數學家 Carl Friedrich Hindenburg 用可調節(jié)的滑動條在整頁表格上一次排除所有倍數。另一種技術含量低但非常有效的方法是用漏字板來查找倍數的位置。到了19世紀中葉，數學家 Jakob Kulik 開始了一項雄心勃勃的計劃，他要找出1億以內的所有質數。

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數的研究，這個19世紀的“大數據”或許只能作為一張參考表。在有了這張包含300萬以內所有質數的列表之后，高斯開始著手數它們，每次以1000為分界點分組。他找出1000以內的質數，然后再找出1000到2000之間的質數，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，隨著數值的增高，質數出現的頻率會遵循“反對數”定律逐漸下降。雖然高斯定律沒確切地給出質數的數量，但它給出了一個非常好的估計。例如他預測了從1,000,000至1,001,000之間大約有72個質數;而正確的計數是75個，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世紀里，他的定律在“質數定理”中得到了證明。在數值越大的質數范圍內，它的誤差百分比接近于零。作為世界七大數學難題之一的黎曼假設，也描述了高斯估算的準確程度。

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

質數定理和黎曼假設都得到了應有的關注和資金，但這兩者都是在早期不那么迷人的數據分析中得到的。

Modern prime mysteries

現代質數之謎

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

現在，我們的數據集來自計算機程序而非手工切割的漏字模板，但數學家仍在努力尋找質數中的新模式。

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，所有質數都以數字1、3、7、9結尾。在19世紀，數學家證明了這些可能的結尾數字有著同樣的出現頻率。 換句話說，如果數100萬以內的質數，會發(fā)現大約25%的質數以1結尾，25%以3結尾，25%以7結尾，以及25%以9結尾。

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，斯坦福大學的數論學家 Robert Lemke Oliver 和 Kannan Soundararajan 在一個觀察質數和下一個質數的最后一位數字的實驗中，發(fā)現了質數的結尾數的奇異之處。例如質數23之后的下一個質數是29，它們的結尾數字分別是3和9。那么是否在質數的結尾數中，3和9的出現要多過于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.

數論學家預計會有一些變化，但他們的發(fā)現遠遠超出預期。質數與質數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結尾的質數比先以7再以3結尾的質數要普遍得多，盡管這兩種質數組合的間隔都是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.

雖然數學家很快找到了合理的解釋。但是，在研究連續(xù)質數時，數學家大多能做的僅限于數據分析和盡力說服。而數學家用以解釋某事物為何為真的黃金標準——證明，似乎仍距我們數十年之遠。