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繫，這種聯繫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。可以說，相關的研究構成了數學史上最大最古老的數據集。朗蘭茲說他著迷於質數的歷史和最近的進展，並熱衷於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。現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.
數學家歐幾裡德(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.
歐幾裡德證明了質數的無限性，但歷史表明是埃拉託色尼(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。這就是埃拉託色尼篩法的力量。
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，一位致力於創建有用數字的表格的英國數學家。他的動力來源於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。