解決世界上最著名數學難題的男人：英語精讀講義

In 1994,

1994年，

Andrew Wiles solved a mystery that had stumped mathematicians for 357 years. 用詞

Andrew Wiles解開了一個困擾數學家357年的謎題。

The 17th century French mathematician Pierre de Fermat states that you can't find numbers

17世紀的法國數學家Pierre de Fermat提出，你無法找到這樣的一些數字，

where x to the n + y to the n

使得x的n次方加y的n次方

is equal to z to the n

等於z的n次方，

when n is a whole number greater than two.

前提是n為大於2的整數。

For example,

例如，

3 to the power of 3 + 4 to the power of 3 equals 91, 文化背景

3的3次方加4的3次方等於91，

which is not the cube of any whole number.

這並不是任何整數的立方。

Fermat claims to have proved his theorem, 文法

Fermat聲稱自己已經證明了他的定理，

writing in his copy of Arithmetica,

他在自己那本《算術》（Arithmetica）

an ancient Greek text on number theory,

——一本古希臘數論文獻的空白處寫道：

"I have discovered a truly marvelous proof of this,

「我已經發現了一個真正絕妙的證明，

which this margin is too narrow to contain." 文法

但這裡的空白太小，寫不下。」

Unfortunately,

不幸的是，

he never wrote down his proof,

他從未將證明寫下來，

not anywhere that anyone has ever found.

在任何地方都無人發現。

For over 300 years,

在三百多年的時間裡，

mathematicians puzzled over this perplexing problem, 片語動詞

數學家們對這個令人費解的難題百思不得其解，

which is similar to the Pythagorean theorem,

它類似於畢氏定理，

but applies to exponents greater than two.

但適用於大於2的指數。

Andrew Wiles was a young boy when he stumbled upon Fermat's Last Theorem 片語動詞

Andrew Wiles在1963年偶然發現了費馬最後定理時，

at a library in Cambridge,

還是個小男孩，

England in 1963.

當時他在英國劍橋的一家圖書館裡。

As he recalled in an interview later featured in the book Fermat's Enigma

正如他後來在Simon Singh撰寫的

by Simon Singh,

《費馬最後定理》（Fermat's Enigma）一書

which I highly recommend,

（我強烈推薦這本書）的一篇採訪中所回憶的那樣：

"Here was a problem that I,

「這就是一個我，

a 10-year-old,

一個10歲的孩子，

could understand,

能夠理解的難題，

and I knew from that moment

從那一刻起我就知道

that I would never let it go. 片語動詞

我絕對不會放棄它。

I had to solve it."

我必須解決它。」

His determination was extraordinary,

他的決心非同尋常，

considering the brightest minds in math

要知道數學界最頂尖的頭腦

had failed to crack it for centuries. 用詞

幾個世紀以來都沒能攻克它。

The 18th century Swiss mathematician Leonhard Euler

18世紀的瑞士數學家Leonhard Euler

proved a specific case

證明了一個特定情況，

where there are no solutions

即當n等於3時

for n equals 3,

沒有解，

but his work didn't extend to other exponents.

但他的研究未能擴展到其他指數。

Euler even asked a friend to search Fermat's house

Euler甚至請朋友去搜查Fermat的家，

just in case he had scribbled down his proof somewhere, 片語搭配

以防他把證明潦草地寫在了某個地方，

with no luck.

但一無所獲。

The wealthy German industrialist Paul Wolfskehl

富有的德國工業家Paul Wolfskehl

was so miserable because the woman he loved 文法

曾因為深愛的女人

didn't love him back

不愛他而痛苦不堪，

that he planned to end his life,

以至於計劃結束自己的生命，

only to be reinvigorated by Fermat's puzzle. 文法

卻因Fermat的這個難題而重新振作。

He never solved it himself,

他自己從未能解開它，

but he offered a reward of 100,000 German marks

但他懸賞了十萬德國馬克，

to anyone who could,

獎勵給任何能解開它的人，

motivating the next generation of mathematicians.

從而激勵了下一代的數學家們。

When Andrew Wiles was a grad student at the University of Cambridge

當Andrew Wiles在劍橋大學讀研究所時，

in the 1970s,

那是在20世紀70年代，

his supervisor John Coates

他的指導教授John Coates

suggested he study elliptic curves,

建議他研究橢圓曲線，

which proved crucial for solving Fermat's Last Theorem. 片語搭配

這被證明對解決費馬最後定理至關重要。

In the 1950s,

在20世紀50年代，

Japanese mathematicians Yutaka Taniyama and Goro Shimura

日本數學家谷山豐（Yutaka Taniyama）和志村五郎（Goro Shimura）

suggested that for every elliptic curve,

提出，對於每一條橢圓曲線，

there was a corresponding modular form.

都對應著一個模形式。

The Taniyama-Shimura conjecture connected two distinct areas of mathematics.

「谷山-志村猜想」將數學中兩個不同的領域聯繫了起來。

Later,

後來，

Ken Ribet,

加州大學柏克萊分校的教授

a professor at UC Berkeley,

Ken Ribet，

linked their conjecture to Fermat's Last Theorem.

將他們的猜想與費馬最後定理聯繫在了一起。

How?

怎麼做到的？

He showed that if Fermat's Last Theorem were false, 文法

他證明了如果費馬最後定理是錯誤的，

certain elliptical curves would have to exist.

那麼必然存在某些特定的橢圓曲線。

However,

然而，

the Taniyama-Shimura conjecture implied that those curves could not exist.

谷山-志村猜想意味著這些曲線不可能存在。

Therefore,

因此，

if the Taniyama-Shimura conjecture were true,

如果谷山-志村猜想是真的，

Fermat's Last Theorem would also be true.

費馬最後定理也必為真。

Proving the conjecture would prove Fermat's Last Theorem.

證明了這個猜想也就證明了費馬最後定理。

The problem was Taniyama and Shimura

問題在於谷山和志村

never proved their own conjecture.

從未證明過他們自己的猜想。

Tragically,

悲劇的是，

Taniyama committed suicide in 1958,

谷山在1958年自殺身亡，

leaving a note that read,

留下了一張字條，上面寫著：

"I am in the frame of mind that I lost confidence in my future."

「我處於對未來失去信心的心境中。」

His fiancee took her own life shortly after,

不久之後，他的未婚妻也結束了自己的生命，

writing,

她寫道：

"We promised each other that no matter where we went, 文法

「我們曾互相承諾，無論去哪裡，

we would never be separated.

我們永遠都不會分開。

Now that he is gone,

既然他已經走了，

I must go too in order to join him."

我也必須去陪伴他。」

In 1986,

1986年，

Andrew Wiles was sipping iced tea at a friend's house

Andrew Wiles正在朋友家喝著冰茶，

when his friend mentioned Ken Ribet

朋友提到Ken Ribet

had proved the link between Taniyama-Shimura and Fermat's Last Theorem.

已經證明了谷山-志村猜想和費馬最後定理之間的聯繫。

Andrew recalled,

Andrew回憶說：

"I was electrified. 慣用語

「我被電到了。

I knew that moment that the course of my life was changing,

那一刻我就知道我人生的軌跡改變了，

because this meant that to prove Fermat's Last Theorem,

因為這意味著要想證明費馬最後定理，

all I had to do was prove the Taniyama-Shimura conjecture. 文法

我只需證明谷山-志村猜想。

It meant that my childhood dream was now a respectable thing to work on.

這意味著我的童年夢想現在成了一項值得去做的正經工作。

I just knew that I could never let that go."

我只知道我絕不能放手。」

By this time,

到這個時候，

Andrew had moved to the US

Andrew已經搬到了美國，

where he was a mathematics professor at Princeton University.

在普林斯頓大學擔任數學教授。

When he wasn't lecturing,

當他不講課的時候，

he was in his attic office

他就在閣樓的辦公室裡，

trying to prove that every single elliptic equation

試圖證明每一個橢圓方程式

could be correlated with a modular form.

都能與一個模形式相關聯。

Only his wife knew what he was up to. 慣用語

只有他的妻子知道他在幹什麼。

He kept his work a secret,

他對自己的工作保密，

not wanting to be distracted.

不想被打擾。

Though,

不過，

there may have been another reason for his secrecy.

他的保密可能還有另一個原因。

He may have feared that if his progress leaked,

他可能擔心，如果自己的進展洩露出去，

a rival mathematician might beat him to the punch and claim the glory. 慣用語

競爭對手可能會捷足先登搶走榮耀。

He cleverly threw his colleagues off track 慣用語

他巧妙地轉移了同事們的注意力，

by publishing his ongoing research on elliptic equations every six months,

每半年發表一次自己關於橢圓方程式的階段性研究，

making it seem like he was keeping busy 表達脈絡

讓人覺得他一直很忙，

while actually devoting his time to the age-old enigma.

而實際上他把時間都傾注在了那個古老的謎題上。

He began by proving the connection for a specific class of elliptic curves,

他首先證明了一類特定橢圓曲線的關聯，

like knocking down the first domino in a long line. 表達脈絡

就像推倒了長長一排骨牌中的第一張。

The challenge was to ensure that if one domino fell,

挑戰在於要確保如果一張骨牌倒下，

it would topple the next.

它能推倒下一張。

Andrew had to show that if one element of the elliptic curve series

Andrew必須證明，如果橢圓曲線序列中的一個元素

matched the corresponding element in the modular form series,

與模形式序列中的對應元素相匹配，

then the next elements would also match.

那麼隨後的元素也會匹配。

Yet,

然而，

he struggled to topple the rest of the dominoes.

他在試圖推倒剩餘骨牌時陷入了困境。

Five years after working in secret,

在秘密研究了五年之後，

he decided to emerge from his hermit-like state

他決定從隱居狀態中走出來，

in hopes of finding new ideas.

希望能找到新的靈感。

In 1991,

1991年，

he attended a major math conference in Boston

他在波士頓參加了一個大型數學會議，

where his PhD supervisor John Coates mentioned that a student of his,

他的博士指導教授John Coates提到，他的一個學生，

Matthias Flach,

Matthias Flach，

was working on a powerful way of analyzing elliptic equations,

正在研究一種分析橢圓方程式的強力方法，

building on the work of Russian mathematician Viktor Kolyvagin. 片語動詞

該方法基於俄羅斯數學家Viktor Kolyvagin的研究。

Andrew realized this Kolyvagin-Flach method

Andrew意識到這個Kolyvagin-Flach方法

could help him topple the rest of the dominoes,

能幫他推倒剩下的骨牌，

ultimately proving every elliptic equation

最終證明每個橢圓方程式

could be matched to a modular form.

都能與一個模形式相匹配。

But this involved incredibly hard mathematics

但這涉及極其艱深的數學，

and took several more years.

並又花了好幾年的時間。

After seven years of working all by himself,

在獨自一人研究了七年之後，

he finally decided he needed some help.

他終於決定自己需要一些幫助。

In 1993,

1993年，

he confided in his colleague at Princeton, 片語搭配

他向他在普林斯頓的同事、

mathematician Nick Katz.

數學家Nick Katz吐露了秘密。

They arranged a secret weekly meeting disguised as a graduate course

他們安排了每週一次的秘密會議，偽裝成研究所課程，

where Nick meticulously checked and verified Andrew's calculations

Nick在那裡細緻地檢查和驗證Andrew的計算，

and assessed that the Kolyvagin-Flach method worked perfectly.

並評估Kolyvagin-Flach方法是否完美奏效。

There was still one missing piece of the puzzle.

但拼圖中仍然缺失一塊。

Andrew used a key insight gleaned from a paper by American mathematician Barry Mazur

Andrew運用從美國數學家Barry Mazur的一篇論文中獲得的關鍵啟示，

to complete the puzzle in May 1993.

在1993年5月完成了這塊拼圖。

It was time to tell the world he had solved the world's hardest math problem.

是時候向世界宣告他已解開了世界上最難的數學題了。

He chose to do so at a conference at the Isaac Newton Institute

他選擇在牛頓研究所的一場會議上宣布，

in June 1993

時間是1993年6月，

in his hometown of Cambridge.

地點在他的家鄉劍橋。

An air of excitement filled the room

房間裡瀰漫著興奮的氣氛，

as rumors spread of his breakthrough

因為關於他在

on the more than 300-year-old mystery.

這個有著300多年歷史的謎題上取得突破的謠言已經傳開。

Andrew recalled,

Andrew回憶說：

"There was a typical dignified silence

「在我唸出證明過程時，

while I read out the proof,

現場有一種典型的莊嚴寂靜，

and then I just wrote up the statement of Fermat's Last Theorem.

然後我剛好寫下了費馬最後定理的陳述。

I said,

我說：

"I think I'll stop here."

『我想我就停在這裡吧。』

And then there was sustained applause."

接著便響起了經久不息的掌聲。」

Mathematicians usually don't make headlines,

數學家通常不會上頭條，

but this was an exception.

但這次是個例外。

The excitement was palpable,

興奮之情溢於言表，

yet the proof still needed to be verified.

但證明依然需要得到驗證。

He submitted his draft to the journal Inventiones Mathematicae,

他把手稿提交給了《數學發明》期刊，

where the editor Barry Mazur

編輯Barry Mazur

selected six referees to pore over Andrew's 200-page work. 片語動詞

挑選了六位審稿人來研讀Andrew長達200頁的論文。

This was subject matter that perhaps only a half dozen people in the world could fully grasp,

正如屢獲殊榮的紀錄片《費馬最後定理》的導演John Lynch

as John Lynch, director of the award-winning documentary Fermat's Last Theorem, 表達脈絡

所言，

put it.

這是一門全世界大概只有六七個人能完全掌握的學科。

During this rigorous scrutiny,

在這種嚴格的審查中，

a flaw emerged.

出現了一個破綻。

One of the referees,

其中一位審稿人，

Andrew's Princeton colleague Nick Katz,

也就是Andrew在普林斯頓的同事Nick Katz，

discovered a gap in the part of the argument 用詞

在涉及Kolyvagin-Flach方法的論證部分

involving the Kolyvagin-Flach method.

發現了一個漏洞。

Nick and Andrew had missed this during their earlier secret meetings disguised as lectures.

Nick和Andrew在之前偽裝成講課的秘密會議上漏掉了這一點。

Was Andrew's proof about to collapse like a house of cards? 慣用語

Andrew的證明難道要像紙牌屋一樣倒塌了嗎？

Despite the setback,

儘管遇到了挫折，

he remained optimistic.

他依然保持樂觀。

He was confident that Kolyvagin-Flach just needed a little tweaking.

他堅信Kolyvagin-Flach方法只需要一點點修正。

He retreated to his attic once again,

他再次退回到閣樓裡，

but it was slow going. 慣用語

但進展緩慢。

His wife Nada had told him she wanted everything corrected by her birthday on October 6,

他的妻子Nada告訴他，希望他在10月6日她的生日之前把一切都改正過來，

two weeks away.

距離現在只有兩週了。

But as time went on with no manuscript published,

但隨著時間推移，論文遲遲未能發表，

rumors began circulating that there were issues with his proof.

開始有傳言說他的證明出了問題。

Andrew had no choice but to address the speculation publicly in a forum at the end of 1993. 文法

Andrew別無選擇，只能在1993年底的一個論壇上公開回應這些猜測。

"During the review process,

「在審查過程中，

a number of problems emerged,

出現了許多問題，

most of which have been resolved, 文法

其中大部分已經解決，

but one in particular I have not yet settled.

但有一個特定的問題我還沒有處理好。

I believe that I will be able to finish this in the near future

我相信我會在不久的將來

using the ideas explained in my Cambridge lectures."

利用我在劍橋講座中解釋的思想來完成它。」

Winter passed and he was still nowhere close to completing the proof. 片語搭配

冬天過去了，他離完成證明還差得很遠。

He decided to bounce ideas off of someone he trusted. 慣用語

他決定找個值得信賴的人交流想法。

His former student Richard Taylor,

他以前的學生Richard Taylor，

now a Cambridge lecturer and one of the referees who looked over his proof.

現在是劍橋講師，也是審查他證明的審稿人之一。

They worked through the summer of 1994

他們整個1994年夏天都在工作，

without any progress.

但沒有任何進展。

Andrew was at wit's end and thought of giving up. 慣用語

Andrew束手無策，甚至想過放棄。

But Richard persuaded him to keep working on until Richard returned to Cambridge in September.

但Richard說服他繼續努力，直到Richard九月份返回劍橋。

That extra month made all the difference. 慣用語

那多出來的一個月帶來了翻天覆地的改變。

The morning of Monday,

1994年9月19日，

September 19, 1994,

星期一的早晨，

Andrew was again intensely examining the Kolyvagin-Flach method,

Andrew再次專注地審視Kolyvagin-Flach方法，

trying desperately to understand why it didn't work.

拚命想弄明白它為什麼不奏效。

Suddenly,

突然間，

he had an incredible revelation.

他獲得了一個不可思議的啟示。

Although the Kolyvagin-Flach method wasn't entirely successful,

雖然Kolyvagin-Flach方法並不完全成功，

it provided enough insight to revive his original approach

但它提供了足夠的洞見來復活他最初的方案

using Iwasawa theory,

——運用岩澤理論（Iwasawa theory），

a theory of analyzing elliptic equations

這是他在劍橋當學生時學到的一種

that he had learned as a student at Cambridge.

分析橢圓方程式的理論。

Combining the two methods,

將這兩種方法結合起來，

which were inadequate on their own,

它們單獨使用時都不夠充分，

provided the eureka moment.

但結合在一起帶來了頓悟的時刻。

"It was the most uh the most important moment of my working life.

「那是我，呃，我職業生涯中最重要的時刻。

Nothing I ever do again will

我以後做的任何事情都不會再……

I'm sorry."

抱歉。」

After eight years of dedicating himself to one task,

經過八年致力於這項單一的任務，

Andrew's 129-page proof was published in May 1995,

Andrew長達129頁的證明在1995年5月發表，

putting to rest the greatest mystery in math. 慣用語

讓數學界最大的未解之謎塵埃落定。

He reflected,

他反思道：

"Having solved this problem,

「解決了這個問題，

there's certainly a sense of loss. 片語搭配

當然有一種失落感。

But at the same time,

但與此同時，

there is this tremendous sense of freedom.

也有一種巨大的自由感。

That particular odyssey is now over. 慣用語

那段漫長的奧德賽之旅如今已經結束。

My mind is at rest."

我的心安定了。」

If you've been inspired by Andrew's remarkable achievement

如果你被Andrew的非凡成就所啟發，

and want to improve your math skills,

並想提高自己的數學技能，

and who knows,

甚至，誰知道呢，

solve the next big mathematical mystery,

解決下一個重大的數學難題，

Brilliant is the perfect place to start.

Brilliant就是完美的起點。

If you've ever felt that math is hard,

如果你曾覺得數學很難，

Brilliant is a website and app

Brilliant就是能讓

that makes math easy and intuitive.

數學變得簡單直觀的網站和應用程式。

In their measurement course,

在他們的測量課程中，

you learn to find angles, side lengths, and areas

你將透過識別形狀之間的關係，

by recognizing relationships between shapes.

學會求角度、邊長和面積。

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無需複雜的計算。

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Brilliant的向量課程將提升你在多維空間中視覺化

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Brilliant has something for everyone.

Brilliant為每個人都提供了適合的內容。

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如果你對程式設計感興趣，

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你可以熟悉Python，

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並透過內建的拖曳編輯器，

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從第一天起就開始構建程式。

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你可以透過影片描述裡的專屬連結註冊，

by signing up with my custom link in the description.

免費試用Brilliant 30天，

brilliant.org/newsthink

訪問 brilliant.org/newsthink

or scan the QR code on your screen.

或掃描螢幕上的QR碼。

By signing up with my link or code,

透過我的連結或代碼註冊，

you'll also get 20% off your annual premium subscription,

你還可以獲得年度進階訂閱的8折優惠，

giving you access to thousands of courses.

讓你能訪問數以千計的課程。

Thanks for watching.

感謝觀看。

For Newsthink, I'm Cindy Pom.

這裡是Newsthink，我是Cindy Pom。