Speaking of this year's Internet buzzwords, "three sentences" must be on the list.
The reason is that the "lecturer" on Douyin talks to the camera and brags about how he can make others willing to spend money for themselves with three sentences. But until this person's account was blocked, I didn't know which three sentences had such a powerful ability.
But if you look at the history of mathematics, there has indeed been a mathematician who has actually said only three sentences, and brought a completely new theorem and a difficult puzzle to the world, so that for more than 300 years, countless mathematicians have been fascinated by it, and even spared no expense to solve it.
Today, I will tell you about this "three-sentence story" - Fermat's Last Theorem.
Cause: Thinking triggered by the "Pythagorean theorem"
If there is any mathematical theorem inscribed in my DNA in history, it must be Pythagorean theorem, the famous Pythagorean theorem: the sum of squares of the two right-angled sides of a right triangle is equal to the square of the hypotenuses.
This familiar theorem was recorded in a book called Arithmetica by the "father of algebra" Diophantu.
Arithmetic was the first major book on algebra, and it had a major influence not only on Greek mathematics, but also on Arabic and later Western mathematics. In addition to using symbols for unknowns, Diophantu also uses symbols to represent "equals", but not the "=" we use today.
Centuries after the publication of Arithmetic, a man named Pierre German · Fermat's French lawyer flipped it open and was mesmerized. Apparently Pythagorean theorem provoked his thinking, causing him to leave scrawled notes on the margins:
A cubic number cannot be represented by the sum of two cubic numbers, nor can a quaternary number be represented by the sum of two quaternions. In general, no number greater than two powers can be written as the sum of two of the same power.
In short, Fermat extended the formula for Pythagorean's theorem to:
And asserts that if n is greater than 2, then the equation has no integer solution.
And then he went on to write: I have found a wonderful way to prove it, but the page on this side is too narrow to write.
So, let's briefly summarize Fermat's three sentences:
| Write a copycat: I can write 10W+, just not enough time
This passage was written around 1637 and was not published publicly by Fermat nor told to anyone. He used to make assertions without evidence, but he was often right, because although he was a lawyer, Fermat was also a celebrity in the mathematical world, known as the "king of amateur mathematicians".
Fermat died in 1665, and in 1670 his son published a collection of his writings. Among them, the text marked next to the Pythagorean theorem has attracted attention.
For a time, mathematicians all over the world were caught in the trap of this problem and began to look for proofs, and this troublesome puzzle became known as "Fermat's Last Theorem".
Development: Three sentences, let people spend 2 million for this
For later mathematicians, this little marginal hint has an incredible appeal. Although some even think that Fermat is just making it up, or at best discovering an imperfect method of proof.
But Fermat's other ideas on the edges of the page are confirmed one by one, and only this one overwhelms everyone, so that it becomes "Fermat's final theorem/ Fermat's Last Theorem", proving (or refuting) this theorem as if it were the "holy grail" for number theorists.
The holy grail created by these three sentences has really succeeded in making people spend millions on it, and really gives this "holy grail" practical value.
It is a dramatic story in which the protagonist is a wealthy German industrialist and amateur mathematician, Paul Schwarzenegger. Wolfski.
| Paul · Wolfski
It is said that he was emotionally injured by a girl and was preparing to shoot himself in the head at midnight. But before that, he went to the library and read a paper on Fermat's Last Theorem. He discovered one of these mistakes, immediately began to calculate the solution, indulged in it, and completely forgot about suicide (so knowledge can change fate ah friends!). ）。
But he still failed to solve the puzzle, and he was not reconciled. So, in 1906, when Wolfski died, he made a will: to reward the first person to prove Fermat's Last Theorem with 100,000 marks (about $2 million at the time), a promise valid for 100 years.
Ending: A mathematical dream of a 10-year-old boy
Although this reward makes Fermat's Last Theorem more attractive, no one can solve the puzzle.
In 1963, a man named Andrew M. A 10-year-old boy from Wiles showed up, who was passionate about math and borrowed a book from the local library in Cambridge. The author of the book is the mathematician Eric Temple Bell, who discusses this theorem and pessimistically predicts that no one will be able to prove Fermat's Last Theorem even if humanity is destroyed by nuclear war. Little Andrew immediately made up his mind to prove Bell wrong.
In 1986, Andrew M. Wiles has become a professor at Princeton University, but he has been unable to put down the proof of Fermat's theorem and has worked on it for many years.
Finally, he made a breakthrough, and on June 23, 1993, he published his research at a conference in his hometown of Cambridge. When he presented his paper, the entire attendees were mesmerized. In the end, Andrew throws a "bombshell": this is the proof of Fermat's Last Theorem, and I think I will stop here.
| Andrew · Wiles
The media went crazy about it. But when Andrew examines his own large and complex evidence, ready to send it to validators as usual, he discovers an error. To prove that this theorem is correct in all cases, Andrew can only prove that a confirmed case can infinitely deduce the next, like the domino effect.
The problem is that the case has not been confirmed. Andrew collapsed, and instead of defeating the "demon" of Fermat's Last Theorem, he declared himself to the world that he had done it.
Andrew only asked his former student, Richard M. Taylor confided his thoughts and then went back to research and correct the error. On September 19, 1994, he suddenly had a stroke of genius: what if this error was not a flaw but a path to proof?
This was soon proven, and Andrew finally handed over his answer sheet, which was verified by his peers over the next 3 years. Finally, on June 27, 1997, Andrew finally proved Fermat's Last Theorem and successfully received a prize of 100,000 marks.
Andrew · Wiles's final 130-page proof, which used many modern mathematical theorems and operations, may have been a solution that Fermat had not thought of more than 300 years ago.
It is hard not to wonder, if what Fermat said at that time was true, then how concise and ingenious should his way of proving be?
Or maybe Fermat just suddenly decided to make a joke with people, but the joke showed us so many top talents coming after him.
Perhaps Fermat's Last Theorem is indeed too far away from us, and in daily life, it is easy for us to fall into the idea of "mathematical uselessness". But if you think about it, the formula we use today may come from the conjecture of a mathematician, and after that, countless mathematicians have used tens or hundreds of pages of proof to turn "conjecture" into "theorem", so that we can use it with confidence, and these theorems continue to push people to understand the world around them, and inspire people to explore deeper into the universe...
All these discoveries will continue to develop and extend outward on the basis of the research of predecessors, and because of this, mathematics has a broader territory waiting for people to explore and discover.
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Author: Twilight Ma
The Recommended Title for this issue: The Fibonacci Rabbit: Fifty Discoveries That Changed Mathematics, unread
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First sentence: Why is there 60 seconds in 1 minute? Can monkeys also write Shakespeare? This book uses postmodern illustrations and easy-to-understand language to take you through 50 groundbreaking mathematical discoveries over the past 20,000 years, and learn the anecdotes of mathematicians.
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