A Mathematical Legend Spanning 300 Years: Fermat’s Last Theorem

  The recently aired TV series “Basic Law of Genius” has greatly stimulated the enthusiasm of young audiences to learn mathematics, and the famous Fermat’s Last Theorem is specifically mentioned in the plot. Fermat’s Last Theorem is a glorious page in the history of mathematics. Let’s learn more about this theorem together!
“King of Amateur Mathematicians”: Fermat

  It is generally believed that the founder of modern number theory was Fermat (1601-1665). He is a French mathematician, but his main profession is a lawyer, and mathematics is his hobby. Because of his fundamental contributions to many branches of mathematics such as number theory, calculus, analytic geometry, and probability, he was crowned as the “King of Amateur Mathematicians” by later generations.
  In fact, at the beginning of the 17th century, the subject of mathematics had just woken up from the millennia-long dark ages of the Middle Ages in Europe and was not taken seriously. At that time, only Oxford University had a dedicated mathematics professorship in the entire European country (started in 1619), so most mathematicians in the 17th century were amateurs. And Fermat is one of the most outstanding.
Birth of Fermat’s Last Theorem

  The source of Fermat’s research work on number theory comes from a book called “Arithmetic” by the ancient Greek mathematician Diophantus (about 200-284 years). There are 13 volumes of this book, but only 6 of them survive.
  In 1621, the French mathematician Bechet (1581-1638) translated the “Arithmetic” into Latin. In addition to this, Bechet has compiled a collection of interesting math puzzles, including many well-known puzzles, one of which is the weight problem: “How many weights are needed to weigh 1 gram on a balance? The weight of any whole number of grams between 40 grams?”
  Diophantus’ “Arithmetic” is actually a collection of problems. There are more than 100 questions, and Diophantus gives detailed answers to each question. But Fermat is interested in going beyond the original topic, deviating from the analogy, and thinking about and solving some related, more subtle problems.

Schematic diagram of the Pythagorean theorem

  As Fermat said, in the process of “thinking what no one had thought before,” he “discovered many very beautiful theorems.” This is how the famous Fermat’s Last Theorem was born.
What does Fermat’s last theorem say?

  In Book 2 of Arithmetic, Diophantus discusses Pythagorean numbers, the positive integer solutions to the equation x2+y2=z2. For example: (x, y, z) = (3, 4, 5) is the simplest set of Pythagorean numbers, which is what we often call “three hooks, four strings and five”.
  When Fermat read this discussion of Diophantus, he wondered what would happen if the exponent 2 in this equation was replaced by 3? Fermat found that the equation x3+y3=z3 no longer has a positive integer solution at this time. Going a step further, he replaced the exponent 3 in the equation with any larger integer n, resulting in the equation xn+yn=zn.
  Fermat came to the following general conclusion: When n is greater than 2, the equation xn+yn=zn has no positive integer solution.
  He wrote in the blank of the paragraph of “Arithmetic”: “I have an excellent proof for this proposition, but unfortunately the blank space here is too small to write down.” This short comment has aroused the strong interest of many mathematicians in later generations. . But no one knows what Fermat’s “brilliant proof” is. Fermat himself gave the proof for n=4 in 1640, and the proof for n=3 was given by the Swiss mathematician Euler (1707-1783) more than 100 years later (1758-1770).
Why is there a conclusion after 350 years?

  So what about when n is equal to some other positive integer? Since then, almost all famous mathematicians in history have participated in the proof of this proposition, but it has remained unresolved for more than 350 years.
  It was not until 1995 that Fermat’s above thesis was proved by the British mathematician Wiles (1953-). But Wiles’ splendid proof was by no means the kind of proof Fermat could understand at the time. Because Wiles used the most advanced thinking, language and technology of modern number theory, it took 8 years and 130 pages to prove Fermat’s Last Theorem.
  Proving Fermat’s Last Theorem was Wiles’ childhood dream. As he said: “For everyone, the realization of childhood dreams is full of wonderful magic. Only very few people can make dreams come true, and I am one of the lucky ones.”
  Some people may ask, so many mathematicians spend a lot of money Time proves a proposition, what is the meaning? In fact, the process of proving Fermat’s Last Theorem is a process of constantly opening up new fields of mathematics. For example, when Euler proved Fermat’s Last Theorem n=3, he invented imaginary numbers and expanded the field of numbers. After that, imaginary numbers played an important role in many fields such as informatics.
  When Fermat wrote that paragraph in the blank space, he didn’t necessarily know what the use of his research was, but later researchers have used it to a greater extent. Therefore, to this day, Fermat’s Last Theorem has passed through nearly 400 years and still has its unique charm!