Starting from Fermat's Last Theorem (I): Background

从费马大定理谈起（一）：背景简介 苏剑林 · 2014-08-15 · Mathematics number theory Diophantine equation fermat's last theorem

Fermat's Last Theorem, also known simply as Fermat's Last Theorem, states that

Let $n$ be an integer greater than 2. Then the indeterminate equation $x^n+y^n=z^n$ has no integer solutions in which all of the variables are nonzero.

Pierre_de_Fermat

Anyone who has read a little about the history of mathematics will know that this theorem was first noted by the French amateur mathematician Pierre de Fermat in 1637, in the margin next to Proposition 8 of Book XI while reading a Latin translation of Diophantus's Arithmetica. He added: "I have discovered a truly marvelous proof of this, but there is not enough space here to write it down." Later scholarship suggests that Fermat may have had a way of proving the cases n=3, 4, 5, but it is unlikely that he had a general proof, since it took Andrew Wiles 130 pages and a great deal of sophisticated modern theory to fully prove Fermat's Last Theorem in the 1990s. So Fermat's assertion at the time was more likely just an inductive conjecture.more

In this series of posts, I will try to start from Fermat's Last Theorem and introduce some knowledge and history related to indeterminate equations, number fields, and number rings, and I will present proofs of Fermat's Last Theorem for the cases $n=3$ and $n=4$. Both of these proofs make use of number fields that go beyond the ordinary integers (the Eisenstein integers and the Gaussian integers), and both illustrate how powerful the idea of extending number fields can be. Through them we can catch a glimpse of the modern ideas behind the proof of Fermat's Last Theorem, since the tools Wiles used in his proof are, in essence, the same kind of tools — just pushed much deeper and much further. This post is my attempt to build a bridge between amateur math enthusiasts and Wiles's proof of Fermat's Last Theorem — though, to be clear, I do not intend to actually write out Wiles's proof of Fermat's Last Theorem here, and frankly I'm not capable of doing so.

Let me briefly sketch the historical development of the proof of Fermat's Last Theorem. For a long stretch of time after Fermat proposed the conjecture, people could only prove it for certain specific values of $n$. In the mid-19th century, the mathematician Kummer made a major advance in the study of the conjecture: he proved that for every regular prime $p$, the equation $x^p+y^p=z^p$ has no nonzero integer solutions. Kummer's ideas were later developed further into "Iwasawa theory" ["泽岩理论" in the original — a rendering of the term], which Wiles further extended, adding in a great many other mathematical ideas and techniques, and finally succeeded in proving Fermat's Last Theorem. The proofs of the two special cases of Fermat's Last Theorem that this series will present are based on the ideas Kummer used in his proof.

Reference sites:

http://fermatslasttheorem.blogspot.com/

http://zh.wikipedia.org/zh/费马大定理