Fermat’s Last Theorem, also known as Fermat’s Final Theorem, states that:
Let n be a positive integer greater than 2. Then the Diophantine equation x^n + y^n = z^n has no integer solutions such that x, y, z are all non-zero.
Friends who have read a bit of mathematical history should know that this theorem was first noted by the French amateur mathematician Pierre de Fermat in 1637. While reading the Latin translation of Diophantus’ Arithmetica, he wrote in the margin of the 8th proposition of the 11th volume: “I have discovered a truly marvelous proof of this, which this margin is too narrow to contain.” According to later research and verification, Fermat might have had a way to prove the cases for n=3, 4, 5, but it is highly unlikely that he could have provided a general proof. This is because in the 1990s, Andrew Wiles required 130 pages and utilized complex modern theories to completely prove Fermat’s Last Theorem. Therefore, Fermat’s assertion at the time was more likely just an inductive conjecture.
In this series of articles, I will attempt to start from Fermat’s Last Theorem to introduce knowledge and history related to Diophantine equations, number fields, and rings of integers. I will also provide proofs for Fermat’s Last Theorem for the cases n=3 and n=4. Both of these proofs utilize number fields that extend beyond the scope of rational integers (specifically, Eisenstein integers and Gaussian integers). These examples demonstrate that the idea of extending number fields is quite powerful. From them, we can catch a glimpse of the modern ideas used to prove Fermat’s Last Theorem, because the tools Wiles used are essentially the same in nature, just deeper and more advanced. This article aims to build a bridge between amateur mathematics enthusiasts and Wiles’ proof of Fermat’s Last Theorem—though, of course, I do not intend to write out Wiles’ full proof process, nor do I have the capability to do so.
Here is a brief overview of the development of the proof of Fermat’s Last Theorem: For a long period after Fermat proposed the conjecture, people could only prove it for specific values of n. In the mid-19th century, the mathematician Kummer greatly advanced the research of this conjecture. He proved that for all regular primes p, the equation x^p + y^p = z^p has no non-zero integer solutions. Kummer’s ideas were further developed into “Iwasawa theory.” Wiles further developed Iwasawa theory and incorporated many other mathematical ideas and techniques to finally prove Fermat’s Last Theorem. The proofs of the two special cases of Fermat’s Last Theorem to be introduced in this series are based on Kummer’s ideas for proving the theorem.
Reference websites:
http://fermatslasttheorem.blogspot.com/
https://en.wikipedia.org/wiki/Fermat’s_Last_Theorem
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