Understanding Wiles' Proof of Fermat's Last Theorem: A Beginner's Guide

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Discussion Overview

The discussion revolves around the prerequisites and understanding necessary to study Andrew Wiles' proof of Fermat's Last Theorem, particularly from the perspective of beginners in number theory. Participants share their experiences, recommend resources, and outline the mathematical concepts involved in Wiles' proof.

Discussion Character

  • Exploratory
  • Technical explanation
  • Conceptual clarification
  • Debate/contested
  • Homework-related

Main Points Raised

  • One participant inquires about the knowledge required to grasp Wiles' proof, expressing a desire for accessible explanations.
  • Another suggests starting with Simon Singh's book on Fermat's Last Theorem as a helpful resource.
  • A participant summarizes that Wiles' proof involves elliptic curves rather than directly addressing Fermat's theorem, mentioning the contributions of Frey and Ribet.
  • Discussion includes the complexity of Wiles' proof, noting the extensive mathematics involved, including elliptic curves and modular forms.
  • One participant emphasizes the historical context of Wiles' work, stating that he built upon previous research and conjectures in the field.
  • Another participant lists various advanced mathematical topics that would be beneficial for understanding Wiles' proof, including complex analysis and group theory.
  • A participant reflects on the time commitment required to gain a working knowledge of the necessary mathematics, suggesting it could take about ten years.
  • There is a mention of the relationship between number theory and other fields, such as physics, highlighting the broader implications of open questions in number theory.
  • Some participants identify themselves as mathematicians or professors, indicating a blend of professional and personal interest in mathematics.
  • A light-hearted exchange occurs regarding the passion for mathematics among participants, regardless of their professional status.
  • Questions arise about the relevance of proofs for special cases of Fermat's Last Theorem, indicating an ongoing exploration of the topic.

Areas of Agreement / Disagreement

Participants generally agree on the complexity of Wiles' proof and the extensive background knowledge required. However, there is no consensus on the specific resources or paths to take for beginners, as different suggestions are made regarding introductory materials.

Contextual Notes

Limitations include the varying levels of mathematical background among participants, which affects their ability to engage with the discussion. The discussion also reflects differing opinions on the accessibility of the mathematics involved in Wiles' proof.

Who May Find This Useful

This discussion may be useful for beginners in number theory, educators seeking resources for teaching, and those interested in the historical and mathematical context of Fermat's Last Theorem.

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Hi everybody,
I wonder what knowledge is required to study Wiles proof of the last theorem of Fermat. Of course, i don't mean fully understand it but just get to understand some of his thoughts and how he actually approached the problem and found the solution. I would really like to hear the experience of anyone that has studied it for a while.
Just make something clear: I wouldn't like very complicated explanations as I am really new in number theory. I have until now studied mostly analysis and linear algebra but i am now finding number theory a really interesting field.
Thanks
 
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Well, you should start with Simon Singh's "Fermat's last theorem" book.
 
I have not read Wiles paper but have read Singh's book, and I recommend it highly.

I know a little about the proof though, as follows.

Wiles did not even look at Fermat's last theorem. He proved a theorem about elliptic curves, which had been conjectured by Frey, and proved by Ribet, to imply Fermat's theorem.

An elliptic curve is a curve with equation of form y^2 = (x-a)(x-b)(x-c). So you see there are three constants in there, a,b,c.

There are also three constants in Fermat's last theorem, p^n + q^n = r^n, namely p,q,r.

Frey realized that if p,q,r, satisfied Fermat's theorem, then a simple recombination of them, would give numbers a,b,c, that defined a very strange elliptic curve, one that did not arise from a "modular form".

Then Wiles proved no such elliptic curves could exist. Actually he assumed another condition, "stability", but that was enough to imply Fermat.

You can get an idea of how much mathematics is involved in Wiles proof by looking at his papers or at some of the books that have been written to explain the proof since then. There is a lot, but you might start by finding out something about elliptic curves.
 
The idea of how he found the solution is simple.

Over the last 50 years various people proved small things that led to the Taniyama-Shimura conjecture, and its proof. (It states that every semi-stable elliptic curve is modular, or equivalently they give the same L-series, but that isn't important). The point is that he didn't sit down and prove "it" he proved a classification theorem which indicated something that someone else had shown would do something if it were to exist... That is he had an explicit problem to solve and lots of tools that had been developed over the years to solve it: he did what all research mathematics is, he stood on the shoulders of giants.


NB. If you want to understand anything more than that about the actual mathematics then you need to be familiar with:

Complex Analysis, such as holomorphic functions, upper half planes, laurent expansions, reimann surfaces might help, Fourier series type results. How these apply to Number Theory, via L-series, and so on, Modular forms, which are approximately functions of complex variables that take values on a torus. Elliptic curves, which are equations of the form y^2 = x^3 + ax +b, and their groups, and (galois) representations of them.

That ought to take you about 10 years to get a working knowledge of.
 
After reading Matt's catalog of topics, it occurred to me that a "beginning" book might be Serre's "Course of Arithmetic". That discusses L - functions, and modular forms, and their use in number theory. But you probably need complex analysis and some group theory first. You could try it though. The first page or so already has some nice stuff.

Here is an outline of some topics from my course from the more advanced part of Serre in 1997.
II) Holomorphic Functions
A. Taylor Series Expansions: Holomorphic versus Analytic
B. Order of Zeroes and Poles / Removable Singularities
C. Principle of Isolated zeroes / Analytic Continuation
D. Uniform Limits of Holomorphic Functions
E. Properties of Logarithmic and Exponential Functions

III) Group Characters
A. Dual group, Double Duality
B. Orthogonality Relations

IV) Dirichlet Series: Four Convergence theorems
A. Prop.6: Maximal Half plane of convergence
B. Prop.7: Convergence vs removable singularities
C. Prop.8: When ordinary D - series converge for Re(s) > 1
D. Prop.9: When ordinary D - series converge for Re(s) > 0

V) Zeta Function and L - Functions
A. Definitions and Euler Product Expansions
B. Prop. 10, z(s) has a simple pole at s=1
C. Prop.10, Cor 2: Behavior of ∑p p-s, ∑p,k≥2 1/kpks, s-->1+
D. Properties of LX = ∑ X(n)/ns, zm(s) = ∏X LX(s)
E. X≠1 implies LX(1) ≠ 0.

VI) Density and Dirichlet Theorem
A. Concept of (Analytic) Density of a set of primes
B. Lemmas 7,8: Behavior of fX(s) = ∑p X(p)/ps, as s-->1+
C. Lemma 9: ƒ(m)(∑p=a(mod m) p-s) = ∑X X(a-1) fX(s)
D. Density (Pa = {p : p=a(mod m)}) = 1/ƒ(m).
 
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You should not that mathwonk's (x-a)(x-b)(x-c) is equivalent to my x^3+ax+b, though the a's and b's are different. He has three constants, and whilst I have two I have "moved" the frame of reference.
 
Thanks for your answers. I understood that a person that doen't study mathematics(although i do have lots of mathematics in my studies-computer engineering is my subject) can't probably get to understand all these topics in order to reach Fermat because of time. I don't think that i will have much time to study all this staff while also having many other things to study. I also didn't know that number theory is so much related to other fields of mathematics. Anyway, thanks for your help- I will do as much as i can. Just a small question: are you mathematicians? I can't believe that this is just your hobby!
 
Yes, I'm a mathematician.

Number theory can surprise: some important open questions in number theory have important implications in physics. See eg Riemann's Zeroes.
 
I am a college prof of math.

But math is also a hobby for people like me and Matt, i.e. we love it.

Look how many posts he has logged, helping people for free.
 
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  • #10
As one of my Colleagues said to me recently (He's a professor in the English sense, which approximately means distinguished research professor in the US sense), "... I get paid to do something I'd've done for nothing."
 
  • #11
matt grime said:
"... I get paid to do something I'd've done for nothing."
Heh. That's pretty cool. :smile:
 
  • #12
What do we do with all the proofs of the special cases of FLT?
 
  • #13
What do you mean?
 
  • #14
matt grime said:
"... I get paid to do something I'd've done for nothing."

Just hope his boss wasn't listening. Might take him up on it. :smile:
 

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