# Is there any book/resource which explain the general idea of the proof of Fermat's last theorem?

I look for a book/resource which display the general idea of the proof of Fermat last theorem in a simple manner for the public.

I mean, books which is not for mathematicians but for the general public. Books like:

Gödel's Theorem: An Incomplete Guide to Its Use and Abuse by Torkel Franzén

Do anyone now any books of this kind? Also, articles or any resources are good.

• There may very well be a few books that "claim" to do what you ask. I doubt any on this planet actually succeed. – PVAL-inactive Sep 18 '14 at 2:15
• A similar question of which an answer would grab my interest is, "What topics would one need to master in order to understand the proof of Fermat's Last Theorem?". It would be quite the journey to attempt it. – user109879 Sep 18 '14 at 4:48
• I had a list of such books, but it's too long to fit in the margin of this page... :-J – keshlam Sep 18 '14 at 21:51

It's very hard, if not impossible, to write such a book. Gödel's theorems are about the foundations of math, and so they don't actually use any significant mathematical results other than what they introduce. The main ideas, such as Gödel numbering, are essentially self-contained, and you can explain them to an audience of mathematically-minded people with enough patience.

On the other hand the proof of Fermat's Last Theorem depends crucially on various huge machineries developed throughout the 20th century, which are in turn based on and motivated by ideas from the 19th century. The new ideas involved all occur in a pre-existing conceptually advanced part of the mathematical spectrum, almost the polar opposite of logic and foundations in its reliance on earlier work.

One can try to give some ideas, but unfortunately it becomes extremely vague and confusing very fast. If the audience doesn't know what an elliptic curve is, one may be tempted to to speak vaguely of "doughnut shapes". For representation theory of Galois groups one could think of 'possible manifestations of the symmetry inherent in numbers', and instead of modular forms you might say 'highly symmetric functions'. But then you end up talking nonsense:

The proof involves showing that doughnut shapes are in some sense the same as highly symmetric functions. It had been shown beforehand that every doughnut surface has a highly symmetric function associated to it, but that this might be a bijection was a relatively new idea. It was established by recasting the problem in the world of possible manifestations of the symmetry inherent in numbers. Each doughnut shape results in such a manifestation, as does each highly symmetric function. One can then reformulate the bijection as a relation between the manifestations of symmetries of numbers associated to doughtnut shapes and to highly symmetric functions. Now one key idea is to consider these objects in a number system where numbers that have the same residue modulo $p$ are identified. It had been conjectured that certain manifestations of number symmetry in this world actually come from the regular world of manifestations of ...

It starts to sound like complete non-sense very quickly, and it would have to go on like that for dozens and dozens of pages.

On the other hand if you try to actually define what these things are and proceed rigorously you will not get very far. The Galois group is typically defined at the end of a year-long abstract algebra course. Even if you're willing to just consider examples and definitions, with no proofs for any theorems, you still need to define what a group is, what fields are, what field extensions are, what is a representation, etc. etc.

It's probably more fruitful to read about the general ideas involved instead, as opposed to how they were actually used. If a book can give a public audience some idea of what elliptic curves and modular forms are, that's already a very impressive accomplishment.

Another way to approach it is to read about the history of the problem, the people involved, and slowly the various ideas they introduced to tackle it. This would itself be a good introduction to modern number theory.

I like the Nova documentary 'The Proof'. It's fit for a public audience and does give them some ideas. I doubt you can expect much more, but I'd love to be proved wrong.

• I've often thought that one could design a very compelling (and complete) NT course just "trying to solve FLT". Even just getting to where we were around 1900 would present the student with a huge range of topics and techniques. – Kieren MacMillan Sep 18 '14 at 2:23
• An analogy that springs to my physics brain. Explaining Godel's theorem is like talking about the basic postulates of special relativity: their consequences may be hard to accept, but the presentation is more-or less self-contained. Explaining FLT is like trying to explain the Higgs boson: You can give some of the big picture, but the machinery and context requires years of background. – Semiclassical Sep 18 '14 at 4:52
• It's interesting that your "dumbed-down" paragraph would make no less sense at all if you just said "elliptic curves" rather than "doughnut shapes" because, of course, it doesn't appeal to any intuitive property of doughnuts anyway. The dumbing down has got you nowhere. But for the Goedel case, a bit of hand-waving about the precise details of Goedel numbering is a welcome relief to the lay reader, because once they have the basic idea you can easily state results that they understand and they're perfectly happy to believe without seeing every line of the proof. – Steve Jessop Sep 18 '14 at 8:39
• It's a little unfair to use the impossibility of describing the proof in a paragraph as a demonstration that it's impossible to write an accessible book about it, but I take your point. But, mostly, I wanted to say, "I have a truly marvellous description of the proof of FLT which this Stack Exchange post is too small to contain." – David Richerby Sep 18 '14 at 11:32
• @David: I'm afraid I've failed to communicate my point. It's not that one can't describe the proof in a paragraph or a book. It's that in order to speak to an audience with no background in modern math one is forced to speak in approximate terms, and these don't lend themselves to any sort of intricate description of detail. You can speak about the proof as much as you want, and fill an entire book with either pleasant anecdotes or vague analogies, but the longer a description of something complicated with oversimplified terms goes on the less sense it makes. – Zavosh Sep 18 '14 at 17:16

Simon Singh's book, creatively titled "Fermat's Last Theorem" ("Fermat's Enigma" in the US), was a very fun read for a 15 year-old me many years ago :) It gives a lot of the history, as well as a rough outline of the proof (at the "using-lines-of-dominoes-to-explain-proof-by-induction" level).

• This book is a nice light read, but it tells you very little actual math. Mostly it's transcripts of interviews and reprints of e-mails sent when the proof was announced. In addition, I find the actual mathematical content is a bit misleading. It basically teaches the reader what induction is, and pretends Fermat's last theorem was proved that way. It's decent as a historical account about people, but not at as a book about math. – Zavosh Sep 18 '14 at 7:35
• In the same vein, there's also the book by Aczel. – Frunobulax Sep 18 '14 at 12:37
• (+1): The OP asked for "general idea of the proof of Fermat last theorem in a simple manner for the public". I think Simon Singh's does a good job at that. As others have pointed out, it is not an easy task to explain some mathematical concepts to the public. – Thomas Sep 18 '14 at 12:55

Fearless symmetry by Avner Ash and Robert Gross is not specifically about Fermat's theorem, but goes through some theory behind the proof. It's quite involved for being a pop science book and I applaud the authors for even trying to present topics such as quadratic reciprocity, elliptic curve theory and Frobenius groups to a broad audience.

• Absolutely, what a great book. Also "Elliptic Tales", and there is a new one coming out shortly about number theory by the same authors. – user50229 Jun 6 '16 at 13:26

Perhaps the closest thing is this article:

"A marvelous proof", by Fernando Gouvêa,
The American Mathematical Monthly, 101 (3), March 1994, pp. 203–222.

This article got the MAA Lester R. Ford Award in 1995.

This and other papers (with various degrees of difficulty) can be found at Bluff your way in Fermat's Last Theorem.

• A quote from this paper: "The $p$-adic numbers are an extension of the field of rational numbers which are, in many ways, analogous to the real numbers. Like the real numbers, they can be obtained by defining a notion of distance between rational numbers and passing to the completion with respect to that distance." This is not my idea of something written for the general public. In fact this is fully a mathematical expository paper written for people with at least the equivalent of a good undergraduate degree in pure math, including Galois theory and such. – Zavosh Sep 18 '14 at 7:28

The Epilogue in Ribenboim's book Fermat's Last Theorem for Amateurs is about the closest you'll get, I think. Of course, it still requires some mathematical skill just to hang with the "overview" — but, as Prometheus pointed out, you can't talk about FLT without using "real math".

For the record, the whole Ribenboim is a great book.

• Which index? there are A and B but none are about "general idea" only about references and wrong and incomplete proofs. – Fawzy Hegab Sep 18 '14 at 3:36
• Sorry, the Epilogue (XI 2 & 3). – Kieren MacMillan Sep 18 '14 at 11:24

Two books which might be what you're looking for (haven't read them myself) are Invitation to the Mathematics of Fermat-Wiles by Yves Hellegouarch and the 3rd edition of Algebraic Number Theory and Fermat's Last Theorem by Ian Stewart and David Tall.

• Hellegouarch's book suffers (in much the same way as other accounts) in spending a lot of time laying the easier foundations (or perhaps the less very difficult) eg elliptic curves, but then speeds up once he hits the more difficult parts (eg Hecke Operators). I think it would be very hard for someone without a lot of mathematical sophistication to use it. to get all the way, but a nice start. I've read quite a few such books, but thought they weren't elementary enough for this question. – Francis Davey Sep 18 '14 at 21:41

The upcoming book Summing It Up by Robert Gross and Avner Ash seems to be the type of book that you want. I believe the third section of the book will go into the topics relevant to understanding Fermat's Last Theorem, according to the description on the linked page.