## Book review: Fermat’s Enigma

After having it recommended to me several times, I finally picked up this book when I happened to see it at our favorite local used bookstore. I think I was a bit reluctant because I previously read another book about Fermat’s Last Theorem which I didn’t like at all. I needn’t have worried; this one is fantastic. It does a wonderful job conveying the mathematical landscape surrounding this (in)famous theorem and telling a gripping story about people who also happen to be mathematicians. In fact, I think that’s one of the things I like best about it—the degree to which it brings out the essential humanity of the characters, their desires, struggles, and triumphs. I think even readers without much background in mathematics will have no trouble connecting with the characters and getting drawn into the story.

So, what is Fermat’s Last Theorem? It is the claim that the simple-looking equation

$\displaystyle x^n + y^n = z^n$

has no solutions when $x$, $y$, and $z$ are positive integers and $n$ is a positive integer greater than $2$. For example, it is definitely not true that $1047955^3 + 2978302^3 = 3020937^3$, and furthermore no equation of this type will be true, no matter what values we substitute in (as long as $n > 2$). When $n = 2$, however, there are lots of solutions (infinitely many, in fact!), which are called Pythagorean triples since they correspond to the side lengths of right triangles. So it’s a bit surprising, perhaps, that by increasing $n$ we suddenly go from infinitely many solutions to none! Pierre de Fermat claimed he had a proof—which, famously, the margin of the book in which he was writing was too small to contain—but it took a few hundred years for someone else to finally find a proof, using lots of difficult mathematics that was a long way from being invented in Fermat’s time. (So, did Fermat really have a proof? No one knows—but probably not.)

What makes this simple claim so difficult to handle? And why is it worth writing a whole book about the story of its solution? Well, to learn the answers to those questions, you’ll have to read it!

This entry was posted in books, open problems, proof, review and tagged , , , . Bookmark the permalink.

### 10 Responses to Book review: Fermat’s Enigma

1. Jason Dyer says:

I don’t suppose you could capsule review the Fermat book you didn’t like?

• Brent says:

Well, I read it many years ago (even before I started writing this blog, I think), and at this point I don’t remember anything other than the fact that I didn’t like it. Unsurprisingly I am not really inclined to go reread a book I didn’t like just to refresh myself on why I didn’t like it!

2. Juan Valera says:

I liked this book very much.

But, I really *loved* “The Code Book”, also from Simon Singh. I recommend it a lot.

• Brent says:

Yes, I love that book too!

• I basically came here to say the same thing. The Code Book is one of the few books I’ve read multiple times; it’s just so entertaining. As much as I enjoyed these two books I still haven’t found the motivation to read Big Bang though.

3. suevanhattum says:

I read this book years ago and liked it, but much of the math was beyond me. I think I’ll learn more of the math from this delightful book, coming out next week: Math Girls 2: Fermat’s Last Theorem, by Hiroshi Yuki.

4. fsharkey05 says:

Suppose I think I can identify patterns in Power number sequences that show Fermat might well have had a simple proof. How do I avoid being labelled crank number ……..? and get binned with the other hundreds.

• Brent says:

The simple answer is humility. Find someone who you think understands math better than you (it works best if you know them in real life, or have some sort of personal connection—emailing random well-known mathematicians is the perfect way to be labelled as a crank). Ask them, “I think I have a simple proof of Fermat’s Last Theorem but I am probably wrong. Can you help me figure out where my mistake is?” And then adopt an open-minded attitude of learning.

• fsharkey05 says:

Thanks for your helpful comments. I should have made it more clear that I am not thinking I have a simple proof. Only that I have identified patterns that seem to support the possibility that there could be a simple proof. My comment was prompted by a suggestion that it was unlikely that Fermat really had a proof. Best wishes for the New Year.