Fermat

Fermat's Last Theorem: Unlocking the Secret of an Ancient Mathematical Problem

Amir D. Aczel

You've probably heard the story: Pierre de Fermat, a seventeenth-century mathematician, is reading a problem in a centuries-old Greek textbook related to breaking down a squared number into two squares (like 3^² + 4^² = 5^², or 12^² + 5^² = 13^²) and in a sudden burst of inspiration scribbles a note in Latin in the margin:

On the other hand, it is impossible to separate a cube into two cubes, or a biquadrate into two biquadrates, or generally any power except a square into two powers with the same exponent. I have discovered a truly marvelous proof of this, which, however, the margin is not large enough to contain.

Don't you hate it when that happens?

This statement, which became known as "Fermat's Last Theorem", was enough to send three centuries of mathematicians into frenzies: Was Fermat's claim provable? Or was it (like dozens of his other so-called "theorems") just wild conjecture?

This little book (130 pages) takes us from Babylon to Princeton University in an effort to convey to the general public why Fermat's Last Theorem captured the attention of so many learned minds over 300 years. To some extent it fails in this mission; telling brief stories of unknown geniuses who, over the years, developed some niche of number theory that would later be used to prove the theorem but telling these stories is a brief, disconnected fashion that leaves the reader wondering what happened to the rest of the story. Yet in another respect it succeeds; showing how this simply-stated theorem reaches back to the most fundamental principles of mathematics (what you know as the Pythagorean Theorem, best stated by the Scarecrow in the Wizard of Oz, "The square of the hypotenuse of a right triangle is equal to the sum of the squares of the two sides," but known to the Babylonians and Egyptians thousands of years before Pythagoras -- or the Scarecrow) yet requires the application of the most sophisticated fields of math to resolve.

The crowning moment

The most compelling portion of the book is the story of the man who spent seven years in isolation building a proof of the theorem, finally presenting it at a math symposium, only to be shown to be incorrect; and who then spent another year re-proving it using another technique. While it's difficult to imagine a dramatic moment at a convention of mathematicians (yawn), the story of Andrew Wiles standing at the chalkboard over three hours building his proof builds to just such a moment. After leading the gathering crowd through hundreds of lines of equations without announcing his final goal, he adds a final line and announces, "And this proves Fermat's Last Theorem. I think I'll stop here." Three hundred years of futile attempts comes to an end at a remote math conference in 1993.

I think you can read this book and skip the math -- there's not much of it -- and still get the picture. If you're into the math you'll be disappointed with the lack of detail, though I doubt many readers have the sophistication to comprehend the proof in its entirety. This book does a good job of breathing life into a subject most consider to be quite dull. More appropriately, it reveals the life in this subject.