FERMAT'S ENIGMA: The Quest to Solve the World's Greatest Mathematical Problem

By Simon Singh

Walker & Company

315 pp., $22

The history of the physical sciences reveals again and again that each successive explanation must be built on the ruins of the previous one. No scientist today would seriously argue the earth-centered cosmology of Ptolemy, or the four humors of the ancient physician Galen.

Yet every schoolchild learns the Pythagorean theorem, equating the square of the hypotenuse of the right triangle to the sum of the squares of its two sides.

Mathematics uses a different standard of proof than the other sciences. In physics or chemistry, a theorem is considered proven if it describes some known phenomenon and accurately predicts its consequences. Mathematical proofs, on the other hand, are absolute. If a conjecture is true at all, it must be true forever, under all circumstances. This is the reward of mathematical speculation and also its burden.

Andrew Wiles, a professor at Princeton University, succeeded in 1995 in solving a problem set more than 300 years before by the great French amateur mathematician Pierre de Fermat. By this accomplishment, the unassuming British mathematical researcher has become a media superstar. Most recently, his seven-year effort is featured in a PBS science special entitled: "The Proof." Simon Singh co-produced and directed the BBC film on which the Nova program is based. For those seeking more detail about this stunning mathematical achievement, Singh has written an excellent guide to the mathematics and the history of Fermat's famous "Last Theorem."

The problem can be quickly stated: Prove there is no whole number solution for the equation xn + yn = zn for any value of n greater than 2. (If n equals 2 this is the familiar Pythagorean theorem.) Ever since Fermat scribbled in his copy of a text that he had a truly marvelous proof of this conjecture but did not have room in the margin for the whole proof, mathematicians have tried to figure out what he meant.

Wiles proved the theorem using the full set of modern mathematical tools unknown to Fermat. No one has yet been able to determine whether the problem could have been solved in the 17th century. Fermat, a jurist and civil servant, never published any of the results of his mathematical pastime. He was notorious for rarely bothering even to note the steps by which a conclusion was reached. In the intervening centuries, the world's greatest mathematicians have attempted this intellectual Everest and were defeated. Gauss, perhaps the greatest arithmetician of all time, claimed the problem was unsolvable.

The amazing achievement of Singh's book is that it actually makes the logic of the modern proof understandable to the nonspecialist.

Some simple algebra is to be found in a series of appendixes, easily accessible to any high school student. The body of the work is determinedly nontechnical. Starting with the ancient Greeks, the author patiently and clearly leads the reader through the history of number theory and explains the famous "Last Theorem" in the context of the science of the time. Finally, the author gives some sense of the work since the 17th century necessary for an understanding of Wiles accomplishment.

The actual proof runs to more than 100 pages of densely argued text. Sadly, the details are incomprehensible to all but a few researchers in the field. Yet the reader of this volume is left with the sense at least of understanding how the proof came together.

More important, Singh shows why it is significant that this problem should have been solved. The efforts to solve the problem resulted in the development of techniques that have had a tremendous impact on other fields of knowledge, even though the prize remained just out of reach. The final achievement, like a great novel or a magnificent symphony, must be valued among the profound works of the human imagination.

* Frederick Pratter is a freelance writer living in Hull, Mass.

A Young Boy's Dream TO FIND A SOLUTION

* For over 300 years many of the greatest mathematicians had tried to rediscover Fermat's lost proof and failed. ... In Milton Road Library, ten-year-old Wiles stared at the most infamous problem in mathematics, undaunted by the knowledge that the most brilliant minds on the planet had failed to rediscover the proof. ...Thirty years later Andrew Wiles stood in the auditorium of the Isaac Newton Institute. He scribbled on the board and then, struggling to contain his glee, stared at his audience.

- From 'Fermat's Enigma'