An obsessive search for a solution

As I've noted in this column on other occasions, it's rare to find a book on math that can be read for pleasure. So I was prepared to be disappointed by a book on an abstruse problem called Fermat's Last Theorem. However, I was very pleasantly surprised by the book, Fermat's Enigma: The Epic Quest to Solve the World's Greatest Mathematical Problem, by Simon Singh (Walker & Co., New York, NY, 1997). The book is an enthralling account of the search for a proof of a theorem apparently discovered, but not explained, by a French amateur mathematician, Pierre de Fermat, in 1637.

Fermat's Last Theorem is easy to state but incredibly difficult to prove. The theorem states that there are no whole-number solutions for the equation xn + yn = zn where n > 2. The whole-number solution where n = 2 is familiar to everyone who has taken math in high school as the Pythagorean Theorem, proved by the ancient Greek philosopher Pythagoras (in a right-angled triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides, or x2 + y2 = z2). There is a simple proof of the Pythagorean Theorem, which you can find in many math textbooks. But proving that Fermat's Last Theorem is valid has tried the patience of amateur and professional mathematicians for centuries.

Fermat himself claimed to have a proof for his theorem. While studying a translation of a work by an ancient Greek mathematician, Diophantus of Alexandria, Fermat made a marginal note of his theorem and wrote, in Latin, "I have a truly marvelous demonstration of this proposition which this margin is too narrow to contain." But Fermat never took the time to write down his proof or explain it to any of his contemporaries. He died in 1665 leaving only the enigmatic marginal statement in his copy of Diophantus's Arithmetica.

An intriguing saga

Fermat's Last Theorem, says author Singh, "is at the heart of an intriguing saga of courage, skullduggery, cunning, and tragedy involving the greatest heroes of mathematics." Dozens of eminent minds in mathematics devoted much of their professional careers to attempting to prove Fermat's Last Theorem. A Japanese mathematician, Yutaka Taniyama, killed himself in 1988. A 19th-century French physicist, Sophie Germain, resorted to taking on the identity of a male student at the prestigious but chauvinistic Ecole Polytechnique in Paris in order to pursue her work on number theory. Another French mathematician, Evariste Galois, whose work was a precursor to finding a proof, was killed at the age of 21 in a duel for the hand of a woman. On the opposite side of the coin, Paul Wolfskehl, a 19th-century German industrialist, attributed his obsession with Fermat's Last Theorem to saving him from suicide. In fact, in 1908 Wolfskehl bequeathed a large part of his fortune as a prize to be awarded to anyone who succeeded in proving Fermat's Last Theorem.

The prize was 100,000 marks, the equivalent of a million dollars by today's standards.

The Wolfskehl Prize, worth $50,000 today, was eventually awarded in June 1997 to British mathematician Andrew Wiles. As Singh describes in his book, Wiles was initially captivated by Fermat's Last Theorem as a schoolboy. He eventually became an academic mathematician who discovered a proof of Fermat's Last Theorem in 1993 after working in solitude for seven years. After Wiles presented his proof, there was a flurry of academic emails, several expressing skepticism about Wiles's work. A flaw in Wiles's proof was discovered, so Wiles retired again to isolation at Princeton University. After more than a year of frenzied reworking, Wiles corrected his proof, which was published in two papers, consisting of 130 pages in total, in Annals of Mathematics in May of 1995. The news even made the front page of The New York Times in a story headlined "Mathematician Calls Classic Riddle Solved."

As Wiles says in the book, "I had this very rare privilege of being able to pursue in my adult life what had been my childhood dream." Singh makes Wiles's search come alive in the pages of the book while interweaving the stories of many mathematicians, from Leonhard Euler to Alan Turing. It's a remarkable journey through the history of math?and one you can read without being a mathematician.