Wednesday, August 17, 2011

Pierre de Fermat: Happy 410th

French mathematician Pierre de Fermat was born August 17, 1601. Google dedicates its search logo to the event today, the reason why I know. As in the case of all the great mathematicians, I greatly admire Fermat without grasping anything he tackled—much less why he’d bother. But I’ve looked into his life—as also the lives of several others over the years—because they are in some ways kin. I’ve spent my life pondering great puzzles, but in my case none is solvable whereas, in theirs, many were, if not by ordinary humans.

Fermat is most famed for his Last Theorem. The theorem is that there are no whole number (integer) solutions for this equation:

xn + yn = zn if n is greater than two
…which is what Google’s logo reproduces. Fermat jotted into the margins of Dophantus’ Arithmetica the following famous phrase. “I have discovered a truly marvelous proof of this, which this margin is too narrow to contain.”

The issue goes back to Pythagoras’ theorem, namely: “In a right-angled triangle the square on the hypotenuse is equal to the sum of the squares on the other two sides,” which makes:

x2 + y2 = z2
Finding numbers that satisfy this relationship became a preoccupation in mathematics; there are endlessly many, among them 3, 4, and 5. What Fermat claimed was that no such triplets were possible if the power is greater than two.

Fermat wrote his intriguing teaser in 1637. He never published his “marvelous proof.” It took 358 years before Andrew John Wiles, a British mathematician, published a proof in 1995. The story of that proof is told in Fermat’s Enigma, by Simon Singh, Walker and Company, New York, 1997. I got my copy from Brigitte as a birthday gift, my 62nd, in 1998. It’s a truly fascinating but, in the end, a not very satisfactory story. Wiles’ proof takes more than 100 pages to present and makes use of the most modern techniques of numbers theory—which, believe me, are not at all accessible to mere mortals. The margin of Arithmetica was not enough for Fermat, but two or three pages of parchment presumably would have been…

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