Today, God help us, we reach the mountaintop. The University of Houston's College of Engineering presents this series about the machines that make our civilization run, and the people whose ingenuity created them.

Years ago, on the closing night of Pinafore, I found the contralto lead sitting on a sandbag backstage, weeping. The play was done. Her moment was finished. The problem with any mountaintop experience is, you can only come down off the mountain.

The French mathematician Pierre de Fermat created one of our great intellectual mountaintops in 1637. Fermat read the old arithmetic text by Diophantus of Alexandria -- the part on Pythagoras's theorem: The sum of the squares on legs of a right triangle equals the square on the hypotenuse.

Fermat wondered about the sum of cubes or fifth powers. Finally, he wrote in the margin that he could show it wouldn't work for any whole number power greater than two. "I've found for this a truly wonderful proof, but the margin won't hold it."

For the next 356 years, the best mathematicians have looked for a proof. Computers found no exceptions, yet no one could prove it in general. Did Fermat himself really have a proof?

Enter now Princeton mathematician Andrew Wiles. Wiles's brilliance is legendary. He was a 10-year-old English schoolboy when he ran across Fermat's theorem in a public library. It transfixed him like a cobra's gaze.

He worked on it during his teens. He went on to became one of the world's great mathematicians. And, in secret, he kept wrestling with Fermat. Finally, in June, 1993, he gave three lectures at Cambridge. No mention of Fermat in his titles!

After the first day, e-mail began humming across continents. Wiles was up to something big. Second lecture: The room filled. The third lecture was electric with excitement as Wiles finished by proving something called the Taniyama Conjecture.

Then, almost as an afterthought, he said aloud what his audience already knew: "And that means that Fermat's Last Theorem must also be true." Wiles stood on the mountaintop. He'd ridden there on centuries of human genius -- including his own.

For 356 years, Fermat's impractical little puzzle drove men and women like Wiles. It drove them to create math that eventually served practical physics and astronomy. Wiles has written a 200-page solution that the experts expect to be correct.

Back home, Wiles went off to play with his daughters on the park swings. When reporters finally found him, he allowed he'd ended an era, and he felt a deep sense of loss. They found a man who'd been to the mountaintop. Now neither he, nor anyone else, can ever go back. Still, Fermat has woven his magic. He's led so many people like Wiles, and like you and me, into the joyful work of stretching the mind -- and stretching human capacity.

I'm John Lienhard, at the University of Houston, where we're interested in the way inventive minds work.

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Kolata, G., At Last, Shout of 'Eureka!' In Age-Old Math Mystery, New York Times, Monday, June 28, 1993, pp. A? & A11.

Kolata, G., Math Whiz Who Battled 350-Year-Old Problem, New York Times, Science Times, Tuesday, June 29, 1993, pp. B5 & B7.

Heath, Sir T. L., Diophantus of Alexandria: A Study in the History of Greek Algebra 2nd ed., New York: Dover Pubs., 1964. (This republication of an 1885 edition deals extensively with Fermat's studies of Diophantus.)

Vogel, K., Diophantus of Alexandria (fl. A, D. 250), Dictionary of Scientific Biography. Vol. ??, (C.C. Gilespie, ed.) Chas. Scribner's Sons, 1970-1980. pp. 110-119.

Paulos, J.A., Beyond Numeracy: Ruminations of a Numbers Man, New York: Alfred A. Knopf, see, "Fermat's Last Theorem," pp. 75-77. (Paulos's book gives fine layman's discussions of what mathematics is about. The section on Fermat's Last Theorem is short, but its well-woven mathematical context will be particularly clear and useful to interested non-mathematicians.)

For an interview with Wiles on the television program NOVA, see the following website:

http://www.pbs.org/wgbh/nova/transcripts/2414proof.html

Two of the early greats who made steps toward proving "Fermat's Last Theorem" (or "FLT") were Leonard Euler and Sophie Germain. Germain corresponded with Gauss, who said of FLT, "I could pose a hundred such impossible problems." He clearly felt it was not important. And indeed no one I know has argued that the problem is important of itself. But its value as an intellectual stimulant has been incalculable.

I am grateful to Giles Auchmuty and Neal Amundson at the University of Houston for their counsel.

Pythagoras' equation:      x² + y² = z²      is true for any right triangle where x and y are the length of the legs and z is the length of the hypotenuse. Only certain sets of whole numbers (or integers) will satisfy the equation (e.g.: x = 3, y = 4, and z = 5).

Fermat's equation:    xⁿ + yⁿ= zⁿ      has no solutions in whole numbers values of x, y, z, and n, if n is greater 2.

And finally, if you're interested, you can generate all the solutions you want for the quadratic form in the following way: You simply choose any pair of integers, m and n. Then the following values of x, y, and z will satisy the Pythagoras equation:

x = m² - n²     y = 2mn;     and     z = m² + n²

The Engines of Our Ingenuity is Copyright © 1988-1997 by John H. Lienhard.

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