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.

(Theme music)

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: x^{n} +
y^{n}= z^{n}
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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