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You are familiar with the Pythagorean
theorem. If a and b are the short sides of a right triangle, and c is
the hypotenuse, then a2 + b2 = c2.
There are many so-called Pythagorean triples, sets of three positive
integers that satisfy this theorem. The most commonly considered solution
is a=3, b=4, and c=5. {5, 12, 13} makes up another such set, and in
fact it is a well-known fact that there are an infinite number of such
sets. Fermat was interested in a related problem. What would happen,
he wondered, if the squares were replaced by cubes, or fourth powers,
or any other positive integer powers? Fermat conjectured that if the
exponent is any number other than two, no integer solutions for a, b,
and c are possible. He hinted in his writings that he knew of a proof,
but he was never known to have given one.
Fermat originally presented the
idea around 1635, but it remained unproven until 1996. Legend has it
that Andrew Wile's proof is so complicated that only a dozen or so people
have the knowledge necessary to fully understand it.
Practical value of the theorem
has not yet been determined, although it certainly gained a lot of positive
PR math as an academic discipline.
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