Fermat Fun to the Fourth Power

Pierre de Fermat
Pierre de Fermat | Source
Leonhard Euler (pronounced "oiler")
Leonhard Euler (pronounced "oiler") | Source

The Fermat Saga

Notwithstanding the complexity of the subject matter, this article is reasonably accessible. It should appeal to serious Number Theory aficionados, and to general readers who would appreciate a sketch of the historical and cultural background.

I'm NOT attempting to prove Pierre de Fermat's Last Theorem. Leonhard Euler, one of the most prolific mathematicians of all time, was the first to prove Fermat's Last Theorem for the special case in which n = 3. Fermat himself proved his famous conjecture for the special case in which the exponent n is 4. They both used a method, called Infinite Descent. Since then, a number of famous mathematicians have proven special cases of FLT for other exponents.

Andrew Wiles published a long, complicated general proof for the whole enchilada in back in 1995. However I do not have the Number Theory background--or the intestinal fortitude--to follow it.

A leading mathematician has stated that Wiles killed the problem, rather than solving it. How so? Because Wiles' proof has precious little heuristic value. It's highly unlikely that anyone will build upon it, to solve other problems in Number Theory.

On the other hand, it's very possible that some future mathematician will create a simpler proof. As Mark Twain would have said, rumors about the death of Fermat's Last Theorem are greatly exaggerated.

Here's the theorem. You start out with trio of positive integers a, b and c. Then for positive integer n > 2, it is NOT possible for the following equation to be true:

c^n - a^n = b^n

The 'hat' notation "^" means, "to the power of." For example,

2^3 = 2*2*2 = 8

If you can prove Fermat's Last Theorem for n = 4, and for all of the primes greater than or equal to 3, then you've automatically established that FLT is true for all powers greater than 2. For example, there are no positive integers a, b, and c, such that

c^6 - a^6 = b^6

Why not? Because sixth powers are also cubes (third powers).

For hundreds of years, the general proof of Fermat's Last Theorem was regarded as one of the great unsolved problems of Number Theory. Predictably, there were many attempted general proofs, which did not stand up under scrutiny.

Fermat claimed to have had such a general proof. If so, it has been lost to history. In 1637, Fermat wrote his famous conjecture in the margin of his copy of the book, Arithmetica, and wrote that he had a marvelous proof that was too big to fit in the margin of the book.

It's possible that Fermat did have a general proof. It's also possible that Fermat's argument contained a fatal flaw, like many of the other valiant attempts over the next few hundred years.

One other possibility: Fermat fully realized the great difficulties inherent in proving the general case of his famous conjecture, but he put it 'out there' anyway. Why? Being a world-class mathematician does not prevent one from having an odd sense of humor.

Before Andrew Wiles came onto the scene, Fermat's Last Theorem had even found its way into popular culture. The Devil and Simon Flagg, by Arthur Porges, is a story in which an enterprising mathematician attempts to use FLT to get the better of Old Scratch.

Copyright 2013 by Larry Fields

Andrew Wiles
Andrew Wiles | Source

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