That Damned Frenchman's Last Theorem

If you possess even an amateur interest in the mathematics existing outside of old school textbooks and class tests, you have probably heard of Fermat's Last Theorem; arguably the most famous, baffling puzzle to have ever existed since the 17th century- and not just because the mathematician in question, Pierre de Fermat, gasped it out on his deathbed. It took over three hundred years to prove, during which it stupefied even the greatest minds of the three centuries between the 1600s and the 1900s; and by that I mean minds such as those of Carl Friedrich Gauss and Leonhard Euler. 

(It is important to note that Gauss initially dismissed the Last Theorem as a typical statement that applies to a large amount of numbers that is impossible to prove or disprove, even if it might not actually be true. This is one of the most significant cases of poor mathematical judgement in the history of mathematics....though in Gauss' defense, many mathematicians in the 17th and 18th centuries thought the same way of the equation. Many even disliked Fermat intensely- René Descartes thought of him as a braggart, and John Wallis referred to him as 'that damned Frenchman'.)

If you are unaware of the Last Theorem, I have presented it here below:

(for any integer(s) x, y, and z, and any whole number n>2)

(It is also important to note that this sort of polynomial, which people attempt to solve using integer solutions, is called a Diophantine equation.)

Seems very simple, doesn't it? However, don't be too quick to judge- the simplicity of the equation was why Gauss dismissed it, and I have already stated what a huge blunder it was on his (and tonnes of other mathematicians') part.

Anyway, it is somewhat ironic that Fermat's main claim to fame is this equation, which he and dozens of his descendants couldn't prove. History says that Fermat, just before he died, was reading up on the Pythagoras theorem which states 'x squared plus y squared equals z squared in a right triangle where x and y are the legs of the triangle, and z is the hypotenuse'. He got the idea of proving the same theorem- except he would cube the sides, not square them. After experimentation, he stated-

"It is impossible to divide a cube into two cubes, or a fourth power into two fourth powers, or in general, any power higher than the second into two like powers. I have discovered a truly marvellous proof of this, which this margin is too narrow to contain."

Obviously, he was stating what would become the famous Last Theorem, but the fact that he plausibly found a proof is too significant to ignore. However, he died before ever revealing his proof, which was actually typical of Fermat, who never wrote his proofs down and often presented them as puzzles, greatly infuriating his fellow mathematicians.

It was only after Fermat died that this theorem became truly important and noteworthy. Many tried to solve it, only to end up failing. Later on, Euler proved the theorem unsolvable for n=3 (which was ironically the opposite of what Fermat was originally trying to do- solve the theorem with n=3. Later on, he realised this to be impossible... but this is a moot point for Euler gets the credit for the first published proof anyway). Adrien Marie-Legendre proved it unsolvable for n=5 almost a hundred years later, and Peter Dirichlet proved it unsolvable for n=14.

In 1955, Yutaka Taniyama presented a new idea by introducing elliptic curves (a special kind of Diophantine equation) and the Taniyama-Shimura-Weil conjecture, which states that elliptic curves are modular functions with infinite solutions. In 1975, Yves Hellegouarch noticed a connection between Fermat's Last Theorem and elliptic curves: this connection was the fact that disproving the Last Theorem with a counterexample would lead to an elliptic curve with insanely strange properties. Gerhard Frey stated afterward that such a curve wouldn't even exist, but the idea was kept open.

Andrew Wiles finally cracked the theorem in 1993, proving the Taniyama-Shimura-Weil conjecture for a general elliptic curve, which meant that, considering Frey was right in stating that the Last Theorem's elliptic curve wouldn't exist, the theorem was indeed unsolvable. Quod erat demonstratum.

The puzzling thing is that the Taniyama-Shimura-Weil conjecture was developed two hundred years after Fermat presented his Last Theorem, so how on earth did he find a proof? The sad fact is, we'll never know, given that the damned Frenchman never saw fit to write down his proofs, and has unknowingly made posterity suffer for it.

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