The seventeenth century mathematician Pierre de Fermat is perhaps most remembered for his famed "last theorem," which may be seen as truly remarkable due to the fact that it possesses the properties of being fairly simple to explain, yet excruciatingly difficult to solve.
Anyone familiar with fundamental geometry and the Pythagorian theorem should have no trouble understanding the theorem.
The Theorem
Consider the Pythagorian theorem:
a² + b² = c² (or, a squared plus b squared equals c squared). This is one of the most revolutionary discoveries in the history of mathematics.
In its simplest instances, this theorem can be solved using only non-zero integers. For example a = 3, b = 4 and c = 5. This, and other examples, hold true: 9 + 16 = 25.
But what happens, Fermat asked, if this theorem is written as a^n + b^n = c^n? In other words, if instead of squaring each of these numbers, they are raised to a power higher than two? Fermat's hypothesis was that in any case where n > 2 (n is greater than two), it is mathematically impossible to find three non-zero integers, a, b and c that satisfy this equation.
Finding a Proof
That's it, all there is to one of history's great mathematical mysteries - the hypothesis which spurred on countless mathematicians to endless searches for a way to actually prove that this was so (for in mathematics, it is not enough to simply try all possible combinations of numbers, as such attempts would prove endless), but centuries passed with little to no progress.
What made the process even more frustrating was that Fermat himself claimed to have had developed a proof for the theorem himself, though never got around to actually writing it down.
In fact, the only mention Fermat made to the theorem which would forever bear his name was in the context of a note written in the margin of Diophantus' Arithmetica, where he mentioned, almost non-chelant, that he had found a "truly marvelous proof..."
Yet centuries passed, men were driven to madness, and no proof could be found.
Andrew Wiles Proof
It was not until the nineteen sixties that the first real steps began to be made toward proving Fermat's theorem. It was Yves Hellegouarch who decided to go about the problem in an entirely new manner - by associating the simple algebra of Fermat's theorem with the geometrical principle of elliptic curves.
Though the mathematics involved are far too involved to be explained in detail, it was realized in the ensuing decades that by utilizing the Taniyama-Shimura conjecture which explored the fundamental features of these curves. In this sense, the theorem could be looked at from an entirely different angle - from a fundamental feature of number theory (proving the theorem by way of number patterns), to geometry.
Viewing the theorem in the context of curves reminds the reader once again of the Pythagorean theorem, which used a similar equation in terms of defining a fundamental truth of geometrical space - a triangle. With Fermat's theorem, the mathematics developed in the 1950's and 60's enabled mathematicians to look at different values of n in terms of geometry - the proof, in other words, had to do with fundamental features of spatial geometry.
It was realized in the 60's that if one could prove the Taniyama-Shimura conjecture to be true, they will essentially have proved Fermat's theorem, as well. This opened up a whole new box of tools for mathematicians.
The first proof finally came after years of solitary work by the British-American mathematician Andrew Wiles in 1993. Soon after, however, some fundamental errors arose in his monumental work, and he was forced into working on correcting these errors before submitting a final version of his proof in October of 1996.
Fermat's theorem had been solved. The "simple" problem had resulted in a 100+ page mathematically intense proof.
Such complexity seems perfectly just after four centuries of mathematical struggle.
References:
Singh, Simon. "Fermat's Enigma: The Epic Quest to Solve the World's Greatest Mathematical Problem." Anchor Books, 1998.
"Solving Fermat: Andrew Wiles." PBS.
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