Fermat's Last Theorem

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In 1637 the French lawyer and mathematician Fermat jotted down in the margin of one of his mathematics books a conjecture which is now known as Fermat's Last Theorem and stated that he had found a proof of the theorem but that he was not able to fit it in the margins of the book.

Subsequently mathematicians who came across this statement in the margin of the book tried to find a proof of the statement but none were successful. Over the following three centuries different mathematicians would claim to have found a proof of the conjecture but later an error would be found in their proof. Thus the conjecture stated by Fermat became notorious for being one of the simplest problems in mathematics to state but one of the hardest to solve.

The statement of the theorem is the following:

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Fermat's Last Theorem

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There are no integer solutions to the equation xⁿ + yⁿ = zⁿ for n > 2.

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If n is equal to 1 or 2 then there are an infinite number of solutions to the above equation. When n = 2 the equation becomes x² + y² = z² which you should recognize from the Pythagorean Theorem. The solutions to the case n = 2 are called Pythagorean Triples. An example of a Pythagorean Triple is x = 3, y = 4 and z = 5. The amazing thing is that as soon as n is bigger than 2 we can not find any solutions at all. For example, there are no integers that satisfy the equation x³ + y³ = z³.

You may have noticed that the conjecture that Fermat stated is now known as a theorem. This means a solution to the conjecture has been found and it is now called a theorem. However, it wasn't until over 350 years after this seemingly innocuous statement appeared in the margin of a mathematics book that a correct proof was found. In 1993 a mathematician at Princeton University named Andrew Wiles claimed to have found a proof of the conjecture. The initial proof that Wiles gave was found to have an error in it. With the help of one of his former students named Richard Taylor, the error in the proof was corrected and a complete and correct proof of Fermat's Last Theorem was given.

As a child Wiles had read about this conjecture and became fascinated by it. Years later, after he had established himself as a mathematician, he began working on a proof of Fermat's conjecture. He worked for seven years to find the solution and only told a few souls that he was trying to solve the problem. The proof given by Wiles would be incomprehensible to a mathematician from Fermat's time. This is because Wiles proved a stronger result called the Taniyama-Shimura Conjecture which is a conjecture from the 20th century by two Japanese mathematicians. It turns out that if the Taniyama-Shimura Conjecture was true then it would follow automatically that Fermat's Last Theorem was also true.

Wile's proof is a proof of the Taniyama-Shimura Conjecture on elliptic curves which uses advanced machinery of 20th century mathematics which would not have be available to somebody like Fermat. It is still a curious thought whether Fermat did have a simple proof of his conjecture using the mathematics of his time or if he had made some sort of error like many that followed him. All we know is that Wile's proof at over 100 hundred pages of dense mathematics surely wouldn't fit in the margin of a book.

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References:

https://en.wikipedia.org/wiki/Fermat%27s_Last_Theorem
https://en.wikipedia.org/wiki/Pierre_de_Fermat
https://en.wikipedia.org/wiki/Andrew_Wiles
https://en.wikipedia.org/wiki/Wiles%27s_proof_of_Fermat%27s_Last_Theorem
https://en.wikipedia.org/wiki/Modularity_theorem

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The image of Fermat was taken from here