The Footnote Theorem
All our life we have seen this theorem and solved it, be it in trigonometry or number theory. But there exists another very interesting form of this equation that had geniuses wracking their brains for centuries without much avail. Let's take a trip into that history.
Mathematical genius Pierre de Fermat while studying a book by another genius Diophantus writes something interesting on the footnote. It's important that we take a peek at Diophantus' field of study to understand how the idea came to Fermat.
Diophantus' studied a set of equations that later would be known as diophantine equation. A classic problem of this type would be finding integers x and y where their sum and the sum of their squares are A and B
So while reading a book by the person who studied such equation, inspiration struck Fermat and he wrote in the footnotes of that book, on the page where such a problem was discussed:
"It is impossible to separate 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 marvelous proof of this, which this margin is too narrow to contain."
This was revolutionary, because it meant that we won't have any triplets x,y,z similar to Pythagorean triplets if the power is increased to 3 or any higher integer. This would mean that this is not only true for integer x,y,z but any rational values as rational values can be changed to integers on multiplication with appropriate number. However, the problem was this was not proved and remained as a conjecture. Fermat asked some of his friends to prove this theorem for n=3,4,5,7 etc. but never asked anyone for a complete proof for any integer n. It is considered that he later realized that his proof might not have been true and may also have doubted if the conjecture was true for all integer n as he never discussed it, neither showed any attempt to get the conjecture proved or prove it himself.
Now the problem stood at a strange place, Fermat himself proved the absence of non-zero answers for n = 4, which means that one has to prove further that the theorem doesn't hold for prime number n, these would imply that the theorem is true for all n.
Now comes the next light bearers Sophie Germain followed by Ernst Kummer. Ernst extended on Germain's work to prove the theorem for regular primes( a subset of primes and consist of about 41% of all the primes, conjectured).
Computational methods were used to prove the theorem for all primes below 125,000.
Finally the proof came from a branch of mathematics that was never considered as having any connection to this problem.
The solution came from the study of elliptic curves and something called the modularity theorem.
It was proved that if the modularity theorem can be proved for semi-stable elliptic curves, all semi-stable elliptic curves would be modular. Further, if Fermat's theorem is proved false the solutions of the equation would make it impossible for the curves to be modular.
At this point the hero of our story comes in, the English Mathematician Andrew Wiles, a person who has experience in elliptical curves and also is interested in Fermat's theorem. He heard of the developments and started working in secret on the Fermat theorem( in 1991 ), while publishing parts of his previously done work. He would change method twice and finally publish a paper that he was confident about in June 1993, he would also give lectures on his work, which would attract immense attention, but as luck would have it his proof was found lacking, there was a small error in his calculations that was pointed out in August 1993. Wiles went back to his study again, but this time he was under pressure, the rumour had spread that his proof had failed and the mathematicians from all over the world were forcing him to get his work published so that the community could work on it.
Wiles in the meantime started working with his student Richard Taylor in an effort to repair his proof, a race against the entire world and the more he delved into the problem the more he understood his mistake, what he considered to be a small mistake appeared as a gaping hole in his proof, so large a hole that he was almost sure his approach was wrong and he would probably never be able to complete the proof.
August 1994, this time inspiration struck Wiles, when he was about to give up and was thinking of publishing his unfinished work, he suddenly understood a very interesting thing. Neither of the approaches he thought would be sufficient were enough, both the Iwasawa theory he had discarded as insufficient and the Kolyvagin–Flach approach that was used in his wrong proof were insufficient, but when combined, he was able to get a hold of how to solve the problem.
On 24th October 1994 Wiles submitted 2 manuscripts "Modular elliptic curves and Fermat's Last Theorem" and "Ring theoretic properties of certain Hecke algebras" , the second being co-authored with his student Taylor. These proved that certain conditions were met that justified the steps he took in his main paper.The two papers were vetted and published as the entirety of the May 1995 issue of the Annals of Mathematics. These papers established the modularity theorem for semistable elliptic curves, the last step in proving Fermat's Last Theorem, 358 years after it was conjectured.
Wiles became the first mathematician to provide a proof, a wild goose chase that had branched off and given rise to entirely new problems and absolutely remarkable branches of mathematics. His proofs were extended to further prove conjectures that had been thought of as impossible, like the full Taniyama–Shimura–Weil conjecture, later known as the modularity theorem( he only proved a part of it ). He would also go on to win the coveted Abel prize in 2016. The theorem, with the thousands of wrong proofs and the story stands as a testament to the passion that mathematicians have for their field and the absolute geniuses that belong to the community. The only thing that we can consider a loss in this story would be the man who started this chase. Fermat predated all these advanced math, so if he had a proof, it probably was much more elementary, much basic, much simple, and thus maybe much more powerful, even if he didn't have a proof, one cannot not marvel at the sheer intuition this man had, a lawyer who also happened to be a part time mathematician and an absolute genius.