# Steps to The Grand Proof

The movie we watched in class about Andrew Wiles and the eventual proof of Fermat’s Last Theorem inspired me to dig deeper into the previous trials and failures at solving this problem. As shown in the video, a passionate mathematician named Andrew Wiles solved the theorem 1995. But where did it start, and who failed in the attempts to prove of this simple problem?

Pierre de Fermat was a 17th century mathematician who was a French lawyer and an amateur mathematician who drew his inspiration from ‘Diophantus’ Arithmetica’ an ancient Alexandrian Greek text. Fermat wrote his observations and theorems in the margins of a copy of that book. He is most famous for what is now known as Fermat’s Last Theorem. He drew inspiration from the page of Arithmetica that talked about Pythagoras’s Theorem and found that if you change the square to a cube it has no solutions. Likewise a power of three to a power of 4 and onward until he came to the conclusion that-

xn + yn = zn  has no solutions in positive integers for n > 2

The theorem had the reputation of being unsolvable as many of math’s greatest mathematicians tried their hand at finding a proof. In fact it is called Fermat’s Last Theorem because it was the last of his theorems to not be proven or disproven. While Fermat says that he proved it, it has been for the most part speculated as being incorrect.

This problem was unsolved for three centuries, spanning across some of the greatest mathematical minds we have ever had on earth ranging from Euler, Legendre, Gauss, Abel, Germain to Dirichlet, Kummer and Cauchy. While people have been able to make generic proofs about certain numbers like n = 3, n = 4, and n = 5; none of the great mathematicians could find a general proof of the theorem.

The first big step in the right direction of a general proof came from female French mathematician Sophie Germain in 1819. In a letter to Carl Friedrich Gauss, a German mathematician, she explained her grand plan to formulate a proof for the theorem. Her goal was to prove that for each odd prime exponent p, there are an infinite number of auxiliary primes in the form N * p + 1 such that the set of non-zero p-th power residues of x to the p mod N * p +1 does not contain any consecutive integers. Her end game was to prove that every odd prime p has infinitely many auxiliary primes satisfying the p-th power residue condition. Germain found her problem as she was unable to prove the existence of infinitely many auxiliary prime exponents. If she succeeded than it would mean that the numbers in Fermat’s Last Theorem would have infinitely many divisors.

The longer this problem went unsolved the more important it became. So important that it drove some mathematicians mad with obsession. In 1856, a man by the name of Paul Wolfskehl was so entranced by it he put up a prize of over 100,000 German marks for the person who was able to give a rigorous proof of FLT.

The age of computers came around after WW2 and Alan Turing inventions, and with that came the incredible system of automatic computation. This didn’t give a rigorous proof, but it did prove that for a certain subset of real numbers FLT holds true. Raising the value of n = 25000 showed that FLT held true for those exponents, but as Professor John Conway(bearded man from the video) said “You are still infinitely many away”. This is the lesson that carried over from Euler.

But, coincidentally, the biggest step towards the proof of FLT was something that was created with no direct correlation to Fermat’s Last Theorem; the biggest step was created by Japanese mathematicians Yutaka Taniyama and Goro Shimura. They started by relating Elliptic curves, which can be described as

Y^2 = (x + a)(x + b)(x + c) where a, b, & c can be any whole number not zero.

To modular forms, which are highly symmetrical functions. A summary of their conjecture was that between these two different areas of mathematics that it seemed that the solutions for any one of the infinite numbers of elliptic curves could be derived from one of the other infinite number of modular forms. To put it simply, each elliptic curve has a modular form. This was known as the Shimura-Taniyama conjecture.

It wasn’t until 1986 that CAL Berkley’s own Kenneth A. Ribet pieced it together and used this unproven conjecture to link the Shimura-Taniyama conjecture to Fermat’s Last Theorem. He stated that if there were a solution to Fermat’s equation would mean that elliptic curves could not be modular, and thus the Shimura-Taniyama conjecture is false.

This is where Andrew Wiles comes in. Backtracking, Ribet’s conjecture means that if you prove the Shumura-Taniyama conjecture then you could have a rigorous proof that Fermat’s Last Theorem was true. But keep in mind that this conjecture had yet to be proven since its creation in the 1950’s. When Andrew Wiles learned of this conjecture he would then drop all of his previous work and work for 7 years in almost complete isolation to find a proof.

He had to prove a conjecture that covered and infinite series of elliptic curves, and he employed a very new strategy from number theory modeling a toppling domino affect. Meaning, he couldn’t prove all of the Shimura-Taniyama conjecture as one grand proof but rather if he could prove that each one was true and could ensure the one after could also be true than he would by induction, prove the conjecture. This idea was presented around the nation at a conference in Cambridge and people rejoiced that he really did prove the theorem. But he could not really guarantee his domino theory to prove always correct.

It was then that Wiles talked in confidence with Richard Taylor, a former student under Wiles, that he would hope could inspire a different tactic to solving or fixing Wiles proof. They then realized that one of his old tactics was the correct tactic, a tactic that it would take him a year later to have a “revelation” about. To quote Andrew

“It was the most important moment in my working life. Nothing I ever do again will be the same.”

2 years later Wiles had won the Wolfskehl Prize and took home his 2 million dollar reward. But all of that was little to Andrew, because he had the accomplishment of a childhood dream and the feeling of success after the culmination of a decade of effort.

So what did Fermat have? Skeptics believe that Fermat had a slip-up in his long great career and only made a flawed proof. Other people think that he did have a proof only based on 17th century techniques and so witty and cunning that no other person could create it.

Copyright C. J. Mozzochi, Princeton N.J.

“There’s no other problem that will mean the same to me. This was my childhood passion. There’s nothing to replace that. I had this very rare privilege of being able to pursue in my adult life what had been my childhood dream. I know it’s a rare privilege, but if you can tackle something in adult life that means that much to you, then it’s more rewarding than anything imaginable.”

-Sir Andrew Wiles (Princeton University)

http://simonsingh.net/books/fermats-last-theorem/the-whole-story/

-Simon Singh-Long article outlining the History of Fermat’s Last Theorem. MAIN SOURCE

http://www.agnesscott.edu/lriddle/women/germain-FLT/SGandFLT.htm

-Larry Riddle- Outline of Sophie Germain’s contributions to solving FLT

http://simonsingh.net/books/fermats-last-theorem/sophie-germain/

-Johannes Reitzenstein: Talks about Diaphantus and its relation to FLT

https://vimeo.com/18216532

-Nova: Video watched in class giving interviews with Sir Andrew Wiles and Shimura.

https://www.awesomestories.com/asset/view/Fermat-s-Last-Theorem-Professor-Andrew-Wiles

-Carole Bos: Found quotes from the NOVA movie to use in this paper. Also gives a quick summary of movie

http://www.public.iastate.edu/~kchoi/time.htm

-Barry Taylor: Rough outline of the TimeLine of FLT.