Fermat's Last Theorem

            Today’s blog post will about Fermat’s Last Theorem. I know right off the bat I am going to get a comment saying that Fermat’s last theorem has been solved, and you would be correct. However, it is still a great mathematical problem that was unsolved for a couple hundred years. So no this is not an unsolved problem, but it is one of the simpler problems to explain, that was once an unsolved problem. First I would like to talk about some background information, such as the origin of the problem, the problems contributors, etc. Then I want to talk about the problem itself, and how it works. Then I want to talk about how common it comes up in society. I will not be talking about the proof in this post because, it is too long for a single post, in the future I might post another, or maybe two more posts to explain it in depth but that will not be happening in this post.

            It seems only right to start with Fermat himself. Pierre de Fermat was a French man who lived during the 17^th century in France. He himself was not a mathematician, in fact he was a judge. He was a judge that had an interest in math during his spare time. He would read these published mathematical texts and write notes in the margin. On one occasion he was reading a book called Arimetica by Diophantus. He stumbled on an equation that was very similar to that of the Pythagoras, that being X^2+Y^2=Z^2. He was curious to an alteration of the problem he wanted to know if the equation had any other solutions that would satisfy a higher order of this equation. He thought about it, and then in the margin wrote something to the extent of, “I can prove that there is no solution to a higher order, but there is not room for me to write it in the margin.” His book was littered with these little marginal notes. He would write,” I have a proof to this, but I am too busy to write it out.” Fermat then without proving anything died. His son found his annotated copy of the book and had it republished with his father’s notes added to it. Because of this, mathematicians started working on these proofs that Fermat claimed to have been able to prove. Now proofs are flying in left and right, one after another, the marginal notes are being ratified. That was all good until, they got to this one involving Pythagoras’ equation. The theorem could not be solved.  This problem remained un solved for centuries afterward. The problem itself is not that hard to explain, but by the 20^th century it is understood that this problem does not have a simple solution. Many mathematicians have tried a crack at the problem, but to no avail, until Andrew Wiles comes along. Andrew Wiles grows up learning and loving the problem. So after going through college getting a PHD and going on with life. Someone proves that if you can solve The Taniyama Shimura conjecture, you would also be proving Fermat’s last Theorem as well in doing so. Now it is not important to know about the Conjecture other than by solving it Wile’s would get what he wanted. He spent several years working on it alone, then gave a lecture solving the conjecture. Unfortunately, he had made a mistake. It took him another year, but then he fixed his mistake. After 8 years he had gotten what had been trying to do since the age of ten.

            The problem itself is rather straight forward. If you have had junior high math you know Pythagoras’ equation X^2+Y^2=Z^2. Well Fermat thought to himself, "I wonder if there are solutions to equations like X^3+Y^3=Z^3 or to the fourth power." He could not find any whole number solutions. He said that there were no whole number solutions to any power greater than two so in math terms, there is no solution to X^N+Y^N=Z^N where X,Y and Z are not equal, and N is greater than two. That is all the problem is. I said it was a simple problem. If you do not believe me, go ahead try to find a solution.

            Now I mentioned that this problem actually comes up in the media. There are a few famous instances of this. The main one being the Simpsons. In one the episodes when Homer is writing on the chalk board doing calculations it flashes an apparent solution to Fermat’s unsolvable equation. The equation shown on the board is obviously not correct, but rather to pay tribute to the mathematician. Another example in a book called, The Devil and Simon Flagg, where Simon makes a deal with the devil that he gets to keep his soul if the devil can prove Fermat’s Last theorem. Also the problem crops up in star trek. The point is, the problem is more common than you think it is. So maybe next time someone makes reference to the problem you will be able to spot it.