Fermat’s Last Theorem

Fermat’s last theorem is one of the best known mathematical puzzles ever posed. It is very easy to understand yet it eluded a proof for 350 years. Fermat stated in the margin of Arithmetica that he had the most marvellous proof of the conjecture, but it was too long to fit in the margin. It has always been known as Fermat’s last theorem even though it has only been a conjecture for 350 years. Pierre de Fermat stated that

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 marvellous proof of this, which this margin is too narrow to contain.

In other words

does not have solutions for n > 2.

For n = 2 there exist finitely many solutions and we have been dealing with them in problem 9 of Project Euler.

The problem was finally solved in 1995 by Andrew Wiles after he dedicated 8 years struggling to prove the theorem. In order to prove the theorem he had to prove several other conjectures and not least use results and methods in many branches of mathematics developed within the last 100 years. So there is no way that Fermat could have proven it the same way as Andrew Wiles did. Wikipedia has a section on Fermat’s Last Theorem where they briefly go through the history and the content of the proof.

My story on the theorem

I was first introduced to Fermat’s last theorem when I went in high school a mere 4 years after the theorem of proven. Our math teacher (whom I owe a lot of thanks for sparking my curiosity) wanted us to watch the movie on the subject made by Simon Singh and John Lynch. Many of my fellow students giggled at Andrew Wiles and thought he was a complete nut job, but I saw something different. I saw a man with a burning passion for solving this problem, and by the end of the movie I was so touched that I was almost crying. To me it was a real story and a treasure hunt for the truth.

This movie sparked something in me, and inspired me in many ways. I don’t claim to be good at mathematics and I am not rigorous enough to prove many things. But my passion and curiosity for math was ignited and will burn forever after this movie. It has been aired on television in many countries and until Google video was closed it was available through that service. Today it might be available through other means on the internet, but I haven’t found a source for it. If you a legal source for the movie I would be very interested in hearing from you.

Simon Singh is also the author of a book on the subject called Fermat’s Last Theorem. Simon Singh is a great story teller and manages to take the reader through the story in a way that most people can follow. The book takes you all the way from Fermat’s life and achievement and through the long history of Fermat’s Theorem which many people have spend many hours trying to prove without success until Wiles finally did his brilliant work and finally proved it. So if you love a good story I will highly recommend you to read the book and watch the movie.

On a side note Simon Singh has written other great books on different subjects such as The Code Book.

But did he prove it?

The big question is; did he prove it? Not Wiles of course but Fermat – did he prove it? Most sources believe not. Wolfram has the very good argument that he later on looked for proofs of n=4 and n=5, which would have been meaningless if he had already proven it.

Due to my personal pride I hope and doubt he in fact did not prove it, because that means he would have had an insight that has eluded the rest of humanity for 350 years even though the mathematics has evolved incredibly since then.