Conjectured by: Pierre de Fermat in 1637
Proven by: Andrew Wiles in 1995
Fermat's Last Theorem was a conjecture (now proven) in number theory which states that there are no integer solutions that satisfy the equation a^n + b^n = c^n for any integer n greater than two. Fermat proved specific cases for when n = 3 and when n = 4, but there was no proof by him or anyone else to prove the general case until Andrew Wiles became interested in the problem. Of course when it was conjectured in a letter by Fermat, he had written that he did have a proof for this theorem because the margin was too small to fit it!
If we begin by looking at some examples, right away we can see that this works when we remove the restriction for n greater than two.
3^2 + 4^2 = 5^2
or
5^2 + 12^2 = 13^2
But once we focus our attention for values greater than two, we cannot find any solutions. It is teasing in a way to think that there are no integer solutions for x^3 + y^3 = z^3 or for greater powers.
Even though it looks simple and elegant, it stumped mathematicians for over 300 years. Andrew Wiles first saw the problem when he was ten years old and became fascinated with it. After graduating and obtaining a bachelors degree in mathematics, he moved on to his PhD specializing in number theory. While working on his graduate research, Wiles dealt with elliptic curves which would be the foundation to proving Fermat's Last Theorem.
In order to prove Fermat's Last Theorem, Wiles needed to prove a conjecture that implied a relationship between the conjecture and Fermat's Last Theorem. This conjecture is called the Taniyama-Shimura conjecture. In effect, the conjecture says that every rational elliptic curve is a modular form in disguise. Wiles then established a correspondence between the set of elliptic curves and the set of modular elliptic curves by showing that the number of each was the same. He did this by counting Galois representations and comparing them with the number of modular forms.
In 1993, Wiles was ready to submit the proof of the Taniyama-Shimura Conjecture and upon doing so an error was found. Digging deeper this is what Wiles had said in an interview about the error:
"The error is so abstract that it can't really be described in simple terms. Even explaining it to a mathematician would require the mathematician to spend two or three months studying that part of the manuscript in great detail."
But this did not set him back at all. With the help of a former student Richard Taylor, the two had patched up the error and were ready to submit the correct proof to Princeton in 1995. Later that year, it had been recognized as a correct proof and finally after 300 years Fermat's Last Theorem had been proven true.
A common question that is asked is whether or not this would be the same proof that Fermat claimed he had. The answer is no. Wiles had used 20th century techniques to prove the theorem while Fermat living in the 17th century did not have access to. So Wiles not only had the correct proof, but it was unique from Fermat's which was too large to fit in a margin.
This is a great example of why my passion lies within the field of mathematics. Take a problem that has been unsolved for 300 years, apply current or modern mathematics to this problem, and voilà! Unfortunately it is not that easy as any mathematician will tell you. But what I would like to illustrate is that if you have a problem that you want to solve, you can use properties, and theorems, and the like that were solved and presented to you before your time to continue and add to our ever growing bank of knowledge. This can be seen as a snowball starting at the top of an infinite hill. A long time ago, someone kicked that snowball down and it has been growing ever since.
Sources:
http://www-groups.dcs.st-and.ac.uk/~history/Biographies/Fermat.html
http://www.pbs.org/wgbh/nova/proof/
http://www.storyofmathematics.com/17th_fermat.html
Nice post. You cover the feel, the theorem, and the significance.
ReplyDeleteRemember to cite your sources.
Otherwise 5C's+