Briefly, there are things called elliptic curves, and there are things called modular forms.
Elliptic curves are in fact equations in two variables. They are generally a bit like the quadratics that we know and love. Well, that we know. They look like this:
- y^2 = a*x^3 + b*x^2 + c*x + d
There are usually some special conditions equivalent to a quadratic not being a perfect square. The solutions (when you allow complex numbers (and
ComplexNumbersAreYourFriends so you should)) are actually a surface (a curve over the complex numbers) shaped like a bagel (or donut (or torus)) but in a four dimensional space (two dimensional over the complex numbers).
Modular forms kind of defy description, but they:
- are functions from the complex plane to the complex plane;
- are unreasonably symmetrical;
- satisfy some rather spectacular and special properties because of their symmetries.
The Taniyama-Shimura-Weil theorem says that these two things from totally different areas of mathematics are in fact the same as each other, only in disguise. A clever argument can then be used to show that if
FermatsLastTheorem is false, then this can't be true. Since it is true,
FermatsLastTheorem is also true.
Proofs are starting to filter out on the web, but it's very hard going. One of the reasons people believed that Wiles had proved FermatsLastTheorem was because he'd used advanced techniques invented in the past 20 years from a huge array of subject areas in mathematics.
[There are numerous non-specific mentions of these "advanced techniques", but what are they?]
For a "discussion" arising from this overview see
TaniyamaShimuraWeilArgument.
Taniyama Shimura Weil Theorem