Elliptical Curve Photo

Click here to see a picture of an elliptical curve mentioned in the Andrew Wiles video

3-5 switch

Andrew Wiles was having trouble making some connections with his proof, until he figured out this little "trick" the three-five switch. This is covered in chapter 11 of Nigel Boston's book about the proof of Fermat's last theorem. When dealing with an infinite number of possibilities you have to be able to count them. The idea of counting something infinite may sound like it is not possible, but it is as long as what you are using is countable. For example the integers and natural numbers are infinite countable sets, where the reals are an infinite uncountable set. The three-five switch aided in the countability of sets.

Pictures of some of the important people involved.

Yutaka Taniyama

Goro Shimura

Andrew Wiles

Kenneth Alan Ribet

Taniyama–Shimura–Weil conjecture

Andrew Wiles proof that all that all rational semi stable elliptic curves are modular which, in particular, implies Fermat's Last Theorem. This is the proof of the Taniyama-Shimura-Weil conjecture which can be found here. Kenneth Alan Ribet is the one who found the link between Taniyama-Shimura-Weil and Fermat's last Theorem. This is contained in his proof of the epsilon conjecture, and thereby proved that Fermat's Last Theorem would follow from the Taniyama-Shimura conjecture.