Fermat's Last Theorem: Infinite Descent
Exploring Fermat's proof of x^n + y^n = z^n using infinite descent, with emphasis on the importance of x, y, and z values.
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1. Fermat’s Last Theorem Jonathan Rigby
2. Fermat’s last theorem • x n +y n =z n • Fermats proof by infinite descent • x 4+ +y 4 =z 4
3. Elliptic curves • Cubic curve who’s solution looks like a donut • All points on the donut is the solution to the initial equation
4. Modular form • A function on the Complex plane, which satisfies a certain kind of function equation and growth condition. • They exhibit many symmetries
5. Tanyama Shimora Conjecture • Claim that all elliptic curves are modular forms
6. Epsilon conjecture • Consider if x n +y n =z n has a solution. • Create an elliptic curve using this function. • The resulting elliptic curve appears to not be modular
7. Problem • Show a particular elliptic equation is paired with a modular form. • Show all elements in E have a corresponding element in M • Mathematical induction • Repeat this process for the infinite number of elliptical and modular equations.
8. Galois representation • Group theory • Groups built on the division of solutions to equations into packets with similar properties. • Transforming the elliptical curve equations into packets that could be matched up to modular forms
9. Class Number Formula • Flach Kalyvargan • Iwasawa Theory