ChatGPT-4 (You Have to Pay for It)
Here’s my first conversation with it.
Find alternative proof of Fermat’s Last Theorem by assuming such an exponent K does exist for A^K+B^K=C^K and finding a contradiction.
ChatGPT:
Fermat’s Last Theorem (FLT) states that no three positive integers �a, �b, and �c can satisfy the equation ��+��=��an+bn=cn for any integer value of �n greater than 2.
The proof for Fermat’s Last Theorem was a long-standing challenge in the history of mathematics. After over 350 years, it was finally proven by Andrew Wiles in 1994. Wiles’ proof utilizes elliptic curves and modular forms in a highly sophisticated manner. His proof is intricate and beyond a quick explanation. However, I can give a broad sketch.
- Linking FLT to Elliptic Curves: Wiles began by linking FLT to the properties of certain elliptic curves. Specifically, he considered the curve �2=�(�−��)(�+��)y2=x(x−an)(x+bn). If the Fermat equation has a solution, then this elliptic curve would have some special properties.
2. Modularity Theorem: Wiles’ key insight was to then show that these elliptic curves have a property called “modularity”. This essentially means that the points on these curves can be counted in a particular way using modular arithmetic.
