### Explicit modular approaches to generalized Fermat equations

- Series
- Algebra Seminar
- Time
- Monday, September 17, 2012 - 15:05 for 1 hour (actually 50 minutes)
- Location
- Skiles 005
- Speaker
- David Zureick-Brown – Emory

Let a,b,c >= 2 be integers satisfying 1/a + 1/b + 1/c > 1. Darmon and
Granville proved that the generalized Fermat equation x^a + y^b = z^c has
only finitely many coprime integer solutions; conjecturally something
stronger is true: for a,b,c \geq 3 there are no non-trivial solutions and
for (a,b,c) = (2,3,n) with n >= 10 the only solutions are the trivial solutions and (+- 3,-2,1) (or (+- 3,-2,+- 1) when n is even). I'll
explain how the modular method used to prove Fermat's last theorem adapts
to solve generalized Fermat equations and use it to solve the equation x^2
+ y^3 = z^10.