MATH Seminar

Title: Explicit modular approaches to generalized Fermat equations
Colloquium: Number Theory
Speaker: David Brown of University of Wisconsin - Madison
Contact: Susan Guppy, sguppy@emory.edu
Date: 2011-02-14 at 4:00PM
Venue: MSC W201
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Abstract: Let $a,b,c \geq 2$ 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 $n \geq 10$ the only solutions are the trivial solutions and $(\pm 3,-2,1)$ (or $(\pm 3,-2,\pm 1)$ 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 $x2 + y3 = z^{10}$.