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Description

A conjectured generalization of Fermat's Last Theorem states that the equation $x^p + y^q = z^r$ has no solutions in non-zero mutually coprime integers $x, y, z$ whenever the integer exponents $p, q, r \geq 3$. Since the proof of Fermat's Last Theorem, it was natural to attempt to study this generalization using a similar approach by Galois representations and modular forms. In this talk, I will survey some of the successes of applying this method, current ongoing approaches, and fundamental challenges in carrying out a complete resolution.