### Asymptotic Fermat's Last Theorem for a family of equations of signature (n, 2n, 2)

Linfoot Number Theory Seminar

28th February 2024, 11:00 am – 12:00 pm

Fry Building, 2.04

Even prior to Wiles’s proof of Fermat’s Last Theorem, many researchers considered generalisations of the Fermat equation, such as Ax^p + By^q = Cz^r. While it is expected that these equations would almost always have finitely many integer solutions, proving this is a very hard question. Clearly, a necessary step towards proving that only finitely many solutions exist is showing that the exponents p, q and r are bounded. If this happens, we say that Asymptotic Fermat’s Last Theorem (AFLT) holds.

In this talk, we shall consider an infinite family of generalised Fermat equations and present an algorithmically testable set of conditions which, if satisfied, imply AFLT. In order to prove these conditions, we shall use the modular method for Diophantine equations developed in the proof of Fermat’s Last Theorem, together with Ribet’s level lowering theorem, Cebotarev’s density theorem and Galois theory.

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