University of Sydney

    School of Mathematics and Statistics

    Algebraic Geometry Seminar

    11:00-12:30 in Carslaw 709 on Wednesday 6 June.

    Nils Bruin

    Solving generalised Fermat equations

    The generalised Fermat equation x^r+y^s=z^t is known to have only finitely many integral solutions with gcd(x,y,z)=1 (primitive solutions) for any fixed triple (r,s,t) of exponents with 1/r+1/s+1/t<1. There are some remarkably large solutions, such as

    1549034^2 + 33^8 = 15613^3.

    If one wants to determine all primitive solutions to a generalised Fermat equation with a given exponent triple, then one has to find the rational points on a finite number of algebraic curves.

    In this talk, I will describe how one can determine those curves in some special situations and how one may proceed to determine the rational points on them. The techniques apply to hyperelliptic curves in general.
     
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