## STNB2019 (33rd edition)

### Local restrictions to asymptotic Fermat conjecture with coefficients

Eduard Soto

#### Abstract

Let $A,B,C$ non-zero integers. Contrary to classical Fermat equation, Fermat equation $Ax^p + B y^p + C z^p =0$ with coefficients might have non-trivial solutions $(x,y,z)$ for big primes $p$. A conjecture of Frey and Mazur implies that there is a bound $k=k(A,B,C)$ so that ${\it essentially \; all}$ solutions $(x,y,z)$ of $Ax^p + By^p + c Z^p=0$ appear for $p < k(A,B,C)$. This is the so-called Asymptotic Fermat Conjecture with coefficients $(A,B,C)$. In this talk I will show that AFC is true for some families of $(A,B,C)$ being divisible by any number of primes. This is joint work with L. Dieulefait.

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