## STNB2020(34th edition)

### On the Asymptotic Fermat Conjecture

Samir Siksek

#### Abstract

The asymptotic Fermat conjecture states that for a number field K there is a constant $B_K$ such that for primes $p \ge B_K$ the only K-rational points on the Fermat curve $X^p+Y^p+Z^p=0$ are the obvious ones. In this talk we survey joint work with Nuno Freitas, Alain Kraus and Haluk \c Seng\"un, on the asymptotic Fermat conjecture. In particular we prove AFC for family of number fields $K=\mathbb{Q}(\zeta_{2^r})^+$.