Recipes to Fermat-type equations of the form x^r + y^r = Cz^p

Abstract

We describe a strategy to attack infinitely many Fermat-type equations of signature $(r,r,p)$, where $r \geq 7$ is a fixed prime and $p$ is a prime allowed to vary. We use a variant of the modular method over totally real subfields of $\mathbb{Q}(\zeta_r)$. In particular, to a solution $(a,b,c)$ of $x^r + y^r =Cz^p$ we will attach several Frey curves $E=E_{(a,b)}$. We prove modularity of all the Frey curves and the exsitence of a constant constant $M_r$, depending only on $r$, such that for all $p>M_r$ the representations $\bar{\rho}_{E,p}$ are absolutely irreducible. Along the way, we also prove modularity of certain elliptic curves that are semistable at all $v \mid 3$.\par Finally, we illustrate our methods by proving arithmetic statements about equations of signature $(7,7,p)$. Among which we emphasize that, using a multi-Frey technique, we show there is some constant $M$ such that if $p > M$ then the equation $x^7 + y^7 = 3z^p$ has no non-trivial primitive solutions.