Asymptotic Fermat for signatures $(r,r,p)$ using the modular approach

Abstract

Let $K$ be a totally real field, and $r\ge 5$ a fixed rational prime. In this paper, we use the modular method as presented in the work of Freitas and Siksek to study non-trivial, primitive solutions $(x,y,z)\in {\mathcal{O}}_K^3$ of the signature $(r,r,p)$ equation $x^r+y^r=z^p$ (where $p$ is a prime that varies). An adaptation of the modular method is needed, and we follow the work of Freitas which constructs Frey curves over totally real subfields of $K({\zeta }_r)$. When $K=\mathbb{Q}$ we get that for most of the primes $5\le r<150$ with $r\equiv 3,5\mathrm{mod}8$ there are no non-trivial, primitive integer solutions $(x,y,z)$ with $2|z$ for signatures $(r,r,p)$ when $p$ is sufficiently large. Similar results hold for quadratic fields, for example when $K=\mathbb{Q}(\sqrt{2})$ there are no non-trivial, primitive solutions $(x,y,z)\in {\mathcal{O}}_K^3$ with $\sqrt{2}|z$ for signatures $(5,5,p),(11,11,p),(13,13,p)$ and sufficiently large $p$.

arXiv commentCorrected Corollaries 7,8,9 and Example 10. Minor corrections resolved