English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Global: Zendy Tools annual

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the summarisation as premium feature

Summarisation

Insights

Presents the pdf analysis as premium feature

PDF Analysis

Presents the zaia usage as premium feature

ZAIA

Global: Zendy Tools monthly

Global: Zendy Plus monthly

Presents the access of premium content as premium feature

Premium Content

Global: Zendy Plus annual

Global: Zendy Free Plan

Let $K$ be a totally real field, and $r\geq 5$ a fixed rational prime. Inthis paper, we use the modular method as presented in the recent work ofFreitas 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 wefollow the recent work of Freitas which constructs Frey curves over totallyreal subfields of $K(\zeta_r)$. When $K=\mathbb{Q}$ we get that there are nonon-trivial, primitive integer solutions $(x,y,z)$ with $2|z$ for signatures$(r,r,p)$ when $r \in \{5,7,11,13,19,23,37,47,53,59,61,67,71,79,83,101,103,107,131,139,149\}$ and $p$ is sufficientlylarge. 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$(5,5,p),(7,7,p)$, $(11,11,p),(13,13,p)$ and sufficiently large $p$.

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