{"paper":{"title":"Generalized Fermat equation over cyclotomic $\\mathbb{Z}_l$-extensions of totally real fields","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Satyabrat Sahoo","submitted_at":"2026-05-20T07:52:51Z","abstract_excerpt":"Let $K$ be a totally real number field of odd degree. Let $l \\geq 5$ be a prime with $l \\nmid [K:\\mathbb{Q}]$ and $\\gcd(\\frac{l-1}{2}, [K:\\mathbb{Q}])=1$. We prove that if $2$ is inert in $K$, $l$ is non-Wieferich, i.e., $2^{l-1} \\not\\equiv 1 \\pmod{l^2}$, and $l$ is totally ramified in $K$, then the asymptotic Fermat's Last Theorem holds over each $n$-th layer $K_{n,l}$ of the cyclotomic $\\mathbb{Z}_l$-extension of $K$. We then prove that the generalized Fermat equation $Ax^p+By^p+Cz^p=0$ has no asymptotic solution over each $n$-th layer $K_{n,l}$ when $A,B,C \\in \\{u2^r : u\\in \\mathcal{O}_K^\\t"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2605.20860","kind":"arxiv","version":1},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2605.20860/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}