{"paper":{"title":"The quartic Fermat equation in Hilbert class fields of imaginary quadratic fields","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Patrick Morton, Rodney Lynch","submitted_at":"2014-10-11T15:54:36Z","abstract_excerpt":"It is shown that the quartic Fermat equation $x^4 +y^4=1$ has nontrivial integral solutions in the Hilbert class field $\\Sigma$ of any quadratic field $K=\\mathbb{Q}(\\sqrt{-d})$ whose discriminant satisfies $-d \\equiv 1$ (mod 8). A corollary is that the quartic Fermat equation has no nontrivial solution in $K=\\mathbb{Q}(\\sqrt{-p})$, for $p$ $( > 7)$ a prime congruent to $7$ (mod 8), but does have a nontrivial solution in the odd degree extension $\\Sigma$ of $K$. These solutions arise from explicit formulas for the points of order 4 on elliptic curves in Tate normal form. The solutions are studi"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1410.3008","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":""},"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"}