{"paper":{"title":"On biquadratic fields: when 5 squares are not enough","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Daniel Dombek","submitted_at":"2025-06-25T20:35:23Z","abstract_excerpt":"In this paper we study the Pythagoras number $\\mathcal{P}(\\mathcal{O}_K)$ for the rings of integers in totally real biquadratic fields $K$. We continue the work of Tinkov\\'a towards proving the conjecture by Kr\\'asensk\\'y, Ra\\v{s}ka and Sgallov\\'a that a biquadratic $K$ satisfies $\\mathcal{P}(\\mathcal{O}_K)\\geq 6$ if and only if it contains neither $\\sqrt{2}$ nor $\\sqrt{5}$, with only finitely many exceptions. We fully solve two out of three remaining classes of fields by proving that all but finitely many $K$ containing $\\sqrt{6}$ or $\\sqrt{7}$ satisfy $\\mathcal{P}(\\mathcal{O}_K)\\geq 6$. Furt"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2506.20820","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/2506.20820/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"}