{"paper":{"title":"For which real quadratic fields is Kim's octonary form universal?","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Scott Duke Kominers","submitted_at":"2026-06-28T10:28:02Z","abstract_excerpt":"Let $K=\\mathbb{Q}(\\sqrt{D})$ with $D>1$ squarefree, and let $\\varepsilon_+$ be the totally positive fundamental unit of $\\mathcal{O}_K$. B. M. Kim proved in 2000 that the octonary diagonal form \\[\n  f=x_1^2+\\cdots+x_4^2+\\varepsilon_+(x_5^2+\\cdots+x_8^2) \\] is universal over $\\mathcal{O}_K$ whenever $D=n^2-1$ is squarefree. We complete Kim's result to an if-and-only-if classification: $f$ is universal if and only if $D=n^2-1$ for some $n\\ge2$, or $D=n^2-4$ for some odd $n\\ge3$, in both cases subject to squarefreeness. The second family appears to be new in this context and contains $\\mathbb{Q}("},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2606.29321","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/2606.29321/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"}