{"paper":{"title":"Arithmetic progressions represented by diagonal ternary quadratic forms","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Hai-Liang Wu, Zhi-Wei Sun","submitted_at":"2018-11-14T15:39:12Z","abstract_excerpt":"Let $d>r\\ge 0$ be integers. For positive integers $a,b,c$, if any term of the arithmetic progression $\\{r+dn:\\ n=0,1,2,\\ldots\\}$ can be written as $ax^2+by^2+cz^2$ with $x,y,z\\in\\mathbb{Z}$, then the form $ax^2+by^2+cz^2$ is called $(d,r)$-universal. In this paper, via the theory of ternary quadratic forms we study the $(d,r)$-universality of some diagonal ternary quadratic forms conjectured by L. Pehlivan and K. S. Williams, and Z.-W. Sun. For example, we prove that $2x^2+3y^2+10z^2$ is $(8,5)$-universal, $x^2+3y^2+8z^2$ and $x^2+2y^2+12z^2$ are $(10,1)$-universal and $(10,9)$-universal, and "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1811.05855","kind":"arxiv","version":2},"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/1811.05855/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"}