{"about":{"site":"https://codewithpapers.app","non_affiliation":"Code with Papers and Syntology are not affiliated with, endorsed by, or sponsored by Papers with Code, Meta, or the pwc-archive mirror.","licence":"CC BY-SA 4.0","licence_url":"https://creativecommons.org/licenses/by-sa/4.0/legalcode","attribution":"https://codewithpapers.app/attribution","modified":"archive material modified by Syntology; see the attribution page"},"url":"/paper/ternary-quadratic-forms-representing-a-given","title":"Ternary quadratic forms representing a given arithmetic progression","arxiv_id":"1906.02538","date":"2019-06-06","proceeding":null,"authors":["Tomáš Hejda","Vítězslav Kala"],"abstract":"A positive quadratic form is $(k,\\ell)$-universal if it represents all the numbers $kx+\\ell$ where $x$ is a non-negative integer, and almost $(k,\\ell)$-universal if it represents all but finitely many of them. We prove that for any $k,\\ell$ such that $k\\nmid\\ell$ there exists an almost $(k,\\ell)$-universal diagonal ternary form. We also conjecture that there are only finitely many primes $p$ for which a $(p,\\ell)$-universal diagonal ternary form exists (for any $\\ell<p$) and we show the results of computer experiments that speak in favor of the conjecture.","url_abs":"https://arxiv.org/abs/1906.02538v2","url_pdf":"https://arxiv.org/pdf/1906.02538v2.pdf","source":{"archive":"pwc-archive (Hugging Face), CC BY-SA 4.0","snapshot":"2025-07-28","licence_url":"https://creativecommons.org/licenses/by-sa/4.0/legalcode","row_kind":"links_only","authors_date_abstract":"arXiv metadata, CC0 1.0 (https://info.arxiv.org/help/license), from the Kaggle arXiv metadata snapshot of 2026-09-12"},"code_links":[{"paper_slug":"ternary-quadratic-forms-representing-a-given","repo_url":"https://github.com/tohecz/ternary","is_official":1,"mentioned_in_paper":1,"mentioned_in_github":1,"framework":"none","reach":null}],"tasks":[],"methods":[],"datasets_introduced":[],"methods_introduced":[],"results":[],"syntology":{"atlas_url":null,"mcp":null,"developers":"https://syntology.ai/developers"},"arxiv_metadata":null,"syntology_extracted_results":null}