{"paper":{"title":"Universal mixed sums of generalized $4$- and $8$-gonal numbers","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Byeong-Kweon Oh, Jangwon Ju","submitted_at":"2018-09-11T04:17:20Z","abstract_excerpt":"An integer of the form $P_m(x)= \\frac{(m-2)x^2-(m-4)x}{2}$ for an integer $x$, is called a generalized $m$-gonal number. For positive integers $\\alpha_1,\\dots,\\alpha_u$ and $\\beta_1,\\dots,\\beta_v$, a mixed sum $\\Phi=\\alpha_1P_4(x_1)+\\cdots+\\alpha_uP_4(x_u)+\\beta_1P_8(y_1)+\\cdots+\\beta_vP_8(y_v)$ of generalized $4$- and $8$-gonal numbers is called universal if $\\Phi=N$ has an integer solution for any nonnegative integer $N$. In this article, we prove that there are exactly 1271 proper universal mixed sums of generalized $4$- and $8$-gonal numbers. Furthermore, the \"$61$-theorem\" is proved, whic"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1809.03673","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"}