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A quaternary sum $\\Phi_{a,b,c,d}(x,y,z,t)=aP_8(x)+bP_8(y)+cP_8(z)+dP_8(t)$ of generalized octagonal numbers is called {\\it universal} if $\\Phi_{a,b,c,d}(x,y,z,t)=n$ has an integer solution $x,y,z,t$ for any positive integer $n$. In this article, we show that if $a=1$ and $(b,c,d)=(1,3,3), (1,3,6), (2,3,6), (2,3,7)$ or $(2,3,9)$, then $\\Phi_{a,b,c,d}(x,y,z,t)$ is universal. These were conjectured by Sun in \\cite {sun}. We also give an effective criterion on the universality of an arbitrary sum"},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"1704.08826","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2017-04-28T07:17:46Z","cross_cats_sorted":[],"title_canon_sha256":"98ae3c960809954a13b475d72fdfa8173eade13e5066b8d6ce4ebaec8b232797","abstract_canon_sha256":"bb57fe1f8845cb596fdcaea52cfc7386d89822b17373722eb985cddd0e1204ca"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:39:46.260941Z","signature_b64":"LkOaqotsnLgRMEnCpnT7ICq/iJfC1Y8l6ZsbsyadTb3LPoaez4wAB/e+RVxLXybLrEUZByZuJPd+hXc6EBGGDg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"ed6618af92eadd391a90042bce66fbbf042dd1e0da48879442ac70577a44b9c0","last_reissued_at":"2026-05-18T00:39:46.260292Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:39:46.260292Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Universal sums of generalized octagonal 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":"2017-04-28T07:17:46Z","abstract_excerpt":"An integer of the form $P_8(x)=3x^2-2x$ for some integer $x$ is called a generalized octagonal number. A quaternary sum $\\Phi_{a,b,c,d}(x,y,z,t)=aP_8(x)+bP_8(y)+cP_8(z)+dP_8(t)$ of generalized octagonal numbers is called {\\it universal} if $\\Phi_{a,b,c,d}(x,y,z,t)=n$ has an integer solution $x,y,z,t$ for any positive integer $n$. In this article, we show that if $a=1$ and $(b,c,d)=(1,3,3), (1,3,6), (2,3,6), (2,3,7)$ or $(2,3,9)$, then $\\Phi_{a,b,c,d}(x,y,z,t)$ is universal. These were conjectured by Sun in \\cite {sun}. 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