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Sun [13] studied for what values of positive integers $a,b,c$ the sum $ap_5+bp_5+cp_5$ is universal over $\\mathbb Z$ (i.e., any $n\\in\\mathbb N=\\{0,1,2,\\ldots\\}$ has the form $ap_5(x)+bp_5(y)+cp_5(z)$ with $x,y,z\\in\\mathbb Z$). We prove that $p_5+bp_5+3p_5\\,(b=1,2,3,4,9)$ and $p_5+2p_5+6p_5$ are universal over $\\mathbb Z$, as conjectured by Sun. Sun also conjectured that any $n\\in\\mathbb N$ can be written as $p_3(x)+p_5(y)+p_{11}(z)$ and $3p_3(x)+p_5(y)+p_7(z)$ with $x,y,z\\in\\mathbb "},"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":"0906.2450","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2009-06-15T03:47:06Z","cross_cats_sorted":["math.CO"],"title_canon_sha256":"bd085cf6f42d03d6ae9736341262c270d0bdb34020ee6fa9ed1737ede1356d8e","abstract_canon_sha256":"e30d8f81fe3c106bce5af4fa8e11d114a283ea552aae8cf6f49731892ecb72f4"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:12:00.646500Z","signature_b64":"FnKwYf/qRHy2HMb4+NIgAJl1CUP2CiD8w/JlI63HRYoDbCFjwREO4RNIGBPTZGwHZHH60NLf/wvqTPV18WxiDw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"44084ae307c9f365dd3aea12e465d2708bcf61076d7880799023be9d77cf6d9e","last_reissued_at":"2026-05-18T01:12:00.646171Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:12:00.646171Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"On some universal sums of generalized polygonal numbers","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.CO"],"primary_cat":"math.NT","authors_text":"Fan Ge, Zhi-Wei Sun","submitted_at":"2009-06-15T03:47:06Z","abstract_excerpt":"For $m=3,4,\\ldots$ those $p_m(x)=(m-2)x(x-1)/2+x$ with $x\\in\\mathbb Z$ are called generalized $m$-gonal numbers. Sun [13] studied for what values of positive integers $a,b,c$ the sum $ap_5+bp_5+cp_5$ is universal over $\\mathbb Z$ (i.e., any $n\\in\\mathbb N=\\{0,1,2,\\ldots\\}$ has the form $ap_5(x)+bp_5(y)+cp_5(z)$ with $x,y,z\\in\\mathbb Z$). We prove that $p_5+bp_5+3p_5\\,(b=1,2,3,4,9)$ and $p_5+2p_5+6p_5$ are universal over $\\mathbb Z$, as conjectured by Sun. Sun also conjectured that any $n\\in\\mathbb N$ can be written as $p_3(x)+p_5(y)+p_{11}(z)$ and $3p_3(x)+p_5(y)+p_7(z)$ with $x,y,z\\in\\mathbb "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"0906.2450","kind":"arxiv","version":3},"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"},"aliases":[{"alias_kind":"arxiv","alias_value":"0906.2450","created_at":"2026-05-18T01:12:00.646222+00:00"},{"alias_kind":"arxiv_version","alias_value":"0906.2450v3","created_at":"2026-05-18T01:12:00.646222+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.0906.2450","created_at":"2026-05-18T01:12:00.646222+00:00"},{"alias_kind":"pith_short_12","alias_value":"IQEEVYYHZHZW","created_at":"2026-05-18T12:26:00.592388+00:00"},{"alias_kind":"pith_short_16","alias_value":"IQEEVYYHZHZWLXJ2","created_at":"2026-05-18T12:26:00.592388+00:00"},{"alias_kind":"pith_short_8","alias_value":"IQEEVYYH","created_at":"2026-05-18T12:26:00.592388+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":0,"internal_anchor_count":0,"sample":[]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/IQEEVYYHZHZWLXJ25IJOIZOSOC","json":"https://pith.science/pith/IQEEVYYHZHZWLXJ25IJOIZOSOC.json","graph_json":"https://pith.science/api/pith-number/IQEEVYYHZHZWLXJ25IJOIZOSOC/graph.json","events_json":"https://pith.science/api/pith-number/IQEEVYYHZHZWLXJ25IJOIZOSOC/events.json","paper":"https://pith.science/paper/IQEEVYYH"},"agent_actions":{"view_html":"https://pith.science/pith/IQEEVYYHZHZWLXJ25IJOIZOSOC","download_json":"https://pith.science/pith/IQEEVYYHZHZWLXJ25IJOIZOSOC.json","view_paper":"https://pith.science/paper/IQEEVYYH","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=0906.2450&json=true","fetch_graph":"https://pith.science/api/pith-number/IQEEVYYHZHZWLXJ25IJOIZOSOC/graph.json","fetch_events":"https://pith.science/api/pith-number/IQEEVYYHZHZWLXJ25IJOIZOSOC/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/IQEEVYYHZHZWLXJ25IJOIZOSOC/action/timestamp_anchor","attest_storage":"https://pith.science/pith/IQEEVYYHZHZWLXJ25IJOIZOSOC/action/storage_attestation","attest_author":"https://pith.science/pith/IQEEVYYHZHZWLXJ25IJOIZOSOC/action/author_attestation","sign_citation":"https://pith.science/pith/IQEEVYYHZHZWLXJ25IJOIZOSOC/action/citation_signature","submit_replication":"https://pith.science/pith/IQEEVYYHZHZWLXJ25IJOIZOSOC/action/replication_record"}},"created_at":"2026-05-18T01:12:00.646222+00:00","updated_at":"2026-05-18T01:12:00.646222+00:00"}