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For positive integers $a,b,c$ and $i,j,k\\ge3$ with $\\max\\{i,j,k\\}\\ge5$, we call the triple $(ap_i,bp_j,cp_k)$ universal if for any $n=0,1,2,\\ldots$ there are nonnegative integers $x,y,z$ such that $n=ap_i(x)+bp_j(y)+cp_k(z)$. We show that there are only 95 candidates for universal triples (two of which are $(p_4,p_5,p_6)$ and $(p_3,p_4,p_{27})$), and conjecture that they are indeed universal triples. For many triples $(ap_i,bp_j,cp_k)$ (including $(p_3,4p_4,p_5),(p_4,p_5,p_6)$ and $"},"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":"0905.0635","kind":"arxiv","version":22},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2009-05-05T18:59:03Z","cross_cats_sorted":["math.CO"],"title_canon_sha256":"9d59ca627f6d24f059331890fe6f0c09eb082bb21ec13d3d6d243002c817b90a","abstract_canon_sha256":"989d6b4e4f943030e9e68ac25474293c494ce3973ed89d642a30cdf76ba7f80d"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T02:09:54.897832Z","signature_b64":"DDx2hMD8kwwHWvTTZI3omY7y4Az/P1/WeHI34m108S8aVux/Pe9wTZ1YWziGrUO4TsiefHJmK6zC6mQhz4aMBQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"dba1ff31ad036e1b58edda61db4f7b9f399ad388dce26704c0526bf1857c5654","last_reissued_at":"2026-05-18T02:09:54.897043Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T02:09:54.897043Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"On universal sums of polygonal numbers","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.CO"],"primary_cat":"math.NT","authors_text":"Zhi-Wei Sun","submitted_at":"2009-05-05T18:59:03Z","abstract_excerpt":"For $m=3,4,\\ldots$, the polygonal numbers of order $m$ are given by $p_m(n)=(m-2)\\binom n2+n\\ (n=0,1,2,\\ldots)$. For positive integers $a,b,c$ and $i,j,k\\ge3$ with $\\max\\{i,j,k\\}\\ge5$, we call the triple $(ap_i,bp_j,cp_k)$ universal if for any $n=0,1,2,\\ldots$ there are nonnegative integers $x,y,z$ such that $n=ap_i(x)+bp_j(y)+cp_k(z)$. We show that there are only 95 candidates for universal triples (two of which are $(p_4,p_5,p_6)$ and $(p_3,p_4,p_{27})$), and conjecture that they are indeed universal triples. For many triples $(ap_i,bp_j,cp_k)$ (including $(p_3,4p_4,p_5),(p_4,p_5,p_6)$ and $"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"0905.0635","kind":"arxiv","version":22},"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":"0905.0635","created_at":"2026-05-18T02:09:54.897168+00:00"},{"alias_kind":"arxiv_version","alias_value":"0905.0635v22","created_at":"2026-05-18T02:09:54.897168+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.0905.0635","created_at":"2026-05-18T02:09:54.897168+00:00"},{"alias_kind":"pith_short_12","alias_value":"3OQ76MNNANXB","created_at":"2026-05-18T12:25:58.018023+00:00"},{"alias_kind":"pith_short_16","alias_value":"3OQ76MNNANXBWWHN","created_at":"2026-05-18T12:25:58.018023+00:00"},{"alias_kind":"pith_short_8","alias_value":"3OQ76MNN","created_at":"2026-05-18T12:25:58.018023+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/3OQ76MNNANXBWWHN3JQ5WT33T4","json":"https://pith.science/pith/3OQ76MNNANXBWWHN3JQ5WT33T4.json","graph_json":"https://pith.science/api/pith-number/3OQ76MNNANXBWWHN3JQ5WT33T4/graph.json","events_json":"https://pith.science/api/pith-number/3OQ76MNNANXBWWHN3JQ5WT33T4/events.json","paper":"https://pith.science/paper/3OQ76MNN"},"agent_actions":{"view_html":"https://pith.science/pith/3OQ76MNNANXBWWHN3JQ5WT33T4","download_json":"https://pith.science/pith/3OQ76MNNANXBWWHN3JQ5WT33T4.json","view_paper":"https://pith.science/paper/3OQ76MNN","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=0905.0635&json=true","fetch_graph":"https://pith.science/api/pith-number/3OQ76MNNANXBWWHN3JQ5WT33T4/graph.json","fetch_events":"https://pith.science/api/pith-number/3OQ76MNNANXBWWHN3JQ5WT33T4/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/3OQ76MNNANXBWWHN3JQ5WT33T4/action/timestamp_anchor","attest_storage":"https://pith.science/pith/3OQ76MNNANXBWWHN3JQ5WT33T4/action/storage_attestation","attest_author":"https://pith.science/pith/3OQ76MNNANXBWWHN3JQ5WT33T4/action/author_attestation","sign_citation":"https://pith.science/pith/3OQ76MNNANXBWWHN3JQ5WT33T4/action/citation_signature","submit_replication":"https://pith.science/pith/3OQ76MNNANXBWWHN3JQ5WT33T4/action/replication_record"}},"created_at":"2026-05-18T02:09:54.897168+00:00","updated_at":"2026-05-18T02:09:54.897168+00:00"}