{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2015:UOT6FK6SLER4M6MRR34AGYPQMF","short_pith_number":"pith:UOT6FK6S","schema_version":"1.0","canonical_sha256":"a3a7e2abd25923c679918ef80361f0614a697fa5912ea2e773fd06f606ab859f","source":{"kind":"arxiv","id":"1507.03485","version":3},"attestation_state":"computed","paper":{"title":"On the number of representations of n as a linear combination of four triangular numbers","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Min Wang, Zhi-Hong Sun","submitted_at":"2015-07-13T14:59:11Z","abstract_excerpt":"Let $\\Bbb Z$ and $\\Bbb N$ be the set of integers and the set of positive integers, respectively. For\n  $a,b,c,d,n\\in\\Bbb N$ let $t(a,b,c,d;n)$ be the number of representations of $n$ by $ax(x-1)/2+by(y-1)/2+cz(z-1)/2\n  +dw(w-1)/2$ $(x,y,z,w\\in\\Bbb Z$). In this paper we obtain explicit formulas for $t(a,b,c,d;n)$ in the cases\n  $(a,b,c,d)=(1,2,2,4),\\ (1,2,4,4),\\ (1,1,4,4),\\ (1,4,4,4)$, $(1,3,9,9),\\ (1,1,3,9)$, $(1,3,3,9)$, $(1,1,9,9),\\ (1,9,9,9)$ and $(1,1,1,9).$"},"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":"1507.03485","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2015-07-13T14:59:11Z","cross_cats_sorted":[],"title_canon_sha256":"c4c65f9d51a2f920c083d9532b2e1a0a28f2fdbb5e54fdb9a0dbf2c35dd6ba22","abstract_canon_sha256":"b1b9b4c55ef73a9a9514cf1a822997482cfee9a854da67d49ab71ae80d177d48"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:28:13.984194Z","signature_b64":"4GAKSiDOKKga44GoXdXhJcD/PxMQ+riO3EEOpac52JNKKB5LDNPreaZamivxfNycwhapW9PyIFVPHhOfkW3jCA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"a3a7e2abd25923c679918ef80361f0614a697fa5912ea2e773fd06f606ab859f","last_reissued_at":"2026-05-18T01:28:13.983598Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:28:13.983598Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"On the number of representations of n as a linear combination of four triangular numbers","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Min Wang, Zhi-Hong Sun","submitted_at":"2015-07-13T14:59:11Z","abstract_excerpt":"Let $\\Bbb Z$ and $\\Bbb N$ be the set of integers and the set of positive integers, respectively. For\n  $a,b,c,d,n\\in\\Bbb N$ let $t(a,b,c,d;n)$ be the number of representations of $n$ by $ax(x-1)/2+by(y-1)/2+cz(z-1)/2\n  +dw(w-1)/2$ $(x,y,z,w\\in\\Bbb Z$). In this paper we obtain explicit formulas for $t(a,b,c,d;n)$ in the cases\n  $(a,b,c,d)=(1,2,2,4),\\ (1,2,4,4),\\ (1,1,4,4),\\ (1,4,4,4)$, $(1,3,9,9),\\ (1,1,3,9)$, $(1,3,3,9)$, $(1,1,9,9),\\ (1,9,9,9)$ and $(1,1,1,9).$"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1507.03485","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":"1507.03485","created_at":"2026-05-18T01:28:13.983670+00:00"},{"alias_kind":"arxiv_version","alias_value":"1507.03485v3","created_at":"2026-05-18T01:28:13.983670+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1507.03485","created_at":"2026-05-18T01:28:13.983670+00:00"},{"alias_kind":"pith_short_12","alias_value":"UOT6FK6SLER4","created_at":"2026-05-18T12:29:44.643036+00:00"},{"alias_kind":"pith_short_16","alias_value":"UOT6FK6SLER4M6MR","created_at":"2026-05-18T12:29:44.643036+00:00"},{"alias_kind":"pith_short_8","alias_value":"UOT6FK6S","created_at":"2026-05-18T12:29:44.643036+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/UOT6FK6SLER4M6MRR34AGYPQMF","json":"https://pith.science/pith/UOT6FK6SLER4M6MRR34AGYPQMF.json","graph_json":"https://pith.science/api/pith-number/UOT6FK6SLER4M6MRR34AGYPQMF/graph.json","events_json":"https://pith.science/api/pith-number/UOT6FK6SLER4M6MRR34AGYPQMF/events.json","paper":"https://pith.science/paper/UOT6FK6S"},"agent_actions":{"view_html":"https://pith.science/pith/UOT6FK6SLER4M6MRR34AGYPQMF","download_json":"https://pith.science/pith/UOT6FK6SLER4M6MRR34AGYPQMF.json","view_paper":"https://pith.science/paper/UOT6FK6S","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1507.03485&json=true","fetch_graph":"https://pith.science/api/pith-number/UOT6FK6SLER4M6MRR34AGYPQMF/graph.json","fetch_events":"https://pith.science/api/pith-number/UOT6FK6SLER4M6MRR34AGYPQMF/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/UOT6FK6SLER4M6MRR34AGYPQMF/action/timestamp_anchor","attest_storage":"https://pith.science/pith/UOT6FK6SLER4M6MRR34AGYPQMF/action/storage_attestation","attest_author":"https://pith.science/pith/UOT6FK6SLER4M6MRR34AGYPQMF/action/author_attestation","sign_citation":"https://pith.science/pith/UOT6FK6SLER4M6MRR34AGYPQMF/action/citation_signature","submit_replication":"https://pith.science/pith/UOT6FK6SLER4M6MRR34AGYPQMF/action/replication_record"}},"created_at":"2026-05-18T01:28:13.983670+00:00","updated_at":"2026-05-18T01:28:13.983670+00:00"}