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For $a,b,c,d,n\\in\\Bbb N$ let $N(a,b,c,d;n)$ be the number of representations of $n$ by $ax^2+by^2+cz^2+dw^2$, and 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 reveal the connections between $t(a,b,c,d;n)$ and $N(a,b,c,d;n)$. Suppose $a,n\\in\\Bbb N$ and $2\\nmid a$. We show that $$t(a,b,c,d;n)=\\frac 23N(a,b,c,d;8n+a+b+c+d)-2N(a,b,c,d;2n+(a+b+c+d)/4)$$ for $(a,b,c,d)= (a,a,2a,8m),\\ (a,3a,8k+2,8m+6),\\ (a"},"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":"1511.00478","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2015-11-02T12:51:36Z","cross_cats_sorted":[],"title_canon_sha256":"cd036b295d6ee217d6d80ec1b880d9e04a57445687b347c6fd88004003328a97","abstract_canon_sha256":"1a2a38c3d9e41b4814cd3bb252cfb54491b05a6fc0109f906c32238dca3a223d"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:24:45.977696Z","signature_b64":"qXBCbZMxEgPIRa6XxV3rfjywSAZbmepLy5lHn6/Lc4a7v2fidkSrGe7fDiUYprQimCSMv5hrRxhH5IsUIrw7Bw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"81ee7bdb5db02e8dc9c4b869ec9e771c55b52a0ba27139768f887df6e240d35f","last_reissued_at":"2026-05-18T01:24:45.977184Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:24:45.977184Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"On the number of representations of n as a linear combination of four triangular numbers II","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":"2015-11-02T12:51:36Z","abstract_excerpt":"Let $\\Bbb Z$ and $\\Bbb N$ be the set of integers and the set of positive integers, respectively. For $a,b,c,d,n\\in\\Bbb N$ let $N(a,b,c,d;n)$ be the number of representations of $n$ by $ax^2+by^2+cz^2+dw^2$, and 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 reveal the connections between $t(a,b,c,d;n)$ and $N(a,b,c,d;n)$. Suppose $a,n\\in\\Bbb N$ and $2\\nmid a$. We show that $$t(a,b,c,d;n)=\\frac 23N(a,b,c,d;8n+a+b+c+d)-2N(a,b,c,d;2n+(a+b+c+d)/4)$$ for $(a,b,c,d)= (a,a,2a,8m),\\ (a,3a,8k+2,8m+6),\\ (a"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1511.00478","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":"1511.00478","created_at":"2026-05-18T01:24:45.977263+00:00"},{"alias_kind":"arxiv_version","alias_value":"1511.00478v3","created_at":"2026-05-18T01:24:45.977263+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1511.00478","created_at":"2026-05-18T01:24:45.977263+00:00"},{"alias_kind":"pith_short_12","alias_value":"QHXHXW25WAXI","created_at":"2026-05-18T12:29:37.295048+00:00"},{"alias_kind":"pith_short_16","alias_value":"QHXHXW25WAXI3SOE","created_at":"2026-05-18T12:29:37.295048+00:00"},{"alias_kind":"pith_short_8","alias_value":"QHXHXW25","created_at":"2026-05-18T12:29:37.295048+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/QHXHXW25WAXI3SOEXBU6ZHTXDR","json":"https://pith.science/pith/QHXHXW25WAXI3SOEXBU6ZHTXDR.json","graph_json":"https://pith.science/api/pith-number/QHXHXW25WAXI3SOEXBU6ZHTXDR/graph.json","events_json":"https://pith.science/api/pith-number/QHXHXW25WAXI3SOEXBU6ZHTXDR/events.json","paper":"https://pith.science/paper/QHXHXW25"},"agent_actions":{"view_html":"https://pith.science/pith/QHXHXW25WAXI3SOEXBU6ZHTXDR","download_json":"https://pith.science/pith/QHXHXW25WAXI3SOEXBU6ZHTXDR.json","view_paper":"https://pith.science/paper/QHXHXW25","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1511.00478&json=true","fetch_graph":"https://pith.science/api/pith-number/QHXHXW25WAXI3SOEXBU6ZHTXDR/graph.json","fetch_events":"https://pith.science/api/pith-number/QHXHXW25WAXI3SOEXBU6ZHTXDR/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/QHXHXW25WAXI3SOEXBU6ZHTXDR/action/timestamp_anchor","attest_storage":"https://pith.science/pith/QHXHXW25WAXI3SOEXBU6ZHTXDR/action/storage_attestation","attest_author":"https://pith.science/pith/QHXHXW25WAXI3SOEXBU6ZHTXDR/action/author_attestation","sign_citation":"https://pith.science/pith/QHXHXW25WAXI3SOEXBU6ZHTXDR/action/citation_signature","submit_replication":"https://pith.science/pith/QHXHXW25WAXI3SOEXBU6ZHTXDR/action/replication_record"}},"created_at":"2026-05-18T01:24:45.977263+00:00","updated_at":"2026-05-18T01:24:45.977263+00:00"}