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When $2\\nmid ab$, we show that any sufficiently large integer can be written as $$\\frac{w(aw+b)}2+\\frac{x(ax+b)}2+\\frac{y(ay+b)}2+\\frac{z(az+b)}2$$ with $w,x,y,z$ nonnegative integers. When $2\\mid a$ and $2\\nmid b$, we prove that any sufficiently large integer can be written as $$w(aw+b)+x(ax+b)+y(ay+b)+z(az+b)"},"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":"2411.14308","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2024-11-21T17:00:03Z","cross_cats_sorted":[],"title_canon_sha256":"55cd16813ba929b93b9e78aaae491c48d6ec014696ac74b35a9a5e0aed1d627c","abstract_canon_sha256":"28fd65bdd06541e4ed1d2bb4fc10313228ffbf14abdac77435deb19f067a7d2a"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T09:44:54.926035Z","signature_b64":"GuzXeoLLUlRUIWMYB7Cnx6FFMTCu7QVniunCQazwEanDkmmNNzQNtTUFIjiRlpQXMmmt7E16UdHYMiESCj4FAw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"6a5fe54fe3e0abd5be7051e2faa9e8094d578cb4477b09e83124d09a52960e65","last_reissued_at":"2026-07-05T09:44:54.925519Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T09:44:54.925519Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"New results similar to Lagrange's four-square theorem","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Zhi-Wei Sun","submitted_at":"2024-11-21T17:00:03Z","abstract_excerpt":"In this paper we establish some new results similar to Lagrange's four-square theorem. For example, we prove that any integer $n>1$ can be written as $w(5w+1)/2+x(5x+1)/2+y(5y+1)/2+z(5z+1)/2$ with $w,x,y,z\\in\\mathbb Z$.\n  Let $a$ and $b$ be integers with $a>0$, $b>-a$ and $\\gcd(a,b)=1$. When $2\\nmid ab$, we show that any sufficiently large integer can be written as $$\\frac{w(aw+b)}2+\\frac{x(ax+b)}2+\\frac{y(ay+b)}2+\\frac{z(az+b)}2$$ with $w,x,y,z$ nonnegative integers. When $2\\mid a$ and $2\\nmid b$, we prove that any sufficiently large integer can be written as $$w(aw+b)+x(ax+b)+y(ay+b)+z(az+b)"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2411.14308","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":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2411.14308/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"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":"2411.14308","created_at":"2026-07-05T09:44:54.925582+00:00"},{"alias_kind":"arxiv_version","alias_value":"2411.14308v3","created_at":"2026-07-05T09:44:54.925582+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2411.14308","created_at":"2026-07-05T09:44:54.925582+00:00"},{"alias_kind":"pith_short_12","alias_value":"NJP6KT7D4CV5","created_at":"2026-07-05T09:44:54.925582+00:00"},{"alias_kind":"pith_short_16","alias_value":"NJP6KT7D4CV5LPTQ","created_at":"2026-07-05T09:44:54.925582+00:00"},{"alias_kind":"pith_short_8","alias_value":"NJP6KT7D","created_at":"2026-07-05T09:44:54.925582+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/NJP6KT7D4CV5LPTQKHRPVKPIBF","json":"https://pith.science/pith/NJP6KT7D4CV5LPTQKHRPVKPIBF.json","graph_json":"https://pith.science/api/pith-number/NJP6KT7D4CV5LPTQKHRPVKPIBF/graph.json","events_json":"https://pith.science/api/pith-number/NJP6KT7D4CV5LPTQKHRPVKPIBF/events.json","paper":"https://pith.science/paper/NJP6KT7D"},"agent_actions":{"view_html":"https://pith.science/pith/NJP6KT7D4CV5LPTQKHRPVKPIBF","download_json":"https://pith.science/pith/NJP6KT7D4CV5LPTQKHRPVKPIBF.json","view_paper":"https://pith.science/paper/NJP6KT7D","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=2411.14308&json=true","fetch_graph":"https://pith.science/api/pith-number/NJP6KT7D4CV5LPTQKHRPVKPIBF/graph.json","fetch_events":"https://pith.science/api/pith-number/NJP6KT7D4CV5LPTQKHRPVKPIBF/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/NJP6KT7D4CV5LPTQKHRPVKPIBF/action/timestamp_anchor","attest_storage":"https://pith.science/pith/NJP6KT7D4CV5LPTQKHRPVKPIBF/action/storage_attestation","attest_author":"https://pith.science/pith/NJP6KT7D4CV5LPTQKHRPVKPIBF/action/author_attestation","sign_citation":"https://pith.science/pith/NJP6KT7D4CV5LPTQKHRPVKPIBF/action/citation_signature","submit_replication":"https://pith.science/pith/NJP6KT7D4CV5LPTQKHRPVKPIBF/action/replication_record"}},"created_at":"2026-07-05T09:44:54.925582+00:00","updated_at":"2026-07-05T09:44:54.925582+00:00"}