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We show that $(a,b,c)$ can be either of the following seven triples: $$(1,2,3),\\ (1,2,4),\\ (1,2,5),\\ (2,2,4),\\ (2,2,5),\\ (2,3,3),\\ (2,3,4),$$ and conjecture that any triple $(a,b,c)$ among $$(2,2,6),\\ (2,3,5),\\ (2,3,7),\\ (2,3,8),\\ (2,3,9),\\ (2,3,10)$$ also has the desired property. For integers $0\\le b\\le c\\le d\\le a$ with $a>2$, we prove that any nonnegative integer can be represented as $x(ax+b)+y(ay+c)+z(az+d)$ with $x,y,z"},"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":"1505.03679","kind":"arxiv","version":4},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2015-05-14T16:00:43Z","cross_cats_sorted":[],"title_canon_sha256":"56c7b8282678e7a961a8d68a43c337367f4af62db8dcd7c3005da131c96cf246","abstract_canon_sha256":"5f59dcec57b9ae65ba230e34f3f958b14024185feb5208588f0f81a1d60ae0b8"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:03:32.881800Z","signature_b64":"BtF1/oY7R/AIg+CNho/bamUplteFqQPh60bGCy9PKqCkNu2Uqz58mWwZznG7q1IYSl5zNkMhO6QU10ktIlgrBA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"3b0badfd5cd0c6092372b109c7c55f1d0ea899640854e0bfc04924030010b1fa","last_reissued_at":"2026-05-18T01:03:32.881106Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:03:32.881106Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"On $x(ax+1)+y(by+1)+z(cz+1)$ and $x(ax+b)+y(ay+c)+z(az+d)$","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Zhi-Wei Sun","submitted_at":"2015-05-14T16:00:43Z","abstract_excerpt":"In this paper we first investigate for what positive integers $a,b,c$ every nonnegative integer $n$ can be represented as $x(ax+1)+y(by+1)+z(cz+1)$ with $x,y,z$ integers. 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For integers $0\\le b\\le c\\le d\\le a$ with $a>2$, we prove that any nonnegative integer can be represented as $x(ax+b)+y(ay+c)+z(az+d)$ with $x,y,z"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1505.03679","kind":"arxiv","version":4},"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":"1505.03679","created_at":"2026-05-18T01:03:32.881210+00:00"},{"alias_kind":"arxiv_version","alias_value":"1505.03679v4","created_at":"2026-05-18T01:03:32.881210+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1505.03679","created_at":"2026-05-18T01:03:32.881210+00:00"},{"alias_kind":"pith_short_12","alias_value":"HMF237K42DDA","created_at":"2026-05-18T12:29:25.134429+00:00"},{"alias_kind":"pith_short_16","alias_value":"HMF237K42DDASI3S","created_at":"2026-05-18T12:29:25.134429+00:00"},{"alias_kind":"pith_short_8","alias_value":"HMF237K4","created_at":"2026-05-18T12:29:25.134429+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/HMF237K42DDASI3SWEE4PRK7DU","json":"https://pith.science/pith/HMF237K42DDASI3SWEE4PRK7DU.json","graph_json":"https://pith.science/api/pith-number/HMF237K42DDASI3SWEE4PRK7DU/graph.json","events_json":"https://pith.science/api/pith-number/HMF237K42DDASI3SWEE4PRK7DU/events.json","paper":"https://pith.science/paper/HMF237K4"},"agent_actions":{"view_html":"https://pith.science/pith/HMF237K42DDASI3SWEE4PRK7DU","download_json":"https://pith.science/pith/HMF237K42DDASI3SWEE4PRK7DU.json","view_paper":"https://pith.science/paper/HMF237K4","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1505.03679&json=true","fetch_graph":"https://pith.science/api/pith-number/HMF237K42DDASI3SWEE4PRK7DU/graph.json","fetch_events":"https://pith.science/api/pith-number/HMF237K42DDASI3SWEE4PRK7DU/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/HMF237K42DDASI3SWEE4PRK7DU/action/timestamp_anchor","attest_storage":"https://pith.science/pith/HMF237K42DDASI3SWEE4PRK7DU/action/storage_attestation","attest_author":"https://pith.science/pith/HMF237K42DDASI3SWEE4PRK7DU/action/author_attestation","sign_citation":"https://pith.science/pith/HMF237K42DDASI3SWEE4PRK7DU/action/citation_signature","submit_replication":"https://pith.science/pith/HMF237K42DDASI3SWEE4PRK7DU/action/replication_record"}},"created_at":"2026-05-18T01:03:32.881210+00:00","updated_at":"2026-05-18T01:03:32.881210+00:00"}