{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2015:HMF237K42DDASI3SWEE4PRK7DU","merge_version":"pith-open-graph-merge-v1","event_count":2,"valid_event_count":2,"invalid_event_count":0,"equivocation_count":0,"current":{"canonical_record":{"metadata":{"abstract_canon_sha256":"5f59dcec57b9ae65ba230e34f3f958b14024185feb5208588f0f81a1d60ae0b8","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2015-05-14T16:00:43Z","title_canon_sha256":"56c7b8282678e7a961a8d68a43c337367f4af62db8dcd7c3005da131c96cf246"},"schema_version":"1.0","source":{"id":"1505.03679","kind":"arxiv","version":4}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1505.03679","created_at":"2026-05-18T01:03:32Z"},{"alias_kind":"arxiv_version","alias_value":"1505.03679v4","created_at":"2026-05-18T01:03:32Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1505.03679","created_at":"2026-05-18T01:03:32Z"},{"alias_kind":"pith_short_12","alias_value":"HMF237K42DDA","created_at":"2026-05-18T12:29:25Z"},{"alias_kind":"pith_short_16","alias_value":"HMF237K42DDASI3S","created_at":"2026-05-18T12:29:25Z"},{"alias_kind":"pith_short_8","alias_value":"HMF237K4","created_at":"2026-05-18T12:29:25Z"}],"graph_snapshots":[{"event_id":"sha256:82ad604721c51a887af2c409ce471ec374c152bb9923aa7eed0f24524e382d0a","target":"graph","created_at":"2026-05-18T01:03:32Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"graph_snapshot":{"author_claims":{"count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","strong_count":0},"builder_version":"pith-number-builder-2026-05-17-v1","claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"paper":{"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. 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","authors_text":"Zhi-Wei Sun","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2015-05-14T16:00:43Z","title":"On $x(ax+1)+y(by+1)+z(cz+1)$ and $x(ax+b)+y(ay+c)+z(az+d)$"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1505.03679","kind":"arxiv","version":4},"verdict":{"created_at":null,"id":null,"model_set":{},"one_line_summary":"","pipeline_version":null,"pith_extraction_headline":"","strongest_claim":"","weakest_assumption":""}},"verdict_id":null}}],"author_attestations":[],"timestamp_anchors":[],"storage_attestations":[],"citation_signatures":[],"replication_records":[],"corrections":[],"mirror_hints":[],"record_created":{"event_id":"sha256:36bac044352856b5e9e5dd99291c3a73af4d656335a65bb31895eb837320e9ca","target":"record","created_at":"2026-05-18T01:03:32Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"attestation_state":"computed","canonical_record":{"metadata":{"abstract_canon_sha256":"5f59dcec57b9ae65ba230e34f3f958b14024185feb5208588f0f81a1d60ae0b8","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2015-05-14T16:00:43Z","title_canon_sha256":"56c7b8282678e7a961a8d68a43c337367f4af62db8dcd7c3005da131c96cf246"},"schema_version":"1.0","source":{"id":"1505.03679","kind":"arxiv","version":4}},"canonical_sha256":"3b0badfd5cd0c6092372b109c7c55f1d0ea899640854e0bfc04924030010b1fa","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"3b0badfd5cd0c6092372b109c7c55f1d0ea899640854e0bfc04924030010b1fa","first_computed_at":"2026-05-18T01:03:32.881106Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T01:03:32.881106Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"BtF1/oY7R/AIg+CNho/bamUplteFqQPh60bGCy9PKqCkNu2Uqz58mWwZznG7q1IYSl5zNkMhO6QU10ktIlgrBA==","signature_status":"signed_v1","signed_at":"2026-05-18T01:03:32.881800Z","signed_message":"canonical_sha256_bytes"},"source_id":"1505.03679","source_kind":"arxiv","source_version":4}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:36bac044352856b5e9e5dd99291c3a73af4d656335a65bb31895eb837320e9ca","sha256:82ad604721c51a887af2c409ce471ec374c152bb9923aa7eed0f24524e382d0a"],"state_sha256":"38ba9e68de5222a27d73ec95ff017b539bfbc9a83c5734b1d1523e7c80c7c4c7"}