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For $(a,b)=(1,1),(2,2),(1,3),(2,4)$, we study whether any sufficiently large integer can be expressed as $$p_{m+2}(x_1) + p_{m+2}(x_2) + ap_{m+2}(x_3) + bp_{m+2}(x_4)$$ with $x_1,x_2,x_3,x_4$ nonnegative integers. We show that the answer is positive if $(a,b)\\in\\{(1,3),(2,4)\\}$, or $(a,b)=(1,1)\\ \\&\\ 4\\mid m$, or $(a,b)=(2,2)\\ \\&\\ m\\not\\equiv"},"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":"1608.02022","kind":"arxiv","version":4},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2016-08-08T15:32:59Z","cross_cats_sorted":[],"title_canon_sha256":"292c7e39e91210ef1edbf0930bc2ccc12aef20dd2470edf3c2f1c0afc897cb67","abstract_canon_sha256":"ea6ca6f5a1a22175218867454c2f860ca2cec2027f3c7a3e1016637f6da957f4"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:33:39.645531Z","signature_b64":"LuX1VBm1QyoYsQUIldXESyQ9bovtSMAT6wpynVDdkUk0Id+69EosUa5yYV+dZhwlkUUqBJlgB4Je8nzT0KjBAg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"2a8f320dab0b88b46b9f11fe2a4c089bc0f4283f211240b1b368b353ce928c51","last_reissued_at":"2026-05-18T00:33:39.644882Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:33:39.644882Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Sums of four polygonal numbers with coefficients","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Xiang-Zi Meng, Zhi-Wei Sun","submitted_at":"2016-08-08T15:32:59Z","abstract_excerpt":"Let $m\\ge3$ be an integer. The polygonal numbers of order $m+2$ are given by $p_{m+2}(n)=m\\binom n2+n$ $(n=0,1,2,\\ldots)$. A famous claim of Fermat proved by Cauchy asserts that each nonnegative integer is the sum of $m+2$ polygonal numbers of order $m+2$. For $(a,b)=(1,1),(2,2),(1,3),(2,4)$, we study whether any sufficiently large integer can be expressed as $$p_{m+2}(x_1) + p_{m+2}(x_2) + ap_{m+2}(x_3) + bp_{m+2}(x_4)$$ with $x_1,x_2,x_3,x_4$ nonnegative integers. We show that the answer is positive if $(a,b)\\in\\{(1,3),(2,4)\\}$, or $(a,b)=(1,1)\\ \\&\\ 4\\mid m$, or $(a,b)=(2,2)\\ \\&\\ m\\not\\equiv"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1608.02022","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":"1608.02022","created_at":"2026-05-18T00:33:39.644969+00:00"},{"alias_kind":"arxiv_version","alias_value":"1608.02022v4","created_at":"2026-05-18T00:33:39.644969+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1608.02022","created_at":"2026-05-18T00:33:39.644969+00:00"},{"alias_kind":"pith_short_12","alias_value":"FKHTEDNLBOEL","created_at":"2026-05-18T12:30:15.759754+00:00"},{"alias_kind":"pith_short_16","alias_value":"FKHTEDNLBOELI247","created_at":"2026-05-18T12:30:15.759754+00:00"},{"alias_kind":"pith_short_8","alias_value":"FKHTEDNL","created_at":"2026-05-18T12:30:15.759754+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/FKHTEDNLBOELI247CH7CUTAITP","json":"https://pith.science/pith/FKHTEDNLBOELI247CH7CUTAITP.json","graph_json":"https://pith.science/api/pith-number/FKHTEDNLBOELI247CH7CUTAITP/graph.json","events_json":"https://pith.science/api/pith-number/FKHTEDNLBOELI247CH7CUTAITP/events.json","paper":"https://pith.science/paper/FKHTEDNL"},"agent_actions":{"view_html":"https://pith.science/pith/FKHTEDNLBOELI247CH7CUTAITP","download_json":"https://pith.science/pith/FKHTEDNLBOELI247CH7CUTAITP.json","view_paper":"https://pith.science/paper/FKHTEDNL","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1608.02022&json=true","fetch_graph":"https://pith.science/api/pith-number/FKHTEDNLBOELI247CH7CUTAITP/graph.json","fetch_events":"https://pith.science/api/pith-number/FKHTEDNLBOELI247CH7CUTAITP/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/FKHTEDNLBOELI247CH7CUTAITP/action/timestamp_anchor","attest_storage":"https://pith.science/pith/FKHTEDNLBOELI247CH7CUTAITP/action/storage_attestation","attest_author":"https://pith.science/pith/FKHTEDNLBOELI247CH7CUTAITP/action/author_attestation","sign_citation":"https://pith.science/pith/FKHTEDNLBOELI247CH7CUTAITP/action/citation_signature","submit_replication":"https://pith.science/pith/FKHTEDNLBOELI247CH7CUTAITP/action/replication_record"}},"created_at":"2026-05-18T00:33:39.644969+00:00","updated_at":"2026-05-18T00:33:39.644969+00:00"}