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We also prove that any $n\\in\\mathbb N$ can be written as $x^2+y^2+z^2+w^2$ with $x,y,z,w\\in\\mathbb N$ such that $P(x,y,z)$ is a square, whenever $P(x,y,z)$ is among the polynomials \\begin{gather*} x,\\ 2x,\\ x-y,\\ 2x-2y,\\ a(x^2-y^2)\\ (a=1,2,3),\\ x^2-3y^2,\\ 3x^2-2y^2, \\\\x^"},"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":"1604.06723","kind":"arxiv","version":14},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2016-04-22T16:00:36Z","cross_cats_sorted":[],"title_canon_sha256":"d681946f9327f36f4db80271930daadfcdd6f5437a72dd4140a3380b478453cd","abstract_canon_sha256":"f0d56e6a123b756f2f935fec50a99e4e7c1d91f722f0d278f55136ff2e7922a1"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:52:50.337415Z","signature_b64":"zea4U2358fdBcjrriSa5hL+lIE5FFGKY2PPKdPwSsBQDVZI5gMK9+KTLDdW6cRdEMnfjOSiTFPhZ6z/qLABkBQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"52ebb67ef591f2d869c12d8b7f82da635a2618ddc4c52c4c3c319b3191253562","last_reissued_at":"2026-05-18T00:52:50.336849Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:52:50.336849Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Refining 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":"2016-04-22T16:00:36Z","abstract_excerpt":"Lagrange's four-square theorem asserts that any $n\\in\\mathbb N=\\{0,1,2,\\ldots\\}$ can be written as the sum of four squares. This can be further refined in various ways. We show that any $n\\in\\mathbb N$ can be written as $x^2+y^2+z^2+w^2$ with $x,y,z,w\\in\\mathbb Z$ such that $x+y+z$ (or $x+2y$, $x+y+2z$) is a square (or a cube). We also prove that any $n\\in\\mathbb N$ can be written as $x^2+y^2+z^2+w^2$ with $x,y,z,w\\in\\mathbb N$ such that $P(x,y,z)$ is a square, whenever $P(x,y,z)$ is among the polynomials \\begin{gather*} x,\\ 2x,\\ x-y,\\ 2x-2y,\\ a(x^2-y^2)\\ (a=1,2,3),\\ x^2-3y^2,\\ 3x^2-2y^2, \\\\x^"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1604.06723","kind":"arxiv","version":14},"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":"1604.06723","created_at":"2026-05-18T00:52:50.336921+00:00"},{"alias_kind":"arxiv_version","alias_value":"1604.06723v14","created_at":"2026-05-18T00:52:50.336921+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1604.06723","created_at":"2026-05-18T00:52:50.336921+00:00"},{"alias_kind":"pith_short_12","alias_value":"KLV3M7XVSHZN","created_at":"2026-05-18T12:30:25.849896+00:00"},{"alias_kind":"pith_short_16","alias_value":"KLV3M7XVSHZNQ2OB","created_at":"2026-05-18T12:30:25.849896+00:00"},{"alias_kind":"pith_short_8","alias_value":"KLV3M7XV","created_at":"2026-05-18T12:30:25.849896+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/KLV3M7XVSHZNQ2OBFWFX7AW2MN","json":"https://pith.science/pith/KLV3M7XVSHZNQ2OBFWFX7AW2MN.json","graph_json":"https://pith.science/api/pith-number/KLV3M7XVSHZNQ2OBFWFX7AW2MN/graph.json","events_json":"https://pith.science/api/pith-number/KLV3M7XVSHZNQ2OBFWFX7AW2MN/events.json","paper":"https://pith.science/paper/KLV3M7XV"},"agent_actions":{"view_html":"https://pith.science/pith/KLV3M7XVSHZNQ2OBFWFX7AW2MN","download_json":"https://pith.science/pith/KLV3M7XVSHZNQ2OBFWFX7AW2MN.json","view_paper":"https://pith.science/paper/KLV3M7XV","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1604.06723&json=true","fetch_graph":"https://pith.science/api/pith-number/KLV3M7XVSHZNQ2OBFWFX7AW2MN/graph.json","fetch_events":"https://pith.science/api/pith-number/KLV3M7XVSHZNQ2OBFWFX7AW2MN/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/KLV3M7XVSHZNQ2OBFWFX7AW2MN/action/timestamp_anchor","attest_storage":"https://pith.science/pith/KLV3M7XVSHZNQ2OBFWFX7AW2MN/action/storage_attestation","attest_author":"https://pith.science/pith/KLV3M7XVSHZNQ2OBFWFX7AW2MN/action/author_attestation","sign_citation":"https://pith.science/pith/KLV3M7XVSHZNQ2OBFWFX7AW2MN/action/citation_signature","submit_replication":"https://pith.science/pith/KLV3M7XVSHZNQ2OBFWFX7AW2MN/action/replication_record"}},"created_at":"2026-05-18T00:52:50.336921+00:00","updated_at":"2026-05-18T00:52:50.336921+00:00"}