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In this article, we prove the following extension of Lagrange's theorem: given a prime number $p$ and $v_1\\in Z^4$, $\\dots$, $v_k\\in Z^4$, $1\\leq k\\leq 3$, such that $\\|v_i\\|^2=p$ for all $1\\leq i\\leq k$ and $\\langle v_i|v_j\\rangle=0$ for all $1\\leq i<j\\leq k$, then there exists $v"},"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":"1805.04353","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2018-05-11T12:18:50Z","cross_cats_sorted":[],"title_canon_sha256":"bdace9641c7fbaaeb5b9b577a9ba9491ebc7b2fa901ca648f5cdb6347e809cc2","abstract_canon_sha256":"108b44f700e7c9a4675cd23c83bb5c3a36ff69f9e87c8234a2a1ced08210f2a4"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:16:12.722704Z","signature_b64":"YxocIN63oMRjOHOWArV5a6N/BSVmE4qu+4xypNg7EVD37oEoMbTdON0Hts58rZg3Jzy2HuZvcgy3jHxzuVOzAQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"dc8a39ab23c6a93ef2fb27a8dd503e57f707bc1d38f7c8a8170199cb09e8d45a","last_reissued_at":"2026-05-18T00:16:12.722052Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:16:12.722052Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Extended Lagrange's four-square theorem","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Jes\\'us Lacalle, Laura N. 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