{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2026:VYKNFWPAJ32PJSPXGNPROQCLWQ","short_pith_number":"pith:VYKNFWPA","schema_version":"1.0","canonical_sha256":"ae14d2d9e04ef4f4c9f7335f17404bb4051fbf7212ae8412893ae68825076fdd","source":{"kind":"arxiv","id":"2607.01695","version":1},"attestation_state":"computed","paper":{"title":"A Pfaffian Proof and Generalization of a Conjecture of Sun Zhiwei","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Fei Liu, Hong-Ge Chen","submitted_at":"2026-07-02T04:39:20Z","abstract_excerpt":"Let $p$ be an odd prime, let $n=(p-1)/2$, and let $\\chi=(\\frac{\\cdot}{p})$, with $\\chi(0)=0$. For $a\\in\\mathbb F_p^\\times$ define \\[\n  D_a(x)=\\det_{1\\le i,j\\le n}(x+\\chi(i^2-aj)),\n  \\qquad\n  D_a^{(0)}(x)=\\det_{0\\le i,j\\le n}(x+\\chi(i^2-aj)). \\] We prove \\[\n  D_a(0)=0\n  \\quad\\Longleftrightarrow\\quad\n  p\\equiv 3 \\pmod 4\n  \\quad\\text{and}\\quad\n  \\chi(a n!)=1. \\] For $p\\equiv3\\pmod4$ we also give explicit Pfaffian-square factorizations of $D_a(x)$ and $D_a^{(0)}(x)$. Let $s_p=(-1)^{\\lfloor(p+1)/8\\rfloor}$. If $\\chi(a n!)=1$, then $s_pD_a(x)/x=s_pD_a^{(0)}(x)$ is a positive integer square. If $\\chi"},"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":"2607.01695","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2026-07-02T04:39:20Z","cross_cats_sorted":[],"title_canon_sha256":"5fd6c7e216cc8c2397c0ef42b2dca2642b66afce6c59fa3811a8694fe0ced750","abstract_canon_sha256":"157c3bad7cc33214e6c0387effa28e1839b1e2ef077d51444204bf226509eeca"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-03T01:17:26.977992Z","signature_b64":"0en7j19u5SQHkvJwoxRQ7iXQIT4Cz3gji1F/WJ/RSYyFJY7VE2XN55w6UiwKW/iHgKG37/xdxQo/EgSV6ygWCg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"ae14d2d9e04ef4f4c9f7335f17404bb4051fbf7212ae8412893ae68825076fdd","last_reissued_at":"2026-07-03T01:17:26.977598Z","signature_status":"signed_v1","first_computed_at":"2026-07-03T01:17:26.977598Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"A Pfaffian Proof and Generalization of a Conjecture of Sun Zhiwei","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Fei Liu, Hong-Ge Chen","submitted_at":"2026-07-02T04:39:20Z","abstract_excerpt":"Let $p$ be an odd prime, let $n=(p-1)/2$, and let $\\chi=(\\frac{\\cdot}{p})$, with $\\chi(0)=0$. For $a\\in\\mathbb F_p^\\times$ define \\[\n  D_a(x)=\\det_{1\\le i,j\\le n}(x+\\chi(i^2-aj)),\n  \\qquad\n  D_a^{(0)}(x)=\\det_{0\\le i,j\\le n}(x+\\chi(i^2-aj)). \\] We prove \\[\n  D_a(0)=0\n  \\quad\\Longleftrightarrow\\quad\n  p\\equiv 3 \\pmod 4\n  \\quad\\text{and}\\quad\n  \\chi(a n!)=1. \\] For $p\\equiv3\\pmod4$ we also give explicit Pfaffian-square factorizations of $D_a(x)$ and $D_a^{(0)}(x)$. Let $s_p=(-1)^{\\lfloor(p+1)/8\\rfloor}$. If $\\chi(a n!)=1$, then $s_pD_a(x)/x=s_pD_a^{(0)}(x)$ is a positive integer square. 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