{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2012:MYG2YY2WCV3IO47UV52OZLJ6US","short_pith_number":"pith:MYG2YY2W","schema_version":"1.0","canonical_sha256":"660dac635615768773f4af74ecad3ea4a491bccc1e15a1b17f693fff75c9c58c","source":{"kind":"arxiv","id":"1210.5237","version":4},"attestation_state":"computed","paper":{"title":"Determining $x$ or $y$ mod $p^2$ with $p=x^2+dy^2$","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.CO"],"primary_cat":"math.NT","authors_text":"Zhi-Wei Sun","submitted_at":"2012-10-18T19:58:29Z","abstract_excerpt":"Let $p$ be an odd prime and let $d\\in\\{2,3,7\\}$. When $(\\frac{-d}p)=1$ we can write $p=x^2+dy^2$ with $x,y\\in\\mathbb Z$; in this paper we aim at determining $x$ or $y$ modulo $p^2$. For example, when $p=x^2+3y^2$, we show that if $p\\equiv x\\equiv 1\\pmod 4$ then $$\\sum_{k=0}^{(p-1)/2}(3[3\\mid k]-1)(2k+1)\\frac{\\binom{2k}k^2}{(-16)^k}\\equiv\\left(\\frac2p\\right)2x\\pmod{p^2}$$ where $[3\\mid k]$ takes $1$ or $0$ according as $3\\mid k$ or not, and that if $-p\\equiv y\\equiv 1\\pmod4$ then $$\\sum_{k=0}^{(p-1)/2}\\left(\\frac k3\\right)\\frac{k\\binom{2k}k^2}{(-16)^k} \\equiv(-1)^{(p+1)/4}y\\equiv\\sum_{k=0}^{(p-"},"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":"1210.5237","kind":"arxiv","version":4},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2012-10-18T19:58:29Z","cross_cats_sorted":["math.CO"],"title_canon_sha256":"0cfd3195a7e3a2ddb1f69ce2df12d0f463163e144e97856e9044f0da87e75a87","abstract_canon_sha256":"1dd0531335276af49270400afae90d732bf1b9b65c81b50e0c599edebb96e88c"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:55:58.450835Z","signature_b64":"kM2Pm8t6xAEZeFb7N3P4mfm7T3fy4IrjbZSjhcXno/+vfVWLFUB+NNq3VXacmROujwXFC03cmKh5BZ6bMWE9AA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"660dac635615768773f4af74ecad3ea4a491bccc1e15a1b17f693fff75c9c58c","last_reissued_at":"2026-05-18T01:55:58.450417Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:55:58.450417Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Determining $x$ or $y$ mod $p^2$ with $p=x^2+dy^2$","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.CO"],"primary_cat":"math.NT","authors_text":"Zhi-Wei Sun","submitted_at":"2012-10-18T19:58:29Z","abstract_excerpt":"Let $p$ be an odd prime and let $d\\in\\{2,3,7\\}$. When $(\\frac{-d}p)=1$ we can write $p=x^2+dy^2$ with $x,y\\in\\mathbb Z$; in this paper we aim at determining $x$ or $y$ modulo $p^2$. For example, when $p=x^2+3y^2$, we show that if $p\\equiv x\\equiv 1\\pmod 4$ then $$\\sum_{k=0}^{(p-1)/2}(3[3\\mid k]-1)(2k+1)\\frac{\\binom{2k}k^2}{(-16)^k}\\equiv\\left(\\frac2p\\right)2x\\pmod{p^2}$$ where $[3\\mid k]$ takes $1$ or $0$ according as $3\\mid k$ or not, and that if $-p\\equiv y\\equiv 1\\pmod4$ then $$\\sum_{k=0}^{(p-1)/2}\\left(\\frac k3\\right)\\frac{k\\binom{2k}k^2}{(-16)^k} \\equiv(-1)^{(p+1)/4}y\\equiv\\sum_{k=0}^{(p-"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1210.5237","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":"1210.5237","created_at":"2026-05-18T01:55:58.450484+00:00"},{"alias_kind":"arxiv_version","alias_value":"1210.5237v4","created_at":"2026-05-18T01:55:58.450484+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1210.5237","created_at":"2026-05-18T01:55:58.450484+00:00"},{"alias_kind":"pith_short_12","alias_value":"MYG2YY2WCV3I","created_at":"2026-05-18T12:27:14.488303+00:00"},{"alias_kind":"pith_short_16","alias_value":"MYG2YY2WCV3IO47U","created_at":"2026-05-18T12:27:14.488303+00:00"},{"alias_kind":"pith_short_8","alias_value":"MYG2YY2W","created_at":"2026-05-18T12:27:14.488303+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/MYG2YY2WCV3IO47UV52OZLJ6US","json":"https://pith.science/pith/MYG2YY2WCV3IO47UV52OZLJ6US.json","graph_json":"https://pith.science/api/pith-number/MYG2YY2WCV3IO47UV52OZLJ6US/graph.json","events_json":"https://pith.science/api/pith-number/MYG2YY2WCV3IO47UV52OZLJ6US/events.json","paper":"https://pith.science/paper/MYG2YY2W"},"agent_actions":{"view_html":"https://pith.science/pith/MYG2YY2WCV3IO47UV52OZLJ6US","download_json":"https://pith.science/pith/MYG2YY2WCV3IO47UV52OZLJ6US.json","view_paper":"https://pith.science/paper/MYG2YY2W","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1210.5237&json=true","fetch_graph":"https://pith.science/api/pith-number/MYG2YY2WCV3IO47UV52OZLJ6US/graph.json","fetch_events":"https://pith.science/api/pith-number/MYG2YY2WCV3IO47UV52OZLJ6US/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/MYG2YY2WCV3IO47UV52OZLJ6US/action/timestamp_anchor","attest_storage":"https://pith.science/pith/MYG2YY2WCV3IO47UV52OZLJ6US/action/storage_attestation","attest_author":"https://pith.science/pith/MYG2YY2WCV3IO47UV52OZLJ6US/action/author_attestation","sign_citation":"https://pith.science/pith/MYG2YY2WCV3IO47UV52OZLJ6US/action/citation_signature","submit_replication":"https://pith.science/pith/MYG2YY2WCV3IO47UV52OZLJ6US/action/replication_record"}},"created_at":"2026-05-18T01:55:58.450484+00:00","updated_at":"2026-05-18T01:55:58.450484+00:00"}