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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-","authors_text":"Zhi-Wei Sun","cross_cats":["math.CO"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2012-10-18T19:58:29Z","title":"Determining $x$ or $y$ mod $p^2$ with $p=x^2+dy^2$"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1210.5237","kind":"arxiv","version":4},"verdict":{"created_at":null,"id":null,"model_set":{},"one_line_summary":"","pipeline_version":null,"pith_extraction_headline":"","strongest_claim":"","weakest_assumption":""}},"verdict_id":null}}],"author_attestations":[],"timestamp_anchors":[],"storage_attestations":[],"citation_signatures":[],"replication_records":[],"corrections":[],"mirror_hints":[],"record_created":{"event_id":"sha256:8b81011ca252ea975a866455546459845481c09b71c8dfa234c8f333861a6230","target":"record","created_at":"2026-05-18T01:55:58Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"attestation_state":"computed","canonical_record":{"metadata":{"abstract_canon_sha256":"1dd0531335276af49270400afae90d732bf1b9b65c81b50e0c599edebb96e88c","cross_cats_sorted":["math.CO"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2012-10-18T19:58:29Z","title_canon_sha256":"0cfd3195a7e3a2ddb1f69ce2df12d0f463163e144e97856e9044f0da87e75a87"},"schema_version":"1.0","source":{"id":"1210.5237","kind":"arxiv","version":4}},"canonical_sha256":"660dac635615768773f4af74ecad3ea4a491bccc1e15a1b17f693fff75c9c58c","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"660dac635615768773f4af74ecad3ea4a491bccc1e15a1b17f693fff75c9c58c","first_computed_at":"2026-05-18T01:55:58.450417Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T01:55:58.450417Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"kM2Pm8t6xAEZeFb7N3P4mfm7T3fy4IrjbZSjhcXno/+vfVWLFUB+NNq3VXacmROujwXFC03cmKh5BZ6bMWE9AA==","signature_status":"signed_v1","signed_at":"2026-05-18T01:55:58.450835Z","signed_message":"canonical_sha256_bytes"},"source_id":"1210.5237","source_kind":"arxiv","source_version":4}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:8b81011ca252ea975a866455546459845481c09b71c8dfa234c8f333861a6230","sha256:f08ae0045024800d7f843ee391bc59e2e8b0a57f2c40c03e77c68fb0f57fb204"],"state_sha256":"d159ca198323bddb450aa33ecbb3338b4f75f7f17c113f67a3b063b6745583aa"}