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For any positive integer $\\alpha$ and $m\\in\\{1,2,3\\}$, we have \\begin{align*} \\sum_{k=0}^{p^{\\alpha}n-1}\\frac{(\\frac12)_k}{k!}\\cdot\\frac{(-4)^k}{m^k}\\equiv\\bigg(\\frac{m(m-4)}{p}\\bigg)\\sum_{k=0}^{p^{\\alpha-1}n-1}\\frac{(\\frac12)_k}{k!}\\cdot\\frac{(-4)^k}{m^k}\\pmod{p^{2\\alpha}}, \\end{align*} where $(x)_k=x(x+1)\\cdots(x+k-1)$ and $\\big(\\frac{\\cdot}{\\cdot}\\big)$ denotes the Legendre symbol. Also, when $m=4$, \\begin{align*} \\sum_{k=0}^{p^{\\alpha}n-1}(-1)^k\\cdot\\frac{(\\frac12)_k}{k!}\\equiv p\\sum_{k=0}^{p^{\\alpha-1}n-1}(-1)^k\\cdot\\frac{(\\frac12"},"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":"1810.09370","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2018-10-22T15:38:35Z","cross_cats_sorted":["math.CO"],"title_canon_sha256":"1e41c611614e05e2a36e0dcedec2cc4619cae9bbc8443723dfff78c013b76530","abstract_canon_sha256":"d85a143987ce6ae3f5187c2995d0f819c5350bf164c98dd74400ea4be87f5eb9"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:02:40.321218Z","signature_b64":"4GIzkt/sUxQmRH0VcF4NOe3FOXU011xLo/39G9v24EFMHIV++LMGfXXYMaZHqyf45VhLhvd/272N1vajQe4bCA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"6627a692dd08bcc6972ac6251742b82b2abf785c3fd7aba817333bf658c32dff","last_reissued_at":"2026-05-18T00:02:40.320612Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:02:40.320612Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"On the Atkin and Swinnerton-Dyer type congruences for some truncated hypergeometric ${}_1F_0$ series","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.CO"],"primary_cat":"math.NT","authors_text":"Hao Pan, Yong Zhang","submitted_at":"2018-10-22T15:38:35Z","abstract_excerpt":"Let $p$ be an odd prime and let $n$ be a positive integer. For any positive integer $\\alpha$ and $m\\in\\{1,2,3\\}$, we have \\begin{align*} \\sum_{k=0}^{p^{\\alpha}n-1}\\frac{(\\frac12)_k}{k!}\\cdot\\frac{(-4)^k}{m^k}\\equiv\\bigg(\\frac{m(m-4)}{p}\\bigg)\\sum_{k=0}^{p^{\\alpha-1}n-1}\\frac{(\\frac12)_k}{k!}\\cdot\\frac{(-4)^k}{m^k}\\pmod{p^{2\\alpha}}, \\end{align*} where $(x)_k=x(x+1)\\cdots(x+k-1)$ and $\\big(\\frac{\\cdot}{\\cdot}\\big)$ denotes the Legendre symbol. Also, when $m=4$, \\begin{align*} \\sum_{k=0}^{p^{\\alpha}n-1}(-1)^k\\cdot\\frac{(\\frac12)_k}{k!}\\equiv p\\sum_{k=0}^{p^{\\alpha-1}n-1}(-1)^k\\cdot\\frac{(\\frac12"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1810.09370","kind":"arxiv","version":1},"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":"1810.09370","created_at":"2026-05-18T00:02:40.320701+00:00"},{"alias_kind":"arxiv_version","alias_value":"1810.09370v1","created_at":"2026-05-18T00:02:40.320701+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1810.09370","created_at":"2026-05-18T00:02:40.320701+00:00"},{"alias_kind":"pith_short_12","alias_value":"MYT2NEW5BC6M","created_at":"2026-05-18T12:32:40.477152+00:00"},{"alias_kind":"pith_short_16","alias_value":"MYT2NEW5BC6MNFZK","created_at":"2026-05-18T12:32:40.477152+00:00"},{"alias_kind":"pith_short_8","alias_value":"MYT2NEW5","created_at":"2026-05-18T12:32:40.477152+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/MYT2NEW5BC6MNFZKYYSROQVYFM","json":"https://pith.science/pith/MYT2NEW5BC6MNFZKYYSROQVYFM.json","graph_json":"https://pith.science/api/pith-number/MYT2NEW5BC6MNFZKYYSROQVYFM/graph.json","events_json":"https://pith.science/api/pith-number/MYT2NEW5BC6MNFZKYYSROQVYFM/events.json","paper":"https://pith.science/paper/MYT2NEW5"},"agent_actions":{"view_html":"https://pith.science/pith/MYT2NEW5BC6MNFZKYYSROQVYFM","download_json":"https://pith.science/pith/MYT2NEW5BC6MNFZKYYSROQVYFM.json","view_paper":"https://pith.science/paper/MYT2NEW5","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1810.09370&json=true","fetch_graph":"https://pith.science/api/pith-number/MYT2NEW5BC6MNFZKYYSROQVYFM/graph.json","fetch_events":"https://pith.science/api/pith-number/MYT2NEW5BC6MNFZKYYSROQVYFM/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/MYT2NEW5BC6MNFZKYYSROQVYFM/action/timestamp_anchor","attest_storage":"https://pith.science/pith/MYT2NEW5BC6MNFZKYYSROQVYFM/action/storage_attestation","attest_author":"https://pith.science/pith/MYT2NEW5BC6MNFZKYYSROQVYFM/action/author_attestation","sign_citation":"https://pith.science/pith/MYT2NEW5BC6MNFZKYYSROQVYFM/action/citation_signature","submit_replication":"https://pith.science/pith/MYT2NEW5BC6MNFZKYYSROQVYFM/action/replication_record"}},"created_at":"2026-05-18T00:02:40.320701+00:00","updated_at":"2026-05-18T00:02:40.320701+00:00"}