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We confirm a conjecture of Sun and prove that $${}_{2r+1}F_{2r}\\bigg[\\begin{matrix}\\alpha&\\alpha&\\ldots&\\alpha\\\\ &1&\\ldots&1\\end{matrix}\\bigg|\\,1\\bigg]_{p-1}\\equiv0\\pmod{p^2},$$ where the truncated hypergeometric series $$ {}_{q+1}F_{q}\\bigg[\\begin{matrix}x_0&x_1&\\ldots&x_{q}\\\\ &y_1&\\ldots&y_q\\end{matrix}\\bigg|\\,z\\bigg]_{n}:=\\sum_{k=0}^n\\frac{(x_0)_k(x_1)_k\\cdots(x_q)_k}{(y_1)_k\\cdot (y_q)_k}\\cdot\\frac{z^k}{k!}. $$"},"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":"1801.02213","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2018-01-07T16:57:43Z","cross_cats_sorted":["math.CO"],"title_canon_sha256":"d33d92f28a75386a7e7dd0486e16f48052e7ebf863ae660381b5f9d2c1f4aab1","abstract_canon_sha256":"3d8caa13fe92aac658fbe263c67506c33578436ab53af963cf39c7601789356a"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:22:02.390326Z","signature_b64":"hAuWWbmNEwoJIJ9MNz7VAq/d2Nmlr/vPBnJsSLy8Xj+jCVcEI/m8O+BmX0hEb7n2s8POn3XIEwuHDbM30KG4CQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"c85f765cb655173384b6ad33a5a681a8334850b35f118df6cde6b4d190890933","last_reissued_at":"2026-05-18T00:22:02.389730Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:22:02.389730Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"On the divisibility of some truncated hypergeometric series","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.CO"],"primary_cat":"math.NT","authors_text":"Guo-Shuai Mao, Hao Pan","submitted_at":"2018-01-07T16:57:43Z","abstract_excerpt":"Let $p$ be an odd prime and $r\\geq 1$. Suppose that $\\alpha$ is a $p$-adic integer with $\\alpha\\equiv2a\\pmod p$ for some $1\\leq a<(p+r)/(2r+1)$. We confirm a conjecture of Sun and prove that $${}_{2r+1}F_{2r}\\bigg[\\begin{matrix}\\alpha&\\alpha&\\ldots&\\alpha\\\\ &1&\\ldots&1\\end{matrix}\\bigg|\\,1\\bigg]_{p-1}\\equiv0\\pmod{p^2},$$ where the truncated hypergeometric series $$ {}_{q+1}F_{q}\\bigg[\\begin{matrix}x_0&x_1&\\ldots&x_{q}\\\\ &y_1&\\ldots&y_q\\end{matrix}\\bigg|\\,z\\bigg]_{n}:=\\sum_{k=0}^n\\frac{(x_0)_k(x_1)_k\\cdots(x_q)_k}{(y_1)_k\\cdot (y_q)_k}\\cdot\\frac{z^k}{k!}. $$"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1801.02213","kind":"arxiv","version":2},"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":"1801.02213","created_at":"2026-05-18T00:22:02.389823+00:00"},{"alias_kind":"arxiv_version","alias_value":"1801.02213v2","created_at":"2026-05-18T00:22:02.389823+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1801.02213","created_at":"2026-05-18T00:22:02.389823+00:00"},{"alias_kind":"pith_short_12","alias_value":"ZBPXMXFWKULT","created_at":"2026-05-18T12:33:04.347982+00:00"},{"alias_kind":"pith_short_16","alias_value":"ZBPXMXFWKULTHBFW","created_at":"2026-05-18T12:33:04.347982+00:00"},{"alias_kind":"pith_short_8","alias_value":"ZBPXMXFW","created_at":"2026-05-18T12:33:04.347982+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/ZBPXMXFWKULTHBFWVUZ2LJUBVA","json":"https://pith.science/pith/ZBPXMXFWKULTHBFWVUZ2LJUBVA.json","graph_json":"https://pith.science/api/pith-number/ZBPXMXFWKULTHBFWVUZ2LJUBVA/graph.json","events_json":"https://pith.science/api/pith-number/ZBPXMXFWKULTHBFWVUZ2LJUBVA/events.json","paper":"https://pith.science/paper/ZBPXMXFW"},"agent_actions":{"view_html":"https://pith.science/pith/ZBPXMXFWKULTHBFWVUZ2LJUBVA","download_json":"https://pith.science/pith/ZBPXMXFWKULTHBFWVUZ2LJUBVA.json","view_paper":"https://pith.science/paper/ZBPXMXFW","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1801.02213&json=true","fetch_graph":"https://pith.science/api/pith-number/ZBPXMXFWKULTHBFWVUZ2LJUBVA/graph.json","fetch_events":"https://pith.science/api/pith-number/ZBPXMXFWKULTHBFWVUZ2LJUBVA/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/ZBPXMXFWKULTHBFWVUZ2LJUBVA/action/timestamp_anchor","attest_storage":"https://pith.science/pith/ZBPXMXFWKULTHBFWVUZ2LJUBVA/action/storage_attestation","attest_author":"https://pith.science/pith/ZBPXMXFWKULTHBFWVUZ2LJUBVA/action/author_attestation","sign_citation":"https://pith.science/pith/ZBPXMXFWKULTHBFWVUZ2LJUBVA/action/citation_signature","submit_replication":"https://pith.science/pith/ZBPXMXFWKULTHBFWVUZ2LJUBVA/action/replication_record"}},"created_at":"2026-05-18T00:22:02.389823+00:00","updated_at":"2026-05-18T00:22:02.389823+00:00"}