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We also establish the following new congruences: \\begin{align*}\\sum_{k=0}^{(p-1)/2}\\frac{\\binom{2k}k\\binom{3k}k}{27^k}\\equiv&\\l(\\frac p3\\r)\\frac{2^p+1}3\\pmod{p^2}, \\\\\\sum_{k=0}^{(p-1)/2}\\frac{\\binom{6k}{3k}\\binom{3k}k}{(2k+1)432^k}\\equiv&\\l(\\frac p3\\r)\\frac{3^p+1}4\\pmod{p^2}, \\\\\\sum_{k=0}^{(p-1)/2}\\frac{\\binom{4k}{2k}\\"},"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":"1601.04782","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2016-01-19T03:18:05Z","cross_cats_sorted":["math.CO"],"title_canon_sha256":"034ec09b246ec7d16fed323e17ad25b1136b4ee1c98f6ebf54f7847c76f6d3b9","abstract_canon_sha256":"33fab6f55b634cdb888b264d2a3de9cf3ac7af7af23a8b0195c2f53ec46b8f17"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:05:09.753020Z","signature_b64":"wLvanRwR2DHi4acbL5dgIOTaGgJfhObwmLus1igLsY85oJ4pmjwkj4CNftG8ZLgkjBadSUHvZ8FS51W0jK4dBA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"ae36375ec248be8e43e2b3d9cf645fecd44f1bc852a093731c2fcc8640ff9306","last_reissued_at":"2026-05-18T00:05:09.752552Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:05:09.752552Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"New congruences involving products of two binomial coefficients","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.CO"],"primary_cat":"math.NT","authors_text":"Guo-Shuai Mao, Zhi-Wei Sun","submitted_at":"2016-01-19T03:18:05Z","abstract_excerpt":"Let $p>3$ be a prime and let $a$ be a positive integer. We show that if $p\\equiv1\\pmod 4$ or $a>1$ then $$\\sum_{k=0}^{\\lfloor\\frac34p^a\\rfloor}\\frac{\\binom{2k}k^2}{16^k}\\equiv\\l(\\frac{-1}{p^a}\\r)\\pmod{p^3}$$ with $(-)$ the Jacobi symbol, which confirms a conjecture of Z.-W. Sun. We also establish the following new congruences: \\begin{align*}\\sum_{k=0}^{(p-1)/2}\\frac{\\binom{2k}k\\binom{3k}k}{27^k}\\equiv&\\l(\\frac p3\\r)\\frac{2^p+1}3\\pmod{p^2}, \\\\\\sum_{k=0}^{(p-1)/2}\\frac{\\binom{6k}{3k}\\binom{3k}k}{(2k+1)432^k}\\equiv&\\l(\\frac p3\\r)\\frac{3^p+1}4\\pmod{p^2}, \\\\\\sum_{k=0}^{(p-1)/2}\\frac{\\binom{4k}{2k}\\"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1601.04782","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":"1601.04782","created_at":"2026-05-18T00:05:09.752624+00:00"},{"alias_kind":"arxiv_version","alias_value":"1601.04782v1","created_at":"2026-05-18T00:05:09.752624+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1601.04782","created_at":"2026-05-18T00:05:09.752624+00:00"},{"alias_kind":"pith_short_12","alias_value":"VY3DOXWCJC7I","created_at":"2026-05-18T12:30:48.956258+00:00"},{"alias_kind":"pith_short_16","alias_value":"VY3DOXWCJC7I4Q7C","created_at":"2026-05-18T12:30:48.956258+00:00"},{"alias_kind":"pith_short_8","alias_value":"VY3DOXWC","created_at":"2026-05-18T12:30:48.956258+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/VY3DOXWCJC7I4Q7CWPM46ZC75T","json":"https://pith.science/pith/VY3DOXWCJC7I4Q7CWPM46ZC75T.json","graph_json":"https://pith.science/api/pith-number/VY3DOXWCJC7I4Q7CWPM46ZC75T/graph.json","events_json":"https://pith.science/api/pith-number/VY3DOXWCJC7I4Q7CWPM46ZC75T/events.json","paper":"https://pith.science/paper/VY3DOXWC"},"agent_actions":{"view_html":"https://pith.science/pith/VY3DOXWCJC7I4Q7CWPM46ZC75T","download_json":"https://pith.science/pith/VY3DOXWCJC7I4Q7CWPM46ZC75T.json","view_paper":"https://pith.science/paper/VY3DOXWC","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1601.04782&json=true","fetch_graph":"https://pith.science/api/pith-number/VY3DOXWCJC7I4Q7CWPM46ZC75T/graph.json","fetch_events":"https://pith.science/api/pith-number/VY3DOXWCJC7I4Q7CWPM46ZC75T/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/VY3DOXWCJC7I4Q7CWPM46ZC75T/action/timestamp_anchor","attest_storage":"https://pith.science/pith/VY3DOXWCJC7I4Q7CWPM46ZC75T/action/storage_attestation","attest_author":"https://pith.science/pith/VY3DOXWCJC7I4Q7CWPM46ZC75T/action/author_attestation","sign_citation":"https://pith.science/pith/VY3DOXWCJC7I4Q7CWPM46ZC75T/action/citation_signature","submit_replication":"https://pith.science/pith/VY3DOXWCJC7I4Q7CWPM46ZC75T/action/replication_record"}},"created_at":"2026-05-18T00:05:09.752624+00:00","updated_at":"2026-05-18T00:05:09.752624+00:00"}