{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2015:QKMFLSVBAZ3726V3SWYHECQ6J6","short_pith_number":"pith:QKMFLSVB","schema_version":"1.0","canonical_sha256":"829855caa10677fd7abb95b0720a1e4faca591f3c1257d098e4b987a239778f3","source":{"kind":"arxiv","id":"1504.05482","version":1},"attestation_state":"computed","paper":{"title":"Proof of a congruence on sums of powers of $q$-binomial coefficients","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.CO"],"primary_cat":"math.NT","authors_text":"Ji-Cai Liu, Victor J. W. Guo","submitted_at":"2015-04-20T16:02:07Z","abstract_excerpt":"We prove that, if $m,n\\geqslant 1$ and $a_1,\\ldots,a_m$ are nonnegative integers, then \\begin{align*} \\frac{[a_1+\\cdots+a_m+1]!}{[a_1]!\\ldots[a_m]!}\\sum^{n-1}_{h=0}q^h\\prod_{i=1}^m{h\\brack a_i} \\equiv 0\\pmod{[n]}, \\end{align*} where $[n]=\\frac{1-q^n}{1-q}$, $[n]!=[n][n-1]\\cdots[1]$, and ${a\\brack b}=\\prod_{k=1}^b\\frac{1-q^{a-k+1}}{1-q^k}$. The $a_1=\\cdots=a_m$ case confirms a recent conjecture of Z.-W. Sun. We also show that, if $p>\\max\\{a,b\\}$ is a prime, then \\begin{align*} \\frac{[a+b+1]!}{[a]![b]!}\\sum_{h=0}^{p-1}q^h{h\\brack a}{h\\brack b} \\equiv (-1)^{a-b} q^{ab-{a\\choose 2}-{b\\choose 2}}[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":"1504.05482","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2015-04-20T16:02:07Z","cross_cats_sorted":["math.CO"],"title_canon_sha256":"89be669cc16dec302584258cb972fc2322d8f5167e7b66273123115141730c9f","abstract_canon_sha256":"da7711c8275e171a998c7e1d83360d0e8cf565463ca358fabe2f6a7438336d55"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T02:18:14.174955Z","signature_b64":"XT86KJH30Y4oYfD3lkWErmWQYR32P3XmpxmCJ9JYVMKF2RVwkC6PwsPXx4Dnmx7LqFtZ2aWHhAZ7o64lkCe5Ag==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"829855caa10677fd7abb95b0720a1e4faca591f3c1257d098e4b987a239778f3","last_reissued_at":"2026-05-18T02:18:14.174531Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T02:18:14.174531Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Proof of a congruence on sums of powers of $q$-binomial coefficients","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.CO"],"primary_cat":"math.NT","authors_text":"Ji-Cai Liu, Victor J. W. Guo","submitted_at":"2015-04-20T16:02:07Z","abstract_excerpt":"We prove that, if $m,n\\geqslant 1$ and $a_1,\\ldots,a_m$ are nonnegative integers, then \\begin{align*} \\frac{[a_1+\\cdots+a_m+1]!}{[a_1]!\\ldots[a_m]!}\\sum^{n-1}_{h=0}q^h\\prod_{i=1}^m{h\\brack a_i} \\equiv 0\\pmod{[n]}, \\end{align*} where $[n]=\\frac{1-q^n}{1-q}$, $[n]!=[n][n-1]\\cdots[1]$, and ${a\\brack b}=\\prod_{k=1}^b\\frac{1-q^{a-k+1}}{1-q^k}$. The $a_1=\\cdots=a_m$ case confirms a recent conjecture of Z.-W. Sun. We also show that, if $p>\\max\\{a,b\\}$ is a prime, then \\begin{align*} \\frac{[a+b+1]!}{[a]![b]!}\\sum_{h=0}^{p-1}q^h{h\\brack a}{h\\brack b} \\equiv (-1)^{a-b} q^{ab-{a\\choose 2}-{b\\choose 2}}[p"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1504.05482","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":"1504.05482","created_at":"2026-05-18T02:18:14.174593+00:00"},{"alias_kind":"arxiv_version","alias_value":"1504.05482v1","created_at":"2026-05-18T02:18:14.174593+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1504.05482","created_at":"2026-05-18T02:18:14.174593+00:00"},{"alias_kind":"pith_short_12","alias_value":"QKMFLSVBAZ37","created_at":"2026-05-18T12:29:37.295048+00:00"},{"alias_kind":"pith_short_16","alias_value":"QKMFLSVBAZ3726V3","created_at":"2026-05-18T12:29:37.295048+00:00"},{"alias_kind":"pith_short_8","alias_value":"QKMFLSVB","created_at":"2026-05-18T12:29:37.295048+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/QKMFLSVBAZ3726V3SWYHECQ6J6","json":"https://pith.science/pith/QKMFLSVBAZ3726V3SWYHECQ6J6.json","graph_json":"https://pith.science/api/pith-number/QKMFLSVBAZ3726V3SWYHECQ6J6/graph.json","events_json":"https://pith.science/api/pith-number/QKMFLSVBAZ3726V3SWYHECQ6J6/events.json","paper":"https://pith.science/paper/QKMFLSVB"},"agent_actions":{"view_html":"https://pith.science/pith/QKMFLSVBAZ3726V3SWYHECQ6J6","download_json":"https://pith.science/pith/QKMFLSVBAZ3726V3SWYHECQ6J6.json","view_paper":"https://pith.science/paper/QKMFLSVB","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1504.05482&json=true","fetch_graph":"https://pith.science/api/pith-number/QKMFLSVBAZ3726V3SWYHECQ6J6/graph.json","fetch_events":"https://pith.science/api/pith-number/QKMFLSVBAZ3726V3SWYHECQ6J6/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/QKMFLSVBAZ3726V3SWYHECQ6J6/action/timestamp_anchor","attest_storage":"https://pith.science/pith/QKMFLSVBAZ3726V3SWYHECQ6J6/action/storage_attestation","attest_author":"https://pith.science/pith/QKMFLSVBAZ3726V3SWYHECQ6J6/action/author_attestation","sign_citation":"https://pith.science/pith/QKMFLSVBAZ3726V3SWYHECQ6J6/action/citation_signature","submit_replication":"https://pith.science/pith/QKMFLSVBAZ3726V3SWYHECQ6J6/action/replication_record"}},"created_at":"2026-05-18T02:18:14.174593+00:00","updated_at":"2026-05-18T02:18:14.174593+00:00"}