{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2009:J2WXTH7XTOZ62SRUVSXQPELCNB","short_pith_number":"pith:J2WXTH7X","canonical_record":{"source":{"id":"0907.2870","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2009-07-16T15:34:57Z","cross_cats_sorted":["math.CO"],"title_canon_sha256":"980761b8d4eaaf5dfc938700118d31d85582eb660ea31a589bf3a73df5045067","abstract_canon_sha256":"b8190313d12ce3ac5d341ef3f47a1f2378351c91509ca45d5af845efe4b6dc3a"},"schema_version":"1.0"},"canonical_sha256":"4ead799ff79bb3ed4a34acaf079162684ec1a147ce56d25793a7cabc21596a15","source":{"kind":"arxiv","id":"0907.2870","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"0907.2870","created_at":"2026-05-18T04:26:01Z"},{"alias_kind":"arxiv_version","alias_value":"0907.2870v1","created_at":"2026-05-18T04:26:01Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.0907.2870","created_at":"2026-05-18T04:26:01Z"},{"alias_kind":"pith_short_12","alias_value":"J2WXTH7XTOZ6","created_at":"2026-05-18T12:26:00Z"},{"alias_kind":"pith_short_16","alias_value":"J2WXTH7XTOZ62SRU","created_at":"2026-05-18T12:26:00Z"},{"alias_kind":"pith_short_8","alias_value":"J2WXTH7X","created_at":"2026-05-18T12:26:00Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2009:J2WXTH7XTOZ62SRUVSXQPELCNB","target":"record","payload":{"canonical_record":{"source":{"id":"0907.2870","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2009-07-16T15:34:57Z","cross_cats_sorted":["math.CO"],"title_canon_sha256":"980761b8d4eaaf5dfc938700118d31d85582eb660ea31a589bf3a73df5045067","abstract_canon_sha256":"b8190313d12ce3ac5d341ef3f47a1f2378351c91509ca45d5af845efe4b6dc3a"},"schema_version":"1.0"},"canonical_sha256":"4ead799ff79bb3ed4a34acaf079162684ec1a147ce56d25793a7cabc21596a15","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T04:26:01.570876Z","signature_b64":"7Lz9M43kIxv5egXIRBP/qV5ecF2MIReW1JN/8abgOoWYv17/RhalX6li28zQ3KcNfEfg8Aa/ZmSeSnrRRG3IAA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"4ead799ff79bb3ed4a34acaf079162684ec1a147ce56d25793a7cabc21596a15","last_reissued_at":"2026-05-18T04:26:01.570418Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T04:26:01.570418Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"0907.2870","source_version":1,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T04:26:01Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"GM8/oP5WUz2etjfgc8mDRhPoGf8pVXFrb0eHaaxrZuQVr+w1GlZzEbHM6eMuBH6rdBGRr0LFUMsGxf7U5QXwAw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-26T09:59:24.721858Z"},"content_sha256":"cda8c0ef7b1c31ca5774861adb902b9e8977ae939d15699943774a7f84826a9f","schema_version":"1.0","event_id":"sha256:cda8c0ef7b1c31ca5774861adb902b9e8977ae939d15699943774a7f84826a9f"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2009:J2WXTH7XTOZ62SRUVSXQPELCNB","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"On the least common multiple of $q$-binomial coefficients","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.CO"],"primary_cat":"math.NT","authors_text":"Victor J. W. Guo","submitted_at":"2009-07-16T15:34:57Z","abstract_excerpt":"In this paper, we prove the following identity $$ \\lcm({n\\brack 0}_q,{n\\brack 1}_q,...,{n\\brack n}_q) =\\frac{\\lcm([1]_q,[2]_q,...,[n+1]_q)}{[n+1]_q}, $$ where ${n\\brack k}_q$ denotes the $q$-binomial coefficient and $[n]_q=\\frac{1-q^n}{1-q}$. This result is a $q$-analogue of an identity of Farhi [Amer. Math. Monthly, November (2009)]."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"0907.2870","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"},"verdict_id":null},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T04:26:01Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"r9glSFetjvj5/2dbSdDs4s/HFzJD1MB4iNFxhLfeZXQVANu96b3YJWjy8HAcoM/YCceNRhw/mTBENbiI7ToSAw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-26T09:59:24.722284Z"},"content_sha256":"b76a73248901b3375329be1142fff46978b086760fda286329f120489b710fa9","schema_version":"1.0","event_id":"sha256:b76a73248901b3375329be1142fff46978b086760fda286329f120489b710fa9"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/J2WXTH7XTOZ62SRUVSXQPELCNB/bundle.json","state_url":"https://pith.science/pith/J2WXTH7XTOZ62SRUVSXQPELCNB/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/J2WXTH7XTOZ62SRUVSXQPELCNB/bundle.json","status":"primary"}],"public_keys":[{"key_id":"pith-v1-2026-05","algorithm":"ed25519","format":"raw","public_key_b64":"stVStoiQhXFxp4s2pdzPNoqVNBMojDU/fJ2db5S3CbM=","public_key_hex":"b2d552b68890857171a78b36a5dccf368a953413288c353f7c9d9d6f94b709b3","fingerprint_sha256_b32_first128bits":"RVFV5Z2OI2J3ZUO7ERDEBCYNKS","fingerprint_sha256_hex":"8d4b5ee74e4693bcd1df2446408b0d54","rotates_at":null,"url":"https://pith.science/pith-signing-key.json","notes":"Pith uses this Ed25519 key to sign canonical record SHA-256 digests. Verify with: ed25519_verify(public_key, message=canonical_sha256_bytes, signature=base64decode(signature_b64))."}],"merge_version":"pith-open-graph-merge-v1","built_at":"2026-05-26T09:59:24Z","links":{"resolver":"https://pith.science/pith/J2WXTH7XTOZ62SRUVSXQPELCNB","bundle":"https://pith.science/pith/J2WXTH7XTOZ62SRUVSXQPELCNB/bundle.json","state":"https://pith.science/pith/J2WXTH7XTOZ62SRUVSXQPELCNB/state.json","well_known_bundle":"https://pith.science/.well-known/pith/J2WXTH7XTOZ62SRUVSXQPELCNB/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2009:J2WXTH7XTOZ62SRUVSXQPELCNB","merge_version":"pith-open-graph-merge-v1","event_count":2,"valid_event_count":2,"invalid_event_count":0,"equivocation_count":0,"current":{"canonical_record":{"metadata":{"abstract_canon_sha256":"b8190313d12ce3ac5d341ef3f47a1f2378351c91509ca45d5af845efe4b6dc3a","cross_cats_sorted":["math.CO"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2009-07-16T15:34:57Z","title_canon_sha256":"980761b8d4eaaf5dfc938700118d31d85582eb660ea31a589bf3a73df5045067"},"schema_version":"1.0","source":{"id":"0907.2870","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"0907.2870","created_at":"2026-05-18T04:26:01Z"},{"alias_kind":"arxiv_version","alias_value":"0907.2870v1","created_at":"2026-05-18T04:26:01Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.0907.2870","created_at":"2026-05-18T04:26:01Z"},{"alias_kind":"pith_short_12","alias_value":"J2WXTH7XTOZ6","created_at":"2026-05-18T12:26:00Z"},{"alias_kind":"pith_short_16","alias_value":"J2WXTH7XTOZ62SRU","created_at":"2026-05-18T12:26:00Z"},{"alias_kind":"pith_short_8","alias_value":"J2WXTH7X","created_at":"2026-05-18T12:26:00Z"}],"graph_snapshots":[{"event_id":"sha256:b76a73248901b3375329be1142fff46978b086760fda286329f120489b710fa9","target":"graph","created_at":"2026-05-18T04:26:01Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"graph_snapshot":{"author_claims":{"count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","strong_count":0},"builder_version":"pith-number-builder-2026-05-17-v1","claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"paper":{"abstract_excerpt":"In this paper, we prove the following identity $$ \\lcm({n\\brack 0}_q,{n\\brack 1}_q,...,{n\\brack n}_q) =\\frac{\\lcm([1]_q,[2]_q,...,[n+1]_q)}{[n+1]_q}, $$ where ${n\\brack k}_q$ denotes the $q$-binomial coefficient and $[n]_q=\\frac{1-q^n}{1-q}$. This result is a $q$-analogue of an identity of Farhi [Amer. Math. Monthly, November (2009)].","authors_text":"Victor J. W. Guo","cross_cats":["math.CO"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2009-07-16T15:34:57Z","title":"On the least common multiple of $q$-binomial coefficients"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"0907.2870","kind":"arxiv","version":1},"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:cda8c0ef7b1c31ca5774861adb902b9e8977ae939d15699943774a7f84826a9f","target":"record","created_at":"2026-05-18T04:26:01Z","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":"b8190313d12ce3ac5d341ef3f47a1f2378351c91509ca45d5af845efe4b6dc3a","cross_cats_sorted":["math.CO"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2009-07-16T15:34:57Z","title_canon_sha256":"980761b8d4eaaf5dfc938700118d31d85582eb660ea31a589bf3a73df5045067"},"schema_version":"1.0","source":{"id":"0907.2870","kind":"arxiv","version":1}},"canonical_sha256":"4ead799ff79bb3ed4a34acaf079162684ec1a147ce56d25793a7cabc21596a15","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"4ead799ff79bb3ed4a34acaf079162684ec1a147ce56d25793a7cabc21596a15","first_computed_at":"2026-05-18T04:26:01.570418Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T04:26:01.570418Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"7Lz9M43kIxv5egXIRBP/qV5ecF2MIReW1JN/8abgOoWYv17/RhalX6li28zQ3KcNfEfg8Aa/ZmSeSnrRRG3IAA==","signature_status":"signed_v1","signed_at":"2026-05-18T04:26:01.570876Z","signed_message":"canonical_sha256_bytes"},"source_id":"0907.2870","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:cda8c0ef7b1c31ca5774861adb902b9e8977ae939d15699943774a7f84826a9f","sha256:b76a73248901b3375329be1142fff46978b086760fda286329f120489b710fa9"],"state_sha256":"8d07f75b79f8a7c047e9ab1969a95ac837c49cb93955d074f71457c75cdd16a7"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"yIFO5lsaaGiFpykP9kAeHb+RfpIF160ffr0NYtRcnKMvcU1mTIb2iDXuoO6HAZNGWNSetXmH/UivK+Ks9CyoBg==","signed_message":"bundle_sha256_bytes","signed_at":"2026-05-26T09:59:24.728440Z","bundle_sha256":"6cef6f383bd444c40fa200b22fec07f266c332c2d7fcb3786889d0c1dc903b4b"}}