{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2016:KVKXZOXEFS6XHSJDQ7GRHI5WUU","short_pith_number":"pith:KVKXZOXE","canonical_record":{"source":{"id":"1604.02305","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2016-04-08T11:11:32Z","cross_cats_sorted":[],"title_canon_sha256":"5002ac8cefcf06d45e167667822f88c90ab05a5ecf8cbf1914d46149d2717069","abstract_canon_sha256":"66f7b5d1c31344a1b2e0e615f538e4a731756d88d71c628eb4097820eb3eeeeb"},"schema_version":"1.0"},"canonical_sha256":"55557cbae42cbd73c92387cd13a3b6a52e1103d8e3a3977da2bf76f3aaf3b870","source":{"kind":"arxiv","id":"1604.02305","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1604.02305","created_at":"2026-05-18T01:17:26Z"},{"alias_kind":"arxiv_version","alias_value":"1604.02305v1","created_at":"2026-05-18T01:17:26Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1604.02305","created_at":"2026-05-18T01:17:26Z"},{"alias_kind":"pith_short_12","alias_value":"KVKXZOXEFS6X","created_at":"2026-05-18T12:30:29Z"},{"alias_kind":"pith_short_16","alias_value":"KVKXZOXEFS6XHSJD","created_at":"2026-05-18T12:30:29Z"},{"alias_kind":"pith_short_8","alias_value":"KVKXZOXE","created_at":"2026-05-18T12:30:29Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2016:KVKXZOXEFS6XHSJDQ7GRHI5WUU","target":"record","payload":{"canonical_record":{"source":{"id":"1604.02305","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2016-04-08T11:11:32Z","cross_cats_sorted":[],"title_canon_sha256":"5002ac8cefcf06d45e167667822f88c90ab05a5ecf8cbf1914d46149d2717069","abstract_canon_sha256":"66f7b5d1c31344a1b2e0e615f538e4a731756d88d71c628eb4097820eb3eeeeb"},"schema_version":"1.0"},"canonical_sha256":"55557cbae42cbd73c92387cd13a3b6a52e1103d8e3a3977da2bf76f3aaf3b870","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:17:26.902945Z","signature_b64":"sQD4Kf8Z01xB9DMU9IjSzJIoqOP1L1MXHtwxwGcGzVNocTNaERnUO8MR416BPJ6Cu6yGEfcg4Csg3AK5TsCmAw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"55557cbae42cbd73c92387cd13a3b6a52e1103d8e3a3977da2bf76f3aaf3b870","last_reissued_at":"2026-05-18T01:17:26.902453Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:17:26.902453Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1604.02305","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":"2026-05-18T01:17:26Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"3TEcGOGR0vcvLTUzdnPGLLwAbcmv4K/QsY2FebnClDjiDgmy0KTnw4pFZHlglXuRjh8IRcx/OYtNfAKg0IsZCg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-08T15:40:38.257970Z"},"content_sha256":"a180df89dcb5ed7d347a15398ba67c1d5f480aeb34ae5408acee3f3efbe8c23c","schema_version":"1.0","event_id":"sha256:a180df89dcb5ed7d347a15398ba67c1d5f480aeb34ae5408acee3f3efbe8c23c"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2016:KVKXZOXEFS6XHSJDQ7GRHI5WUU","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Towards characterising polynomiality of $\\frac{1-q^b}{1-q^a}{n\\brack m}$ and applications","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Mohamed El Bachraoui","submitted_at":"2016-04-08T11:11:32Z","abstract_excerpt":"In this note we shall give conditions which guarantee that $\\frac{1-q^b}{1-q^a}{n\\brack m}\\in\\mathbb{Z}[q]$ holds. We shall provide a full characterisation for $\\frac{1-q^b}{1-q^a}{ka\\brack m}\\in\\mathbb{Z}[q]$. This unifies a variety of results already known in literature. We shall prove new divisibility properties for the binomial coefficients and a new divisibility result for a certain finite sum involving the roots of the unity."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1604.02305","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":"2026-05-18T01:17:26Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"PybG3z82pTFf0yeosAnbvCCoUlI/6efCL5NPy93ESsiKf/wzR+ivqp2RfuXJ7NxHfVdM50bmZHrXcxfWRn9LAg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-08T15:40:38.258699Z"},"content_sha256":"14c094818c5ee1c3c4da0d1b4b7f60300dd9d790065c5fc5c0b63d31977c350f","schema_version":"1.0","event_id":"sha256:14c094818c5ee1c3c4da0d1b4b7f60300dd9d790065c5fc5c0b63d31977c350f"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/KVKXZOXEFS6XHSJDQ7GRHI5WUU/bundle.json","state_url":"https://pith.science/pith/KVKXZOXEFS6XHSJDQ7GRHI5WUU/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/KVKXZOXEFS6XHSJDQ7GRHI5WUU/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":"2026-06-08T15:40:38Z","links":{"resolver":"https://pith.science/pith/KVKXZOXEFS6XHSJDQ7GRHI5WUU","bundle":"https://pith.science/pith/KVKXZOXEFS6XHSJDQ7GRHI5WUU/bundle.json","state":"https://pith.science/pith/KVKXZOXEFS6XHSJDQ7GRHI5WUU/state.json","well_known_bundle":"https://pith.science/.well-known/pith/KVKXZOXEFS6XHSJDQ7GRHI5WUU/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2016:KVKXZOXEFS6XHSJDQ7GRHI5WUU","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":"66f7b5d1c31344a1b2e0e615f538e4a731756d88d71c628eb4097820eb3eeeeb","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2016-04-08T11:11:32Z","title_canon_sha256":"5002ac8cefcf06d45e167667822f88c90ab05a5ecf8cbf1914d46149d2717069"},"schema_version":"1.0","source":{"id":"1604.02305","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1604.02305","created_at":"2026-05-18T01:17:26Z"},{"alias_kind":"arxiv_version","alias_value":"1604.02305v1","created_at":"2026-05-18T01:17:26Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1604.02305","created_at":"2026-05-18T01:17:26Z"},{"alias_kind":"pith_short_12","alias_value":"KVKXZOXEFS6X","created_at":"2026-05-18T12:30:29Z"},{"alias_kind":"pith_short_16","alias_value":"KVKXZOXEFS6XHSJD","created_at":"2026-05-18T12:30:29Z"},{"alias_kind":"pith_short_8","alias_value":"KVKXZOXE","created_at":"2026-05-18T12:30:29Z"}],"graph_snapshots":[{"event_id":"sha256:14c094818c5ee1c3c4da0d1b4b7f60300dd9d790065c5fc5c0b63d31977c350f","target":"graph","created_at":"2026-05-18T01:17:26Z","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 note we shall give conditions which guarantee that $\\frac{1-q^b}{1-q^a}{n\\brack m}\\in\\mathbb{Z}[q]$ holds. We shall provide a full characterisation for $\\frac{1-q^b}{1-q^a}{ka\\brack m}\\in\\mathbb{Z}[q]$. This unifies a variety of results already known in literature. We shall prove new divisibility properties for the binomial coefficients and a new divisibility result for a certain finite sum involving the roots of the unity.","authors_text":"Mohamed El Bachraoui","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2016-04-08T11:11:32Z","title":"Towards characterising polynomiality of $\\frac{1-q^b}{1-q^a}{n\\brack m}$ and applications"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1604.02305","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:a180df89dcb5ed7d347a15398ba67c1d5f480aeb34ae5408acee3f3efbe8c23c","target":"record","created_at":"2026-05-18T01:17:26Z","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":"66f7b5d1c31344a1b2e0e615f538e4a731756d88d71c628eb4097820eb3eeeeb","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2016-04-08T11:11:32Z","title_canon_sha256":"5002ac8cefcf06d45e167667822f88c90ab05a5ecf8cbf1914d46149d2717069"},"schema_version":"1.0","source":{"id":"1604.02305","kind":"arxiv","version":1}},"canonical_sha256":"55557cbae42cbd73c92387cd13a3b6a52e1103d8e3a3977da2bf76f3aaf3b870","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"55557cbae42cbd73c92387cd13a3b6a52e1103d8e3a3977da2bf76f3aaf3b870","first_computed_at":"2026-05-18T01:17:26.902453Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T01:17:26.902453Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"sQD4Kf8Z01xB9DMU9IjSzJIoqOP1L1MXHtwxwGcGzVNocTNaERnUO8MR416BPJ6Cu6yGEfcg4Csg3AK5TsCmAw==","signature_status":"signed_v1","signed_at":"2026-05-18T01:17:26.902945Z","signed_message":"canonical_sha256_bytes"},"source_id":"1604.02305","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:a180df89dcb5ed7d347a15398ba67c1d5f480aeb34ae5408acee3f3efbe8c23c","sha256:14c094818c5ee1c3c4da0d1b4b7f60300dd9d790065c5fc5c0b63d31977c350f"],"state_sha256":"cee632c4b5ea978c41ed0fb9f5a7859b4df88bb1c28df3a8d1f8c466b0754067"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"gcDACHmu8+sU8FuFAkn00ryuMVYh9oQYKKPAcsHxtB7mNZ4eR/k5e6Z0GulpoRBVKTjTS+YnBCWEQAo676+VDg==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-08T15:40:38.262545Z","bundle_sha256":"06b073cb9d132b7806968f0d8916597ca5d1253867db1714fcc56a81f8e79ce2"}}