{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2015:QKOOI2YQPJETFXADNVLVWOREB6","short_pith_number":"pith:QKOOI2YQ","canonical_record":{"source":{"id":"1505.01198","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2015-05-05T21:59:20Z","cross_cats_sorted":[],"title_canon_sha256":"5ffabef90ad7f38a4d2443ec10ee38a07735dd8a35901e1fe3083e8b81d9cbb4","abstract_canon_sha256":"44411310fc84c2513c9e2955c8852df6f39c088738f6fc126d903aa8c01da604"},"schema_version":"1.0"},"canonical_sha256":"829ce46b107a4932dc036d575b3a240fa15c29be60db252cca0b28c685ea74bf","source":{"kind":"arxiv","id":"1505.01198","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1505.01198","created_at":"2026-05-18T02:16:50Z"},{"alias_kind":"arxiv_version","alias_value":"1505.01198v1","created_at":"2026-05-18T02:16:50Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1505.01198","created_at":"2026-05-18T02:16:50Z"},{"alias_kind":"pith_short_12","alias_value":"QKOOI2YQPJET","created_at":"2026-05-18T12:29:37Z"},{"alias_kind":"pith_short_16","alias_value":"QKOOI2YQPJETFXAD","created_at":"2026-05-18T12:29:37Z"},{"alias_kind":"pith_short_8","alias_value":"QKOOI2YQ","created_at":"2026-05-18T12:29:37Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2015:QKOOI2YQPJETFXADNVLVWOREB6","target":"record","payload":{"canonical_record":{"source":{"id":"1505.01198","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2015-05-05T21:59:20Z","cross_cats_sorted":[],"title_canon_sha256":"5ffabef90ad7f38a4d2443ec10ee38a07735dd8a35901e1fe3083e8b81d9cbb4","abstract_canon_sha256":"44411310fc84c2513c9e2955c8852df6f39c088738f6fc126d903aa8c01da604"},"schema_version":"1.0"},"canonical_sha256":"829ce46b107a4932dc036d575b3a240fa15c29be60db252cca0b28c685ea74bf","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T02:16:50.281620Z","signature_b64":"uiy47m6AGWUL3SNh73cEcRwKTBmRU4zvoh5ZhRS8ME++RcQSU7e16r3UaCI2ARenQMnMHwF/XtyydzLCUOw+Aw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"829ce46b107a4932dc036d575b3a240fa15c29be60db252cca0b28c685ea74bf","last_reissued_at":"2026-05-18T02:16:50.280807Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T02:16:50.280807Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1505.01198","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":"2026-05-18T02:16:50Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"mq7hi+40DW0PvOTdI5lX7et4zMed3T+WtC5xGDFBdRhZMpyOBv/XHiv3ewKdJ91O+k0wTLrdHY9+WD8JtcWiAQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-31T23:05:23.069282Z"},"content_sha256":"7f72545a81a72ee6ed847d58ed78095efa6ff64a9eabbc125c6cc60d0869d280","schema_version":"1.0","event_id":"sha256:7f72545a81a72ee6ed847d58ed78095efa6ff64a9eabbc125c6cc60d0869d280"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2015:QKOOI2YQPJETFXADNVLVWOREB6","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Distribution of factorials modulo p","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Marc Munsch, Oleksiy Klurman","submitted_at":"2015-05-05T21:59:20Z","abstract_excerpt":"We prove that the sequence $n!\\,(\\bmod\\,p)$ occupies at least $\\sqrt{\\frac{3}{2}N}$ residue classes in the short interval $H\\le n \\le H+N$ and $N\\gg p^{\\frac{1}{4}}$ improving previously known trivial bound $\\sqrt{N}.$ In the other direction, we estimate the average number of residue classes missed by the sequence $n!\\,(\\bmod\\,p)$ for $p\\le x.$"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1505.01198","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":"2026-05-18T02:16:50Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"iWFuy5xo1Aqwd9NVRyVGS3as8aVSZZZeIpwaTUCCKGhP6kdp1gXRHda3Npb5OnbQQX9D1sYF5FSWdQUAx5CZDQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-31T23:05:23.069922Z"},"content_sha256":"8e1ccf66aa6ec129c7f964a56262f2a07a111b4449631ceb61777bfde2e03e14","schema_version":"1.0","event_id":"sha256:8e1ccf66aa6ec129c7f964a56262f2a07a111b4449631ceb61777bfde2e03e14"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/QKOOI2YQPJETFXADNVLVWOREB6/bundle.json","state_url":"https://pith.science/pith/QKOOI2YQPJETFXADNVLVWOREB6/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/QKOOI2YQPJETFXADNVLVWOREB6/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":"2026-05-31T23:05:23Z","links":{"resolver":"https://pith.science/pith/QKOOI2YQPJETFXADNVLVWOREB6","bundle":"https://pith.science/pith/QKOOI2YQPJETFXADNVLVWOREB6/bundle.json","state":"https://pith.science/pith/QKOOI2YQPJETFXADNVLVWOREB6/state.json","well_known_bundle":"https://pith.science/.well-known/pith/QKOOI2YQPJETFXADNVLVWOREB6/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2015:QKOOI2YQPJETFXADNVLVWOREB6","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":"44411310fc84c2513c9e2955c8852df6f39c088738f6fc126d903aa8c01da604","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2015-05-05T21:59:20Z","title_canon_sha256":"5ffabef90ad7f38a4d2443ec10ee38a07735dd8a35901e1fe3083e8b81d9cbb4"},"schema_version":"1.0","source":{"id":"1505.01198","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1505.01198","created_at":"2026-05-18T02:16:50Z"},{"alias_kind":"arxiv_version","alias_value":"1505.01198v1","created_at":"2026-05-18T02:16:50Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1505.01198","created_at":"2026-05-18T02:16:50Z"},{"alias_kind":"pith_short_12","alias_value":"QKOOI2YQPJET","created_at":"2026-05-18T12:29:37Z"},{"alias_kind":"pith_short_16","alias_value":"QKOOI2YQPJETFXAD","created_at":"2026-05-18T12:29:37Z"},{"alias_kind":"pith_short_8","alias_value":"QKOOI2YQ","created_at":"2026-05-18T12:29:37Z"}],"graph_snapshots":[{"event_id":"sha256:8e1ccf66aa6ec129c7f964a56262f2a07a111b4449631ceb61777bfde2e03e14","target":"graph","created_at":"2026-05-18T02:16:50Z","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":"We prove that the sequence $n!\\,(\\bmod\\,p)$ occupies at least $\\sqrt{\\frac{3}{2}N}$ residue classes in the short interval $H\\le n \\le H+N$ and $N\\gg p^{\\frac{1}{4}}$ improving previously known trivial bound $\\sqrt{N}.$ In the other direction, we estimate the average number of residue classes missed by the sequence $n!\\,(\\bmod\\,p)$ for $p\\le x.$","authors_text":"Marc Munsch, Oleksiy Klurman","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2015-05-05T21:59:20Z","title":"Distribution of factorials modulo p"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1505.01198","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":"sha256:7f72545a81a72ee6ed847d58ed78095efa6ff64a9eabbc125c6cc60d0869d280","target":"record","created_at":"2026-05-18T02:16:50Z","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":"44411310fc84c2513c9e2955c8852df6f39c088738f6fc126d903aa8c01da604","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2015-05-05T21:59:20Z","title_canon_sha256":"5ffabef90ad7f38a4d2443ec10ee38a07735dd8a35901e1fe3083e8b81d9cbb4"},"schema_version":"1.0","source":{"id":"1505.01198","kind":"arxiv","version":1}},"canonical_sha256":"829ce46b107a4932dc036d575b3a240fa15c29be60db252cca0b28c685ea74bf","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"829ce46b107a4932dc036d575b3a240fa15c29be60db252cca0b28c685ea74bf","first_computed_at":"2026-05-18T02:16:50.280807Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T02:16:50.280807Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"uiy47m6AGWUL3SNh73cEcRwKTBmRU4zvoh5ZhRS8ME++RcQSU7e16r3UaCI2ARenQMnMHwF/XtyydzLCUOw+Aw==","signature_status":"signed_v1","signed_at":"2026-05-18T02:16:50.281620Z","signed_message":"canonical_sha256_bytes"},"source_id":"1505.01198","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:7f72545a81a72ee6ed847d58ed78095efa6ff64a9eabbc125c6cc60d0869d280","sha256:8e1ccf66aa6ec129c7f964a56262f2a07a111b4449631ceb61777bfde2e03e14"],"state_sha256":"8dfc7d02ecad88aaa0b8e17acb2c80e14a1430a7656fc94bf689a01ba8d3bcaf"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"fDFm4IqQD7EG8ftsXQ8Jrk12Fe4McddmZ437rJZ197ETd6V5BEYgGPwoGdFWUj1HOW2Q15neLcKcumRT51iTDg==","signed_message":"bundle_sha256_bytes","signed_at":"2026-05-31T23:05:23.072914Z","bundle_sha256":"d7b6470d410c735a51dbf8b7f59f56d79d57696f4fb4979a3ad5bd32ea577463"}}