{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2004:OBSREJYQEZPPV63W5ELPRGRTEJ","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":"254c7e9f637cded531c8ad856f32e8da111f8d32f953a562479e5e81050553bf","cross_cats_sorted":[],"license":"","primary_cat":"math.NT","submitted_at":"2004-05-07T16:55:08Z","title_canon_sha256":"3cac8ec7c168ee02a9024c7f61ea5d279b09edb5d4b196ddbb9f84854dd5d6f4"},"schema_version":"1.0","source":{"id":"math/0405120","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"math/0405120","created_at":"2026-05-18T01:38:28Z"},{"alias_kind":"arxiv_version","alias_value":"math/0405120v2","created_at":"2026-05-18T01:38:28Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.math/0405120","created_at":"2026-05-18T01:38:28Z"},{"alias_kind":"pith_short_12","alias_value":"OBSREJYQEZPP","created_at":"2026-05-18T12:25:52Z"},{"alias_kind":"pith_short_16","alias_value":"OBSREJYQEZPPV63W","created_at":"2026-05-18T12:25:52Z"},{"alias_kind":"pith_short_8","alias_value":"OBSREJYQ","created_at":"2026-05-18T12:25:52Z"}],"graph_snapshots":[{"event_id":"sha256:5b2a8f4c836bfc8654bbc0e0ff864494d1c616fb12768de71245dba98b678d25","target":"graph","created_at":"2026-05-18T01:38:28Z","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 consider the periods of the linear congruential and the power generators modulo $n$ and, for fixed choices of initial parameters, give lower bounds that hold for ``most'' $n$ when $n$ ranges over three different sets: the set of primes, the set of products of two primes (of similar size), and the set of all integers. For most $n$ in these sets, the period is at least $n^{1/2+\\epsilon(n)}$ for any monotone function $\\epsilon(n)$ tending to zero as $n$ tends to infinity. Assuming the Generalized Riemann Hypothesis, for most $n$ in these sets the period is greater than $n^{1-\\epsilon}$ for any","authors_text":"C. Pomerance, P. Kurlberg","cross_cats":[],"headline":"","license":"","primary_cat":"math.NT","submitted_at":"2004-05-07T16:55:08Z","title":"On the period of the linear congruential and power generators"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"math/0405120","kind":"arxiv","version":2},"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":"sha256:48ce69b097538849a652e46f483786a8260a8319972593883b4c1895786703ef","target":"record","created_at":"2026-05-18T01:38:28Z","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":"254c7e9f637cded531c8ad856f32e8da111f8d32f953a562479e5e81050553bf","cross_cats_sorted":[],"license":"","primary_cat":"math.NT","submitted_at":"2004-05-07T16:55:08Z","title_canon_sha256":"3cac8ec7c168ee02a9024c7f61ea5d279b09edb5d4b196ddbb9f84854dd5d6f4"},"schema_version":"1.0","source":{"id":"math/0405120","kind":"arxiv","version":2}},"canonical_sha256":"7065122710265efafb76e916f89a3322650826bec707069980d9cb375bde66bf","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"7065122710265efafb76e916f89a3322650826bec707069980d9cb375bde66bf","first_computed_at":"2026-05-18T01:38:28.366675Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T01:38:28.366675Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"mqnJAzUhhmSH36VtbXNwGAgbmLPJOuKBOk9qqckzOhMvAD/sljyBrFYuoCQBKa5bWoZVf00UVgL9fjmuBDAGCg==","signature_status":"signed_v1","signed_at":"2026-05-18T01:38:28.367292Z","signed_message":"canonical_sha256_bytes"},"source_id":"math/0405120","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:48ce69b097538849a652e46f483786a8260a8319972593883b4c1895786703ef","sha256:5b2a8f4c836bfc8654bbc0e0ff864494d1c616fb12768de71245dba98b678d25"],"state_sha256":"f729921f320679ceba04ed088c4d6b9d34f083f891b9932bf481712044cf032f"}