{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2009:3YRYAP6EWDJNA2H4OZL5MDLOSV","short_pith_number":"pith:3YRYAP6E","schema_version":"1.0","canonical_sha256":"de23803fc4b0d2d068fc7657d60d6e955a3bcfb7f6543245b2e3c04d61f6e825","source":{"kind":"arxiv","id":"0911.2660","version":1},"attestation_state":"computed","paper":{"title":"Maximum GCD Among Pairs of Random Integers","license":"http://creativecommons.org/licenses/publicdomain/","headline":"","cross_cats":["math.PR"],"primary_cat":"math.NT","authors_text":"E. E. Pyle, R. W. R. Darling","submitted_at":"2009-11-13T20:58:51Z","abstract_excerpt":"Fix $\\alpha >0$, and sample $N$ integers uniformly at random from $\\{1,2,\\ldots ,\\lfloor e^{\\alpha N}\\rfloor \\}$. Given $\\eta >0$, the probability that the maximum of the pairwise GCDs lies between $N^{2-\\eta }$ and $N^{2+\\eta}$ converges to 1 as $N\\to \\infty $. More precise estimates are obtained. This is a Birthday Problem: two of the random integers are likely to share some prime factor of order $N^2/\\log [N]$. The proof generalizes to any arithmetical semigroup where a suitable form of the Prime Number Theorem is valid."},"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":"0911.2660","kind":"arxiv","version":1},"metadata":{"license":"http://creativecommons.org/licenses/publicdomain/","primary_cat":"math.NT","submitted_at":"2009-11-13T20:58:51Z","cross_cats_sorted":["math.PR"],"title_canon_sha256":"af13ef25b9f167f5ab94e014b3be91d92fa7bea8971d1afd279a6ec38ba9b12c","abstract_canon_sha256":"0ab7d4694c58debf00315002745df648e4dc79910ecd83a159467bb96500ee5e"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T04:28:28.340551Z","signature_b64":"QLuhUg/kfJrqEvV0eLeLGORDGZbDzSHKEOk8XMEC8KymHMICwwX8OXo82FcZyGRN1NnJ3TBX75nZW79jF9ZyDQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"de23803fc4b0d2d068fc7657d60d6e955a3bcfb7f6543245b2e3c04d61f6e825","last_reissued_at":"2026-05-18T04:28:28.340149Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T04:28:28.340149Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Maximum GCD Among Pairs of Random Integers","license":"http://creativecommons.org/licenses/publicdomain/","headline":"","cross_cats":["math.PR"],"primary_cat":"math.NT","authors_text":"E. E. Pyle, R. W. R. Darling","submitted_at":"2009-11-13T20:58:51Z","abstract_excerpt":"Fix $\\alpha >0$, and sample $N$ integers uniformly at random from $\\{1,2,\\ldots ,\\lfloor e^{\\alpha N}\\rfloor \\}$. Given $\\eta >0$, the probability that the maximum of the pairwise GCDs lies between $N^{2-\\eta }$ and $N^{2+\\eta}$ converges to 1 as $N\\to \\infty $. More precise estimates are obtained. This is a Birthday Problem: two of the random integers are likely to share some prime factor of order $N^2/\\log [N]$. The proof generalizes to any arithmetical semigroup where a suitable form of the Prime Number Theorem is valid."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"0911.2660","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":"0911.2660","created_at":"2026-05-18T04:28:28.340199+00:00"},{"alias_kind":"arxiv_version","alias_value":"0911.2660v1","created_at":"2026-05-18T04:28:28.340199+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.0911.2660","created_at":"2026-05-18T04:28:28.340199+00:00"},{"alias_kind":"pith_short_12","alias_value":"3YRYAP6EWDJN","created_at":"2026-05-18T12:25:58.018023+00:00"},{"alias_kind":"pith_short_16","alias_value":"3YRYAP6EWDJNA2H4","created_at":"2026-05-18T12:25:58.018023+00:00"},{"alias_kind":"pith_short_8","alias_value":"3YRYAP6E","created_at":"2026-05-18T12:25:58.018023+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/3YRYAP6EWDJNA2H4OZL5MDLOSV","json":"https://pith.science/pith/3YRYAP6EWDJNA2H4OZL5MDLOSV.json","graph_json":"https://pith.science/api/pith-number/3YRYAP6EWDJNA2H4OZL5MDLOSV/graph.json","events_json":"https://pith.science/api/pith-number/3YRYAP6EWDJNA2H4OZL5MDLOSV/events.json","paper":"https://pith.science/paper/3YRYAP6E"},"agent_actions":{"view_html":"https://pith.science/pith/3YRYAP6EWDJNA2H4OZL5MDLOSV","download_json":"https://pith.science/pith/3YRYAP6EWDJNA2H4OZL5MDLOSV.json","view_paper":"https://pith.science/paper/3YRYAP6E","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=0911.2660&json=true","fetch_graph":"https://pith.science/api/pith-number/3YRYAP6EWDJNA2H4OZL5MDLOSV/graph.json","fetch_events":"https://pith.science/api/pith-number/3YRYAP6EWDJNA2H4OZL5MDLOSV/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/3YRYAP6EWDJNA2H4OZL5MDLOSV/action/timestamp_anchor","attest_storage":"https://pith.science/pith/3YRYAP6EWDJNA2H4OZL5MDLOSV/action/storage_attestation","attest_author":"https://pith.science/pith/3YRYAP6EWDJNA2H4OZL5MDLOSV/action/author_attestation","sign_citation":"https://pith.science/pith/3YRYAP6EWDJNA2H4OZL5MDLOSV/action/citation_signature","submit_replication":"https://pith.science/pith/3YRYAP6EWDJNA2H4OZL5MDLOSV/action/replication_record"}},"created_at":"2026-05-18T04:28:28.340199+00:00","updated_at":"2026-05-18T04:28:28.340199+00:00"}