{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2019:7BYS6O4NOSJ5ZUZKLAI2KWK36C","short_pith_number":"pith:7BYS6O4N","canonical_record":{"source":{"id":"1908.09503","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2019-08-26T07:30:55Z","cross_cats_sorted":[],"title_canon_sha256":"d008ab069b1f1923a3107bad6aac3ad07fdf5ecda23dd54394923cbf8f211a7a","abstract_canon_sha256":"35433844a6f21833ebf176cc3dc8dfa40a0d983d6f14b7c54e51db8d5fb0f3eb"},"schema_version":"1.0"},"canonical_sha256":"f8712f3b8d7493dcd32a5811a5595bf0a59df35299b042fab9c1d738a9d4f1b8","source":{"kind":"arxiv","id":"1908.09503","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1908.09503","created_at":"2026-07-04T23:59:40Z"},{"alias_kind":"arxiv_version","alias_value":"1908.09503v1","created_at":"2026-07-04T23:59:40Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1908.09503","created_at":"2026-07-04T23:59:40Z"},{"alias_kind":"pith_short_12","alias_value":"7BYS6O4NOSJ5","created_at":"2026-07-04T23:59:40Z"},{"alias_kind":"pith_short_16","alias_value":"7BYS6O4NOSJ5ZUZK","created_at":"2026-07-04T23:59:40Z"},{"alias_kind":"pith_short_8","alias_value":"7BYS6O4N","created_at":"2026-07-04T23:59:40Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2019:7BYS6O4NOSJ5ZUZKLAI2KWK36C","target":"record","payload":{"canonical_record":{"source":{"id":"1908.09503","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2019-08-26T07:30:55Z","cross_cats_sorted":[],"title_canon_sha256":"d008ab069b1f1923a3107bad6aac3ad07fdf5ecda23dd54394923cbf8f211a7a","abstract_canon_sha256":"35433844a6f21833ebf176cc3dc8dfa40a0d983d6f14b7c54e51db8d5fb0f3eb"},"schema_version":"1.0"},"canonical_sha256":"f8712f3b8d7493dcd32a5811a5595bf0a59df35299b042fab9c1d738a9d4f1b8","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-04T23:59:40.834719Z","signature_b64":"/Jx2cFCKiXoA3tUlcHRm46DZ7fG/vLL38p07obZ+/a5QG04xwBfEsL2y28wJZqXk1AKeLD+ndQfrc9sUmyTkAA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"f8712f3b8d7493dcd32a5811a5595bf0a59df35299b042fab9c1d738a9d4f1b8","last_reissued_at":"2026-07-04T23:59:40.834262Z","signature_status":"signed_v1","first_computed_at":"2026-07-04T23:59:40.834262Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1908.09503","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":"2026-07-04T23:59:40Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"Nf4D1fhIEGYqmNBIsn81xr9mKgI638kL2VJ9GF1GvjIZnJb171bw0UKwob8Nj2TzucgD1t+QqJTQR2z2n5ttDQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-16T20:04:53.000073Z"},"content_sha256":"d1fd496b905e473db2ec8d2a83c8983150f65d2ef944e867ffe6e7d8043f6a0b","schema_version":"1.0","event_id":"sha256:d1fd496b905e473db2ec8d2a83c8983150f65d2ef944e867ffe6e7d8043f6a0b"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2019:7BYS6O4NOSJ5ZUZKLAI2KWK36C","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"On the distribution of products of two primes","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Sumaia Saad Eddin, Yuta Suzuki","submitted_at":"2019-08-26T07:30:55Z","abstract_excerpt":"For a real parameter $r$, the RSA integers are integers which can be written as the product of two primes $pq$ with $p<q\\leq rp$, which are named after the importance of products of two primes in the RSA-cryptography. Several authors obtained the asymptotic formulas of the number of the RSA integers. However, the previous results on the number of the RSA integers were valid only in a rather restricted range of the parameter $r$. Dummit, Granville, and Kisilevsky found some bias in the distribution of products of two primes with congruence conditions. Moree and the first author studied some sim"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1908.09503","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":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/1908.09503/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"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":"2026-07-04T23:59:40Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"9IwtgY/XXN5duCoMG9f9RuuBxOKmodyXqyNkWCZIgyTALDU5TIdAZckDPGurZ54u0pZLfi7zqNvkH5SMpP/0BQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-16T20:04:53.000997Z"},"content_sha256":"9efa429a8942f7dda5f9e76bd0ff920a3a2f49c6be3a49f847dd0fc94e4313b9","schema_version":"1.0","event_id":"sha256:9efa429a8942f7dda5f9e76bd0ff920a3a2f49c6be3a49f847dd0fc94e4313b9"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/7BYS6O4NOSJ5ZUZKLAI2KWK36C/bundle.json","state_url":"https://pith.science/pith/7BYS6O4NOSJ5ZUZKLAI2KWK36C/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/7BYS6O4NOSJ5ZUZKLAI2KWK36C/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":"2026-08-16T20:04:53Z","links":{"resolver":"https://pith.science/pith/7BYS6O4NOSJ5ZUZKLAI2KWK36C","bundle":"https://pith.science/pith/7BYS6O4NOSJ5ZUZKLAI2KWK36C/bundle.json","state":"https://pith.science/pith/7BYS6O4NOSJ5ZUZKLAI2KWK36C/state.json","well_known_bundle":"https://pith.science/.well-known/pith/7BYS6O4NOSJ5ZUZKLAI2KWK36C/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2019:7BYS6O4NOSJ5ZUZKLAI2KWK36C","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":"35433844a6f21833ebf176cc3dc8dfa40a0d983d6f14b7c54e51db8d5fb0f3eb","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2019-08-26T07:30:55Z","title_canon_sha256":"d008ab069b1f1923a3107bad6aac3ad07fdf5ecda23dd54394923cbf8f211a7a"},"schema_version":"1.0","source":{"id":"1908.09503","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1908.09503","created_at":"2026-07-04T23:59:40Z"},{"alias_kind":"arxiv_version","alias_value":"1908.09503v1","created_at":"2026-07-04T23:59:40Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1908.09503","created_at":"2026-07-04T23:59:40Z"},{"alias_kind":"pith_short_12","alias_value":"7BYS6O4NOSJ5","created_at":"2026-07-04T23:59:40Z"},{"alias_kind":"pith_short_16","alias_value":"7BYS6O4NOSJ5ZUZK","created_at":"2026-07-04T23:59:40Z"},{"alias_kind":"pith_short_8","alias_value":"7BYS6O4N","created_at":"2026-07-04T23:59:40Z"}],"graph_snapshots":[{"event_id":"sha256:9efa429a8942f7dda5f9e76bd0ff920a3a2f49c6be3a49f847dd0fc94e4313b9","target":"graph","created_at":"2026-07-04T23:59:40Z","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"},"integrity":{"available":true,"clean":true,"detectors_run":[],"endpoint":"/pith/1908.09503/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"For a real parameter $r$, the RSA integers are integers which can be written as the product of two primes $pq$ with $p<q\\leq rp$, which are named after the importance of products of two primes in the RSA-cryptography. Several authors obtained the asymptotic formulas of the number of the RSA integers. However, the previous results on the number of the RSA integers were valid only in a rather restricted range of the parameter $r$. Dummit, Granville, and Kisilevsky found some bias in the distribution of products of two primes with congruence conditions. Moree and the first author studied some sim","authors_text":"Sumaia Saad Eddin, Yuta Suzuki","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2019-08-26T07:30:55Z","title":"On the distribution of products of two primes"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1908.09503","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:d1fd496b905e473db2ec8d2a83c8983150f65d2ef944e867ffe6e7d8043f6a0b","target":"record","created_at":"2026-07-04T23:59:40Z","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":"35433844a6f21833ebf176cc3dc8dfa40a0d983d6f14b7c54e51db8d5fb0f3eb","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2019-08-26T07:30:55Z","title_canon_sha256":"d008ab069b1f1923a3107bad6aac3ad07fdf5ecda23dd54394923cbf8f211a7a"},"schema_version":"1.0","source":{"id":"1908.09503","kind":"arxiv","version":1}},"canonical_sha256":"f8712f3b8d7493dcd32a5811a5595bf0a59df35299b042fab9c1d738a9d4f1b8","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"f8712f3b8d7493dcd32a5811a5595bf0a59df35299b042fab9c1d738a9d4f1b8","first_computed_at":"2026-07-04T23:59:40.834262Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-07-04T23:59:40.834262Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"/Jx2cFCKiXoA3tUlcHRm46DZ7fG/vLL38p07obZ+/a5QG04xwBfEsL2y28wJZqXk1AKeLD+ndQfrc9sUmyTkAA==","signature_status":"signed_v1","signed_at":"2026-07-04T23:59:40.834719Z","signed_message":"canonical_sha256_bytes"},"source_id":"1908.09503","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:d1fd496b905e473db2ec8d2a83c8983150f65d2ef944e867ffe6e7d8043f6a0b","sha256:9efa429a8942f7dda5f9e76bd0ff920a3a2f49c6be3a49f847dd0fc94e4313b9"],"state_sha256":"2b50cbc7120c1d049b073cec9e26debe6c8d5c9912d8c459073a31421b04fd27"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"rM9HgAkZ6i+M1KaFFoc/IifiIuN3is/x/xnxKUWrdZkhFOa4Shliopyo7wdLlIEkBxfBaR6Gt6F8oxfXjHfjDg==","signed_message":"bundle_sha256_bytes","signed_at":"2026-08-16T20:04:53.006457Z","bundle_sha256":"149996b203461b290336ee92d896fed9251222b706fe95d00b3de7272e33d295"}}