{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2013:QFQW3E23AKPAB3GOVIHJAGEPPL","short_pith_number":"pith:QFQW3E23","canonical_record":{"source":{"id":"1311.2527","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2013-11-11T18:49:10Z","cross_cats_sorted":[],"title_canon_sha256":"60c006e32b93b37ae9cdcb37dc62781cfcefed795214bc0864913eba708c632e","abstract_canon_sha256":"67393dbffe626f615d78b7fdcdfaa03166587624b7255e1d9dd5395ec65989ce"},"schema_version":"1.0"},"canonical_sha256":"81616d935b029e00ecceaa0e90188f7ad7ac626dcfdc09555bb3a4a9fb64cd6d","source":{"kind":"arxiv","id":"1311.2527","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1311.2527","created_at":"2026-05-18T00:46:32Z"},{"alias_kind":"arxiv_version","alias_value":"1311.2527v1","created_at":"2026-05-18T00:46:32Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1311.2527","created_at":"2026-05-18T00:46:32Z"},{"alias_kind":"pith_short_12","alias_value":"QFQW3E23AKPA","created_at":"2026-05-18T12:27:57Z"},{"alias_kind":"pith_short_16","alias_value":"QFQW3E23AKPAB3GO","created_at":"2026-05-18T12:27:57Z"},{"alias_kind":"pith_short_8","alias_value":"QFQW3E23","created_at":"2026-05-18T12:27:57Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2013:QFQW3E23AKPAB3GOVIHJAGEPPL","target":"record","payload":{"canonical_record":{"source":{"id":"1311.2527","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2013-11-11T18:49:10Z","cross_cats_sorted":[],"title_canon_sha256":"60c006e32b93b37ae9cdcb37dc62781cfcefed795214bc0864913eba708c632e","abstract_canon_sha256":"67393dbffe626f615d78b7fdcdfaa03166587624b7255e1d9dd5395ec65989ce"},"schema_version":"1.0"},"canonical_sha256":"81616d935b029e00ecceaa0e90188f7ad7ac626dcfdc09555bb3a4a9fb64cd6d","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:46:32.302926Z","signature_b64":"Vl0fkBgjaPGIxFsXHwYIQyNaZIN+DC5JJDPWnjo9os5c2h8hhT2OMDWPLy2l82c7Oz2vjaVE7JSuHyg31xUOAQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"81616d935b029e00ecceaa0e90188f7ad7ac626dcfdc09555bb3a4a9fb64cd6d","last_reissued_at":"2026-05-18T00:46:32.302256Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:46:32.302256Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1311.2527","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":"2026-05-18T00:46:32Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"s8pg90kjj1c5sywD+BsAzpwBKemY28qc3dsZ2FpWzYhDr3MrA6wpHhLxmnsslONKIDNT4T88/crDXbAS0DFaCg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-31T19:17:36.074154Z"},"content_sha256":"070155601d834f1796eddf4c942190d7cf591f8cf8422503b7ebfb3296bd258b","schema_version":"1.0","event_id":"sha256:070155601d834f1796eddf4c942190d7cf591f8cf8422503b7ebfb3296bd258b"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2013:QFQW3E23AKPAB3GOVIHJAGEPPL","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"On shifted primes with large prime factors and their products","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Amalia Pizarro-Madariaga, Florian Luca, Ricardo Menares","submitted_at":"2013-11-11T18:49:10Z","abstract_excerpt":"We estimate from below the lower density of the set of prime numbers p such that p-1 has a prime factor of size at least p^c, where c lies in between 1/4 and 1/2. We also establish upper and lower bounds on the counting function of the set of positive integers n up to x with exactly k prime factors, counted with or without multiplicity, such that the largest prime factor of gcd(p-1 : p | n) exceeds n^{1/2k}."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1311.2527","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":"2026-05-18T00:46:32Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"dp/g8mTm9dvBFFBXX4O0u3RoQBaO4lkzD0RpfiHolCRogRFF/9sIFeKlMEeEaP6eI8mY26bbBV4IIGh5oa5QBw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-31T19:17:36.074989Z"},"content_sha256":"cb1f179aee8c01c8b4e18a9dd1b9c882d8d5e5bf48700461d8db24680ff95595","schema_version":"1.0","event_id":"sha256:cb1f179aee8c01c8b4e18a9dd1b9c882d8d5e5bf48700461d8db24680ff95595"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/QFQW3E23AKPAB3GOVIHJAGEPPL/bundle.json","state_url":"https://pith.science/pith/QFQW3E23AKPAB3GOVIHJAGEPPL/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/QFQW3E23AKPAB3GOVIHJAGEPPL/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":"2026-05-31T19:17:36Z","links":{"resolver":"https://pith.science/pith/QFQW3E23AKPAB3GOVIHJAGEPPL","bundle":"https://pith.science/pith/QFQW3E23AKPAB3GOVIHJAGEPPL/bundle.json","state":"https://pith.science/pith/QFQW3E23AKPAB3GOVIHJAGEPPL/state.json","well_known_bundle":"https://pith.science/.well-known/pith/QFQW3E23AKPAB3GOVIHJAGEPPL/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2013:QFQW3E23AKPAB3GOVIHJAGEPPL","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":"67393dbffe626f615d78b7fdcdfaa03166587624b7255e1d9dd5395ec65989ce","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2013-11-11T18:49:10Z","title_canon_sha256":"60c006e32b93b37ae9cdcb37dc62781cfcefed795214bc0864913eba708c632e"},"schema_version":"1.0","source":{"id":"1311.2527","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1311.2527","created_at":"2026-05-18T00:46:32Z"},{"alias_kind":"arxiv_version","alias_value":"1311.2527v1","created_at":"2026-05-18T00:46:32Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1311.2527","created_at":"2026-05-18T00:46:32Z"},{"alias_kind":"pith_short_12","alias_value":"QFQW3E23AKPA","created_at":"2026-05-18T12:27:57Z"},{"alias_kind":"pith_short_16","alias_value":"QFQW3E23AKPAB3GO","created_at":"2026-05-18T12:27:57Z"},{"alias_kind":"pith_short_8","alias_value":"QFQW3E23","created_at":"2026-05-18T12:27:57Z"}],"graph_snapshots":[{"event_id":"sha256:cb1f179aee8c01c8b4e18a9dd1b9c882d8d5e5bf48700461d8db24680ff95595","target":"graph","created_at":"2026-05-18T00:46:32Z","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 estimate from below the lower density of the set of prime numbers p such that p-1 has a prime factor of size at least p^c, where c lies in between 1/4 and 1/2. We also establish upper and lower bounds on the counting function of the set of positive integers n up to x with exactly k prime factors, counted with or without multiplicity, such that the largest prime factor of gcd(p-1 : p | n) exceeds n^{1/2k}.","authors_text":"Amalia Pizarro-Madariaga, Florian Luca, Ricardo Menares","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2013-11-11T18:49:10Z","title":"On shifted primes with large prime factors and their products"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1311.2527","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":"sha256:070155601d834f1796eddf4c942190d7cf591f8cf8422503b7ebfb3296bd258b","target":"record","created_at":"2026-05-18T00:46:32Z","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":"67393dbffe626f615d78b7fdcdfaa03166587624b7255e1d9dd5395ec65989ce","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2013-11-11T18:49:10Z","title_canon_sha256":"60c006e32b93b37ae9cdcb37dc62781cfcefed795214bc0864913eba708c632e"},"schema_version":"1.0","source":{"id":"1311.2527","kind":"arxiv","version":1}},"canonical_sha256":"81616d935b029e00ecceaa0e90188f7ad7ac626dcfdc09555bb3a4a9fb64cd6d","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"81616d935b029e00ecceaa0e90188f7ad7ac626dcfdc09555bb3a4a9fb64cd6d","first_computed_at":"2026-05-18T00:46:32.302256Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T00:46:32.302256Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"Vl0fkBgjaPGIxFsXHwYIQyNaZIN+DC5JJDPWnjo9os5c2h8hhT2OMDWPLy2l82c7Oz2vjaVE7JSuHyg31xUOAQ==","signature_status":"signed_v1","signed_at":"2026-05-18T00:46:32.302926Z","signed_message":"canonical_sha256_bytes"},"source_id":"1311.2527","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:070155601d834f1796eddf4c942190d7cf591f8cf8422503b7ebfb3296bd258b","sha256:cb1f179aee8c01c8b4e18a9dd1b9c882d8d5e5bf48700461d8db24680ff95595"],"state_sha256":"95bcc4fe12c7ed7ebbad29d3d6bee721fe020e906e2159176e0b055741fb69d0"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"CqsNLBPKD14OfQwe3IMdjPMQMpICo5EXsg8tm+TYSNHmog4JEwp6ndE9PqkVPU9nzCjRw4dN/jMcnLjE1/lWCQ==","signed_message":"bundle_sha256_bytes","signed_at":"2026-05-31T19:17:36.079477Z","bundle_sha256":"89dcd278268fa417f039e3517eed58320d3a66a2fc5d145e29de0fa4094ec845"}}