{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2005:L5I4NULEJIQHA4ATT37K3UPWAX","short_pith_number":"pith:L5I4NULE","canonical_record":{"source":{"id":"math/0507374","kind":"arxiv","version":3},"metadata":{"license":"","primary_cat":"math.NT","submitted_at":"2005-07-18T22:09:34Z","cross_cats_sorted":[],"title_canon_sha256":"2fb3ed95e71d3144432983c5ee3fb841618d5f01609c0b2f1650ac86fb6ee249","abstract_canon_sha256":"022306ea86c2c61bac1a15ac47a7a199de936db9dcc1083a329fe9dae92469f0"},"schema_version":"1.0"},"canonical_sha256":"5f51c6d1644a207070139efeadd1f605ee365397f5c0d099140e59327c983f3d","source":{"kind":"arxiv","id":"math/0507374","version":3},"source_aliases":[{"alias_kind":"arxiv","alias_value":"math/0507374","created_at":"2026-07-04T14:59:20Z"},{"alias_kind":"arxiv_version","alias_value":"math/0507374v3","created_at":"2026-07-04T14:59:20Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.math/0507374","created_at":"2026-07-04T14:59:20Z"},{"alias_kind":"pith_short_12","alias_value":"L5I4NULEJIQH","created_at":"2026-07-04T14:59:20Z"},{"alias_kind":"pith_short_16","alias_value":"L5I4NULEJIQHA4AT","created_at":"2026-07-04T14:59:20Z"},{"alias_kind":"pith_short_8","alias_value":"L5I4NULE","created_at":"2026-07-04T14:59:20Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2005:L5I4NULEJIQHA4ATT37K3UPWAX","target":"record","payload":{"canonical_record":{"source":{"id":"math/0507374","kind":"arxiv","version":3},"metadata":{"license":"","primary_cat":"math.NT","submitted_at":"2005-07-18T22:09:34Z","cross_cats_sorted":[],"title_canon_sha256":"2fb3ed95e71d3144432983c5ee3fb841618d5f01609c0b2f1650ac86fb6ee249","abstract_canon_sha256":"022306ea86c2c61bac1a15ac47a7a199de936db9dcc1083a329fe9dae92469f0"},"schema_version":"1.0"},"canonical_sha256":"5f51c6d1644a207070139efeadd1f605ee365397f5c0d099140e59327c983f3d","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-04T14:59:20.337154Z","signature_b64":"xnY7doulibHriVelxhzmO+FiPZGy2WiuFBIOYOP04dI0YEx+6gqPG85iGLl9YLXf3nGAs+6K+MWfeo5odQs1BQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"5f51c6d1644a207070139efeadd1f605ee365397f5c0d099140e59327c983f3d","last_reissued_at":"2026-07-04T14:59:20.336770Z","signature_status":"signed_v1","first_computed_at":"2026-07-04T14:59:20.336770Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"math/0507374","source_version":3,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-07-04T14:59:20Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"bnQSDKvpOVOnTQl0CDKHIc/2oiln4jV7tIObJC/7M1aeE/b8cIAM94tSSyHbsL/2WcRtnZZJIeNFoDGAUwDSBA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-07-07T10:21:18.412701Z"},"content_sha256":"b85a4c364329cf9b290ff557435011d475ac64338283f9aded3c68807b06c6fa","schema_version":"1.0","event_id":"sha256:b85a4c364329cf9b290ff557435011d475ac64338283f9aded3c68807b06c6fa"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2005:L5I4NULEJIQHA4ATT37K3UPWAX","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Sieving by large integers and covering systems of congruences","license":"","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Carl Pomerance, Gang Yu, Kevin Ford, Michael Filaseta, Sergei Konyagin","submitted_at":"2005-07-18T22:09:34Z","abstract_excerpt":"An old question of Erdos asks if there exists, for each number N, a finite set S of integers greater than N and residue classes r(n) mod n for n in S whose union is all the integers. We prove that if $\\sum_{n\\in S} 1/n$ is bounded for such a covering of the integers, then the least member of S is also bounded, thus confirming a conjecture of Erdos and Selfridge. We also prove a conjecture of Erdos and Graham, that, for each fixed number K>1, the complement in the integers of any union of residue classes r(n) mod n, for distinct n in (N,KN], has density at least d_K for N sufficiently large. He"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"math/0507374","kind":"arxiv","version":3},"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/math/0507374/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":"2026-07-04T14:59:20Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"l2NkXomw5Gn0rRsG/+mxeYtgtcCyCgC9Ufp8BIAoWtZM6pdlCOOnZ5KY1NuG9Zws5GqXSigztE52efKkJHHfDg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-07-07T10:21:18.413097Z"},"content_sha256":"9736f935aa2d59494dbacf10b920bed2568eba493a75cd8c352d04bd8a187484","schema_version":"1.0","event_id":"sha256:9736f935aa2d59494dbacf10b920bed2568eba493a75cd8c352d04bd8a187484"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/L5I4NULEJIQHA4ATT37K3UPWAX/bundle.json","state_url":"https://pith.science/pith/L5I4NULEJIQHA4ATT37K3UPWAX/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/L5I4NULEJIQHA4ATT37K3UPWAX/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":"2026-07-07T10:21:18Z","links":{"resolver":"https://pith.science/pith/L5I4NULEJIQHA4ATT37K3UPWAX","bundle":"https://pith.science/pith/L5I4NULEJIQHA4ATT37K3UPWAX/bundle.json","state":"https://pith.science/pith/L5I4NULEJIQHA4ATT37K3UPWAX/state.json","well_known_bundle":"https://pith.science/.well-known/pith/L5I4NULEJIQHA4ATT37K3UPWAX/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2005:L5I4NULEJIQHA4ATT37K3UPWAX","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":"022306ea86c2c61bac1a15ac47a7a199de936db9dcc1083a329fe9dae92469f0","cross_cats_sorted":[],"license":"","primary_cat":"math.NT","submitted_at":"2005-07-18T22:09:34Z","title_canon_sha256":"2fb3ed95e71d3144432983c5ee3fb841618d5f01609c0b2f1650ac86fb6ee249"},"schema_version":"1.0","source":{"id":"math/0507374","kind":"arxiv","version":3}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"math/0507374","created_at":"2026-07-04T14:59:20Z"},{"alias_kind":"arxiv_version","alias_value":"math/0507374v3","created_at":"2026-07-04T14:59:20Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.math/0507374","created_at":"2026-07-04T14:59:20Z"},{"alias_kind":"pith_short_12","alias_value":"L5I4NULEJIQH","created_at":"2026-07-04T14:59:20Z"},{"alias_kind":"pith_short_16","alias_value":"L5I4NULEJIQHA4AT","created_at":"2026-07-04T14:59:20Z"},{"alias_kind":"pith_short_8","alias_value":"L5I4NULE","created_at":"2026-07-04T14:59:20Z"}],"graph_snapshots":[{"event_id":"sha256:9736f935aa2d59494dbacf10b920bed2568eba493a75cd8c352d04bd8a187484","target":"graph","created_at":"2026-07-04T14:59:20Z","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/math/0507374/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"An old question of Erdos asks if there exists, for each number N, a finite set S of integers greater than N and residue classes r(n) mod n for n in S whose union is all the integers. We prove that if $\\sum_{n\\in S} 1/n$ is bounded for such a covering of the integers, then the least member of S is also bounded, thus confirming a conjecture of Erdos and Selfridge. We also prove a conjecture of Erdos and Graham, that, for each fixed number K>1, the complement in the integers of any union of residue classes r(n) mod n, for distinct n in (N,KN], has density at least d_K for N sufficiently large. He","authors_text":"Carl Pomerance, Gang Yu, Kevin Ford, Michael Filaseta, Sergei Konyagin","cross_cats":[],"headline":"","license":"","primary_cat":"math.NT","submitted_at":"2005-07-18T22:09:34Z","title":"Sieving by large integers and covering systems of congruences"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"math/0507374","kind":"arxiv","version":3},"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:b85a4c364329cf9b290ff557435011d475ac64338283f9aded3c68807b06c6fa","target":"record","created_at":"2026-07-04T14:59:20Z","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":"022306ea86c2c61bac1a15ac47a7a199de936db9dcc1083a329fe9dae92469f0","cross_cats_sorted":[],"license":"","primary_cat":"math.NT","submitted_at":"2005-07-18T22:09:34Z","title_canon_sha256":"2fb3ed95e71d3144432983c5ee3fb841618d5f01609c0b2f1650ac86fb6ee249"},"schema_version":"1.0","source":{"id":"math/0507374","kind":"arxiv","version":3}},"canonical_sha256":"5f51c6d1644a207070139efeadd1f605ee365397f5c0d099140e59327c983f3d","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"5f51c6d1644a207070139efeadd1f605ee365397f5c0d099140e59327c983f3d","first_computed_at":"2026-07-04T14:59:20.336770Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-07-04T14:59:20.336770Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"xnY7doulibHriVelxhzmO+FiPZGy2WiuFBIOYOP04dI0YEx+6gqPG85iGLl9YLXf3nGAs+6K+MWfeo5odQs1BQ==","signature_status":"signed_v1","signed_at":"2026-07-04T14:59:20.337154Z","signed_message":"canonical_sha256_bytes"},"source_id":"math/0507374","source_kind":"arxiv","source_version":3}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:b85a4c364329cf9b290ff557435011d475ac64338283f9aded3c68807b06c6fa","sha256:9736f935aa2d59494dbacf10b920bed2568eba493a75cd8c352d04bd8a187484"],"state_sha256":"394e363744312cc3e103ed2a7ab010617d00b1ad741886ab0ea229cd010b41bb"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"rIa4dZjpimnzyJDIi7/lnUmpssw3tEwFHPMVo0USD2ELXZBNMMqHAkMLLk0iseNhQ6NHcMslGjGo2prrDvgSDg==","signed_message":"bundle_sha256_bytes","signed_at":"2026-07-07T10:21:18.415083Z","bundle_sha256":"87fdf3a8b287939b7ee7e4b42aaecf14fc3d0dd275a470b682c006641ceea605"}}