{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2012:Q67SQY7SB2C4RPB4RYY3SJNCRZ","short_pith_number":"pith:Q67SQY7S","canonical_record":{"source":{"id":"1201.1317","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2012-01-05T22:14:29Z","cross_cats_sorted":[],"title_canon_sha256":"7e50f8a7f44f17cfbd0f888dd5f993efa4d4c8813ab0a4c5108febe9e524d1e5","abstract_canon_sha256":"2eb1454d5eb00ce6204c5587962292cb20c3d1e22afd06cc51c54aff30789c4d"},"schema_version":"1.0"},"canonical_sha256":"87bf2863f20e85c8bc3c8e31b925a28e685b27ce6b4f504921206009b1849edb","source":{"kind":"arxiv","id":"1201.1317","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1201.1317","created_at":"2026-05-18T03:13:58Z"},{"alias_kind":"arxiv_version","alias_value":"1201.1317v1","created_at":"2026-05-18T03:13:58Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1201.1317","created_at":"2026-05-18T03:13:58Z"},{"alias_kind":"pith_short_12","alias_value":"Q67SQY7SB2C4","created_at":"2026-05-18T12:27:18Z"},{"alias_kind":"pith_short_16","alias_value":"Q67SQY7SB2C4RPB4","created_at":"2026-05-18T12:27:18Z"},{"alias_kind":"pith_short_8","alias_value":"Q67SQY7S","created_at":"2026-05-18T12:27:18Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2012:Q67SQY7SB2C4RPB4RYY3SJNCRZ","target":"record","payload":{"canonical_record":{"source":{"id":"1201.1317","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2012-01-05T22:14:29Z","cross_cats_sorted":[],"title_canon_sha256":"7e50f8a7f44f17cfbd0f888dd5f993efa4d4c8813ab0a4c5108febe9e524d1e5","abstract_canon_sha256":"2eb1454d5eb00ce6204c5587962292cb20c3d1e22afd06cc51c54aff30789c4d"},"schema_version":"1.0"},"canonical_sha256":"87bf2863f20e85c8bc3c8e31b925a28e685b27ce6b4f504921206009b1849edb","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:13:58.486588Z","signature_b64":"8geYwm506npqW9AF4br60Df3hVH1+o+gklrGfifXRznxQPl6aLaPU4g4f1WvZKVR8q1d2K/HQrCh3jYziVqLCg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"87bf2863f20e85c8bc3c8e31b925a28e685b27ce6b4f504921206009b1849edb","last_reissued_at":"2026-05-18T03:13:58.486075Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:13:58.486075Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1201.1317","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":"2026-05-18T03:13:58Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"Xa8Mm0SqO0kAtJD0nmS4lh5/IG7tMCt4g91K/VmjZ8NgYUFm+9ds+7ZO79pegfP2W6papk4ZVT08EZsZ4zniBw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-28T04:43:38.138687Z"},"content_sha256":"7396aff3653e2b245f92e8e534d721c646290fc744fda8d3562e482d68486e09","schema_version":"1.0","event_id":"sha256:7396aff3653e2b245f92e8e534d721c646290fc744fda8d3562e482d68486e09"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2012:Q67SQY7SB2C4RPB4RYY3SJNCRZ","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"On sets of integers which are both sum-free and product-free","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Carl Pomerance, Jeffrey C. Lagarias, Par Kurlberg","submitted_at":"2012-01-05T22:14:29Z","abstract_excerpt":"We consider sets of positive integers containing no sum of two elements in the set and also no product of two elements. We show that the upper density of such a set is strictly smaller than 1/2 and that this is best possible. Further, we also find the maximal order for the density of such sets that are also periodic modulo some positive integer."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1201.1317","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":"2026-05-18T03:13:58Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"OdmaK3a3zR6hsIH4xXu5EqYFv6rGEZ0YbQHpRUvwrBNzjS6IGqbPaTtZYjhlGGdsmPd0y9QBo/cSRFxDLkgtAw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-28T04:43:38.139326Z"},"content_sha256":"7577b08b1cfe8885cc34cd4bed468d1aba2ff8e859e28e5bab98473f5214de99","schema_version":"1.0","event_id":"sha256:7577b08b1cfe8885cc34cd4bed468d1aba2ff8e859e28e5bab98473f5214de99"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/Q67SQY7SB2C4RPB4RYY3SJNCRZ/bundle.json","state_url":"https://pith.science/pith/Q67SQY7SB2C4RPB4RYY3SJNCRZ/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/Q67SQY7SB2C4RPB4RYY3SJNCRZ/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":"2026-05-28T04:43:38Z","links":{"resolver":"https://pith.science/pith/Q67SQY7SB2C4RPB4RYY3SJNCRZ","bundle":"https://pith.science/pith/Q67SQY7SB2C4RPB4RYY3SJNCRZ/bundle.json","state":"https://pith.science/pith/Q67SQY7SB2C4RPB4RYY3SJNCRZ/state.json","well_known_bundle":"https://pith.science/.well-known/pith/Q67SQY7SB2C4RPB4RYY3SJNCRZ/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2012:Q67SQY7SB2C4RPB4RYY3SJNCRZ","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":"2eb1454d5eb00ce6204c5587962292cb20c3d1e22afd06cc51c54aff30789c4d","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2012-01-05T22:14:29Z","title_canon_sha256":"7e50f8a7f44f17cfbd0f888dd5f993efa4d4c8813ab0a4c5108febe9e524d1e5"},"schema_version":"1.0","source":{"id":"1201.1317","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1201.1317","created_at":"2026-05-18T03:13:58Z"},{"alias_kind":"arxiv_version","alias_value":"1201.1317v1","created_at":"2026-05-18T03:13:58Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1201.1317","created_at":"2026-05-18T03:13:58Z"},{"alias_kind":"pith_short_12","alias_value":"Q67SQY7SB2C4","created_at":"2026-05-18T12:27:18Z"},{"alias_kind":"pith_short_16","alias_value":"Q67SQY7SB2C4RPB4","created_at":"2026-05-18T12:27:18Z"},{"alias_kind":"pith_short_8","alias_value":"Q67SQY7S","created_at":"2026-05-18T12:27:18Z"}],"graph_snapshots":[{"event_id":"sha256:7577b08b1cfe8885cc34cd4bed468d1aba2ff8e859e28e5bab98473f5214de99","target":"graph","created_at":"2026-05-18T03:13:58Z","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 sets of positive integers containing no sum of two elements in the set and also no product of two elements. We show that the upper density of such a set is strictly smaller than 1/2 and that this is best possible. Further, we also find the maximal order for the density of such sets that are also periodic modulo some positive integer.","authors_text":"Carl Pomerance, Jeffrey C. Lagarias, Par Kurlberg","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2012-01-05T22:14:29Z","title":"On sets of integers which are both sum-free and product-free"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1201.1317","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":"sha256:7396aff3653e2b245f92e8e534d721c646290fc744fda8d3562e482d68486e09","target":"record","created_at":"2026-05-18T03:13:58Z","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":"2eb1454d5eb00ce6204c5587962292cb20c3d1e22afd06cc51c54aff30789c4d","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2012-01-05T22:14:29Z","title_canon_sha256":"7e50f8a7f44f17cfbd0f888dd5f993efa4d4c8813ab0a4c5108febe9e524d1e5"},"schema_version":"1.0","source":{"id":"1201.1317","kind":"arxiv","version":1}},"canonical_sha256":"87bf2863f20e85c8bc3c8e31b925a28e685b27ce6b4f504921206009b1849edb","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"87bf2863f20e85c8bc3c8e31b925a28e685b27ce6b4f504921206009b1849edb","first_computed_at":"2026-05-18T03:13:58.486075Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T03:13:58.486075Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"8geYwm506npqW9AF4br60Df3hVH1+o+gklrGfifXRznxQPl6aLaPU4g4f1WvZKVR8q1d2K/HQrCh3jYziVqLCg==","signature_status":"signed_v1","signed_at":"2026-05-18T03:13:58.486588Z","signed_message":"canonical_sha256_bytes"},"source_id":"1201.1317","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:7396aff3653e2b245f92e8e534d721c646290fc744fda8d3562e482d68486e09","sha256:7577b08b1cfe8885cc34cd4bed468d1aba2ff8e859e28e5bab98473f5214de99"],"state_sha256":"f3ea85810b8b37b4c61317148222fe73aa2d927450f18a0e239a7e0c5b13c293"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"nWmQT9GgR9+O73hoTsief1os2rgtLPSPQFuxusFeLVUA1EJINKNFF2RnzrzbwucfQ0l0iKD1WlN2QZvfD1tpBQ==","signed_message":"bundle_sha256_bytes","signed_at":"2026-05-28T04:43:38.142387Z","bundle_sha256":"7c64b687a9eaaded161f69db2a4d6c982b9fa58e70ded8316ad519342aacb061"}}