{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2018:V2IMF4NIKWU6V4Q2UI4NYIMBK7","short_pith_number":"pith:V2IMF4NI","canonical_record":{"source":{"id":"1808.09247","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2018-08-28T12:22:56Z","cross_cats_sorted":[],"title_canon_sha256":"2a6141f92721069b86d8648635e1aadbb00e70362f61c6a18eb235045f43daa1","abstract_canon_sha256":"e971cb68c85de0c3feb0b00ce707c0801a026933c0cf960dcb228677ffa63572"},"schema_version":"1.0"},"canonical_sha256":"ae90c2f1a855a9eaf21aa238dc218157d417b5f2225e15ce90c53881e24a6766","source":{"kind":"arxiv","id":"1808.09247","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1808.09247","created_at":"2026-05-18T00:07:01Z"},{"alias_kind":"arxiv_version","alias_value":"1808.09247v1","created_at":"2026-05-18T00:07:01Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1808.09247","created_at":"2026-05-18T00:07:01Z"},{"alias_kind":"pith_short_12","alias_value":"V2IMF4NIKWU6","created_at":"2026-05-18T12:32:56Z"},{"alias_kind":"pith_short_16","alias_value":"V2IMF4NIKWU6V4Q2","created_at":"2026-05-18T12:32:56Z"},{"alias_kind":"pith_short_8","alias_value":"V2IMF4NI","created_at":"2026-05-18T12:32:56Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2018:V2IMF4NIKWU6V4Q2UI4NYIMBK7","target":"record","payload":{"canonical_record":{"source":{"id":"1808.09247","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2018-08-28T12:22:56Z","cross_cats_sorted":[],"title_canon_sha256":"2a6141f92721069b86d8648635e1aadbb00e70362f61c6a18eb235045f43daa1","abstract_canon_sha256":"e971cb68c85de0c3feb0b00ce707c0801a026933c0cf960dcb228677ffa63572"},"schema_version":"1.0"},"canonical_sha256":"ae90c2f1a855a9eaf21aa238dc218157d417b5f2225e15ce90c53881e24a6766","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:07:01.932176Z","signature_b64":"IAw+uD2T1mPSjr6ndtCaiiIpc2ZupxQQMBAxHC9AAROK1Ht6q/63rCI7PzNCd+pzlTV5fiXw8Ah+OK6n3F7aAQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"ae90c2f1a855a9eaf21aa238dc218157d417b5f2225e15ce90c53881e24a6766","last_reissued_at":"2026-05-18T00:07:01.931511Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:07:01.931511Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1808.09247","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":"2026-05-18T00:07:01Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"rInKQHTm/OunSCSilMUiu4dC3bJRfim3Fk/dcxw7/++oGZ3khSeCfcqm/D0TintfgKUIma575jG1LnB/FlaBDg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-30T18:53:52.325454Z"},"content_sha256":"77d52e5d716e3a7b7dccbf5ed2e157665fc04ea6d0365b0871195cab05990120","schema_version":"1.0","event_id":"sha256:77d52e5d716e3a7b7dccbf5ed2e157665fc04ea6d0365b0871195cab05990120"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2018:V2IMF4NIKWU6V4Q2UI4NYIMBK7","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Share at least half the numbers in a nontrivial LCM-closed set a nontrivial divisor?","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Tom Fischer","submitted_at":"2018-08-28T12:22:56Z","abstract_excerpt":"For a finite set of non-zero natural numbers that contains at least one element different from 1 and the least common multiple of any of its subsets, there exists a subset of at least half of its members which has a common divisor larger than 1. Utilizing a representation of the natural numbers as an order-theoretical ring of prime power sets, this conjecture is shown to be equivalent to Frankl's union-closed sets conjecture. Some results for cases where the conjecture, which also has meaningful interpretations in graph and lattice theory, is known to hold are provided. An equivalent dual vers"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1808.09247","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":"2026-05-18T00:07:01Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"J3Is5o00P4d7Vn6t33oorGz1Y39Zw5fuTKOyWOum2K856vN1uZjmI3RujX8rXoRwnJw5FlAL1qDD8B4YL1tWAg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-30T18:53:52.325819Z"},"content_sha256":"008ff25873eb3dede54ccde9037a1eea40963acd6019624868a368979e519103","schema_version":"1.0","event_id":"sha256:008ff25873eb3dede54ccde9037a1eea40963acd6019624868a368979e519103"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/V2IMF4NIKWU6V4Q2UI4NYIMBK7/bundle.json","state_url":"https://pith.science/pith/V2IMF4NIKWU6V4Q2UI4NYIMBK7/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/V2IMF4NIKWU6V4Q2UI4NYIMBK7/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":"2026-06-30T18:53:52Z","links":{"resolver":"https://pith.science/pith/V2IMF4NIKWU6V4Q2UI4NYIMBK7","bundle":"https://pith.science/pith/V2IMF4NIKWU6V4Q2UI4NYIMBK7/bundle.json","state":"https://pith.science/pith/V2IMF4NIKWU6V4Q2UI4NYIMBK7/state.json","well_known_bundle":"https://pith.science/.well-known/pith/V2IMF4NIKWU6V4Q2UI4NYIMBK7/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2018:V2IMF4NIKWU6V4Q2UI4NYIMBK7","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":"e971cb68c85de0c3feb0b00ce707c0801a026933c0cf960dcb228677ffa63572","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2018-08-28T12:22:56Z","title_canon_sha256":"2a6141f92721069b86d8648635e1aadbb00e70362f61c6a18eb235045f43daa1"},"schema_version":"1.0","source":{"id":"1808.09247","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1808.09247","created_at":"2026-05-18T00:07:01Z"},{"alias_kind":"arxiv_version","alias_value":"1808.09247v1","created_at":"2026-05-18T00:07:01Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1808.09247","created_at":"2026-05-18T00:07:01Z"},{"alias_kind":"pith_short_12","alias_value":"V2IMF4NIKWU6","created_at":"2026-05-18T12:32:56Z"},{"alias_kind":"pith_short_16","alias_value":"V2IMF4NIKWU6V4Q2","created_at":"2026-05-18T12:32:56Z"},{"alias_kind":"pith_short_8","alias_value":"V2IMF4NI","created_at":"2026-05-18T12:32:56Z"}],"graph_snapshots":[{"event_id":"sha256:008ff25873eb3dede54ccde9037a1eea40963acd6019624868a368979e519103","target":"graph","created_at":"2026-05-18T00:07:01Z","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":"For a finite set of non-zero natural numbers that contains at least one element different from 1 and the least common multiple of any of its subsets, there exists a subset of at least half of its members which has a common divisor larger than 1. Utilizing a representation of the natural numbers as an order-theoretical ring of prime power sets, this conjecture is shown to be equivalent to Frankl's union-closed sets conjecture. Some results for cases where the conjecture, which also has meaningful interpretations in graph and lattice theory, is known to hold are provided. An equivalent dual vers","authors_text":"Tom Fischer","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2018-08-28T12:22:56Z","title":"Share at least half the numbers in a nontrivial LCM-closed set a nontrivial divisor?"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1808.09247","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":"sha256:77d52e5d716e3a7b7dccbf5ed2e157665fc04ea6d0365b0871195cab05990120","target":"record","created_at":"2026-05-18T00:07:01Z","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":"e971cb68c85de0c3feb0b00ce707c0801a026933c0cf960dcb228677ffa63572","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2018-08-28T12:22:56Z","title_canon_sha256":"2a6141f92721069b86d8648635e1aadbb00e70362f61c6a18eb235045f43daa1"},"schema_version":"1.0","source":{"id":"1808.09247","kind":"arxiv","version":1}},"canonical_sha256":"ae90c2f1a855a9eaf21aa238dc218157d417b5f2225e15ce90c53881e24a6766","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"ae90c2f1a855a9eaf21aa238dc218157d417b5f2225e15ce90c53881e24a6766","first_computed_at":"2026-05-18T00:07:01.931511Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T00:07:01.931511Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"IAw+uD2T1mPSjr6ndtCaiiIpc2ZupxQQMBAxHC9AAROK1Ht6q/63rCI7PzNCd+pzlTV5fiXw8Ah+OK6n3F7aAQ==","signature_status":"signed_v1","signed_at":"2026-05-18T00:07:01.932176Z","signed_message":"canonical_sha256_bytes"},"source_id":"1808.09247","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:77d52e5d716e3a7b7dccbf5ed2e157665fc04ea6d0365b0871195cab05990120","sha256:008ff25873eb3dede54ccde9037a1eea40963acd6019624868a368979e519103"],"state_sha256":"2972a78ee82f4e7db896c93631f282a8c0503e8cf00596011862d24351eec11a"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"r2Ny/RC80fl4EsYOGI5i1BySTebmWEbVZz+3wdZGWPyjK3KORZkaYqSW8NRHhJNdGgTd2CW2NjEL1QoImHnGCA==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-30T18:53:52.327896Z","bundle_sha256":"1fbcc64e464a1301e8fd80191f9533b5682918bd9dbd093b478df6f3a1e97ffa"}}