{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2018:TKXSGOOKZDGL36OBOYS6DP4PIO","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":"cb6ba40a28ac4a746d0b588d2b319ccee4857b6c102d768d53c5f0dc4c096794","cross_cats_sorted":["math.CO"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2018-09-22T00:33:13Z","title_canon_sha256":"5dc45ba89de47d2d90c221e3bc2ad4aa865e855e9bfdbb05b7de8808e9c8a443"},"schema_version":"1.0","source":{"id":"1809.08355","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1809.08355","created_at":"2026-05-17T23:54:48Z"},{"alias_kind":"arxiv_version","alias_value":"1809.08355v2","created_at":"2026-05-17T23:54:48Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1809.08355","created_at":"2026-05-17T23:54:48Z"},{"alias_kind":"pith_short_12","alias_value":"TKXSGOOKZDGL","created_at":"2026-05-18T12:32:53Z"},{"alias_kind":"pith_short_16","alias_value":"TKXSGOOKZDGL36OB","created_at":"2026-05-18T12:32:53Z"},{"alias_kind":"pith_short_8","alias_value":"TKXSGOOK","created_at":"2026-05-18T12:32:53Z"}],"graph_snapshots":[{"event_id":"sha256:eee700c06bdbbc63a5d6d2669e5b7cff583f6f54d15c1d57ed82b6fd7512f6c9","target":"graph","created_at":"2026-05-17T23:54:48Z","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 prove the existence of primitive sets (sets of integers in which no element divides another) in which the gap between any two consecutive terms is substantially smaller than the best known upper bound for the gaps in the sequence of prime numbers. The proof uses the probabilistic method. Using the same techniques we improve the bounds obtained by He for gaps in geometric-progression-free sets.","authors_text":"Nathan McNew","cross_cats":["math.CO"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2018-09-22T00:33:13Z","title":"Primitive and geometric-progression-free sets without large gaps"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1809.08355","kind":"arxiv","version":2},"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":"sha256:5d5b0d9a4c7b63aece8ac7ff440b985148f77820eeef4fe032c1e609cb6e096c","target":"record","created_at":"2026-05-17T23:54:48Z","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":"cb6ba40a28ac4a746d0b588d2b319ccee4857b6c102d768d53c5f0dc4c096794","cross_cats_sorted":["math.CO"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2018-09-22T00:33:13Z","title_canon_sha256":"5dc45ba89de47d2d90c221e3bc2ad4aa865e855e9bfdbb05b7de8808e9c8a443"},"schema_version":"1.0","source":{"id":"1809.08355","kind":"arxiv","version":2}},"canonical_sha256":"9aaf2339cac8ccbdf9c17625e1bf8f43a9a53099458a9420ef29a58ce9ff3c66","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"9aaf2339cac8ccbdf9c17625e1bf8f43a9a53099458a9420ef29a58ce9ff3c66","first_computed_at":"2026-05-17T23:54:48.153016Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-17T23:54:48.153016Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"0AeQOQm5GNSjrRQ1kADaKL/4XefJbkbS738CTaZzZ3ulDPkbvvVheRIn0NeRcqoPJ5dWZhavQnKNckJvLWZZDg==","signature_status":"signed_v1","signed_at":"2026-05-17T23:54:48.153660Z","signed_message":"canonical_sha256_bytes"},"source_id":"1809.08355","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:5d5b0d9a4c7b63aece8ac7ff440b985148f77820eeef4fe032c1e609cb6e096c","sha256:eee700c06bdbbc63a5d6d2669e5b7cff583f6f54d15c1d57ed82b6fd7512f6c9"],"state_sha256":"ceee19748ab80c0c5ef62b1f03a44431ee6150bba04572b45dd267f329f7f612"}