{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2026:GKQ7732IZUCMXOG6BYSNUTJ75N","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":"86e0aef062bdce454fa8048a57f386361a21667535de0966ffbd3d183955b641","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2026-06-18T07:22:39Z","title_canon_sha256":"88825fa2ad1876618775caa1afaed7e7c40afacfb92d8dea0e568d97509f77ca"},"schema_version":"1.0","source":{"id":"2606.19863","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2606.19863","created_at":"2026-06-19T16:12:37Z"},{"alias_kind":"arxiv_version","alias_value":"2606.19863v1","created_at":"2026-06-19T16:12:37Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2606.19863","created_at":"2026-06-19T16:12:37Z"},{"alias_kind":"pith_short_12","alias_value":"GKQ7732IZUCM","created_at":"2026-06-19T16:12:37Z"},{"alias_kind":"pith_short_16","alias_value":"GKQ7732IZUCMXOG6","created_at":"2026-06-19T16:12:37Z"},{"alias_kind":"pith_short_8","alias_value":"GKQ7732I","created_at":"2026-06-19T16:12:37Z"}],"graph_snapshots":[{"event_id":"sha256:077d316d447b8a3e5be6cf5b607d7d953ec415ce82bbc7a6c562ca76723a712e","target":"graph","created_at":"2026-06-19T16:12:37Z","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/2606.19863/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"Let $n_k$ denote the least integer $n>2k$ such that $(n-k)(n-k+1)\\cdots(n-1)$ is not divisible by any prime in the interval $(k,2k)$. Confirming a conjecture of Erd\\H{o}s, we prove that, for all sufficiently large $k$, $$ n_k > e^{\\frac{\\log^2 k}{20 \\log \\log k}}. $$","authors_text":"Quanyu Tang, Wouter van Doorn","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2026-06-18T07:22:39Z","title":"Consecutive integers free of certain prime factors"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2606.19863","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":"sha256:985adfdd2ac6c45f7cb5b47485316249cd3f5d6c514b854071c21a3aba292402","target":"record","created_at":"2026-06-19T16:12:37Z","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":"86e0aef062bdce454fa8048a57f386361a21667535de0966ffbd3d183955b641","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2026-06-18T07:22:39Z","title_canon_sha256":"88825fa2ad1876618775caa1afaed7e7c40afacfb92d8dea0e568d97509f77ca"},"schema_version":"1.0","source":{"id":"2606.19863","kind":"arxiv","version":1}},"canonical_sha256":"32a1ffef48cd04cbb8de0e24da4d3feb4eb7f2d453ddb989d3ff34ed232a9e2e","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"32a1ffef48cd04cbb8de0e24da4d3feb4eb7f2d453ddb989d3ff34ed232a9e2e","first_computed_at":"2026-06-19T16:12:37.251771Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-06-19T16:12:37.251771Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"iNpEbJmzhWOiomqms+uL+ZS1PekwiMk6w6UeJn3FQUKnsSZ0Agb2lgktxG0qB0CzV7+tHg18qLuUY02WeJ1wDA==","signature_status":"signed_v1","signed_at":"2026-06-19T16:12:37.252114Z","signed_message":"canonical_sha256_bytes"},"source_id":"2606.19863","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:985adfdd2ac6c45f7cb5b47485316249cd3f5d6c514b854071c21a3aba292402","sha256:077d316d447b8a3e5be6cf5b607d7d953ec415ce82bbc7a6c562ca76723a712e"],"state_sha256":"0ea7b9533edc411a91b4bdd2ddb27e8be78be9a633c4b1860880de101327bc2e"}