{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2026:GKQ7732IZUCMXOG6BYSNUTJ75N","short_pith_number":"pith:GKQ7732I","canonical_record":{"source":{"id":"2606.19863","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2026-06-18T07:22:39Z","cross_cats_sorted":[],"title_canon_sha256":"88825fa2ad1876618775caa1afaed7e7c40afacfb92d8dea0e568d97509f77ca","abstract_canon_sha256":"86e0aef062bdce454fa8048a57f386361a21667535de0966ffbd3d183955b641"},"schema_version":"1.0"},"canonical_sha256":"32a1ffef48cd04cbb8de0e24da4d3feb4eb7f2d453ddb989d3ff34ed232a9e2e","source":{"kind":"arxiv","id":"2606.19863","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"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2026:GKQ7732IZUCMXOG6BYSNUTJ75N","target":"record","payload":{"canonical_record":{"source":{"id":"2606.19863","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2026-06-18T07:22:39Z","cross_cats_sorted":[],"title_canon_sha256":"88825fa2ad1876618775caa1afaed7e7c40afacfb92d8dea0e568d97509f77ca","abstract_canon_sha256":"86e0aef062bdce454fa8048a57f386361a21667535de0966ffbd3d183955b641"},"schema_version":"1.0"},"canonical_sha256":"32a1ffef48cd04cbb8de0e24da4d3feb4eb7f2d453ddb989d3ff34ed232a9e2e","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-06-19T16:12:37.252114Z","signature_b64":"iNpEbJmzhWOiomqms+uL+ZS1PekwiMk6w6UeJn3FQUKnsSZ0Agb2lgktxG0qB0CzV7+tHg18qLuUY02WeJ1wDA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"32a1ffef48cd04cbb8de0e24da4d3feb4eb7f2d453ddb989d3ff34ed232a9e2e","last_reissued_at":"2026-06-19T16:12:37.251771Z","signature_status":"signed_v1","first_computed_at":"2026-06-19T16:12:37.251771Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"2606.19863","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":"2026-06-19T16:12:37Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"dEc1xBzKHu7mWsvEqD/Y0pgf8z6mpnfc+hy1R03QGWs50o7Rwvp9qyQi/SHZziDjkcy3WpdRwgDZ0miMTOvYAw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-06T05:54:27.916311Z"},"content_sha256":"985adfdd2ac6c45f7cb5b47485316249cd3f5d6c514b854071c21a3aba292402","schema_version":"1.0","event_id":"sha256:985adfdd2ac6c45f7cb5b47485316249cd3f5d6c514b854071c21a3aba292402"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2026:GKQ7732IZUCMXOG6BYSNUTJ75N","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Consecutive integers free of certain prime factors","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Quanyu Tang, Wouter van Doorn","submitted_at":"2026-06-18T07:22:39Z","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 $k$, $$ n_k > e^{\\frac{\\log^2 k}{20 \\log \\log k}}. $$"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2606.19863","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":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2606.19863/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":"2026-06-19T16:12:37Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"Ii7G67eA+fZyAzGh/lAuGp0fEWaNyZNIprQlPJ9s3ajxYoFyp3I12J9bBJ+Tyhs5iH854vaBoiNyvBom307jBQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-06T05:54:27.917202Z"},"content_sha256":"077d316d447b8a3e5be6cf5b607d7d953ec415ce82bbc7a6c562ca76723a712e","schema_version":"1.0","event_id":"sha256:077d316d447b8a3e5be6cf5b607d7d953ec415ce82bbc7a6c562ca76723a712e"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/GKQ7732IZUCMXOG6BYSNUTJ75N/bundle.json","state_url":"https://pith.science/pith/GKQ7732IZUCMXOG6BYSNUTJ75N/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/GKQ7732IZUCMXOG6BYSNUTJ75N/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":"2026-08-06T05:54:27Z","links":{"resolver":"https://pith.science/pith/GKQ7732IZUCMXOG6BYSNUTJ75N","bundle":"https://pith.science/pith/GKQ7732IZUCMXOG6BYSNUTJ75N/bundle.json","state":"https://pith.science/pith/GKQ7732IZUCMXOG6BYSNUTJ75N/state.json","well_known_bundle":"https://pith.science/.well-known/pith/GKQ7732IZUCMXOG6BYSNUTJ75N/bundle.json"},"state":{"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"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"ce9Xt7GxKQSfeAqAmEwotm+8v+XRfXNTyihx1rnCWASsMw0R0Gdf9QYjZVcPr0PZvB3Rxa1KP8bWaIUdFgcyAA==","signed_message":"bundle_sha256_bytes","signed_at":"2026-08-06T05:54:27.922532Z","bundle_sha256":"7eac806203dbf9f0da735a4471daebddb0a2074b108ef2ff4f82070d55d9521a"}}