{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2026:GRKEOLS5NHUI7PFLFUMPS5XZGA","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":"a516fb3c9aafdb35233c26c7744db85ce891a6a638ab6be9f6584540aa6d0db8","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2026-07-17T15:07:16Z","title_canon_sha256":"88d992ff4f5cf7603a2dda16be29d90d8629f051a637410925564d8a544408cc"},"schema_version":"1.0","source":{"id":"2607.16032","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2607.16032","created_at":"2026-07-20T02:19:27Z"},{"alias_kind":"arxiv_version","alias_value":"2607.16032v1","created_at":"2026-07-20T02:19:27Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2607.16032","created_at":"2026-07-20T02:19:27Z"},{"alias_kind":"pith_short_12","alias_value":"GRKEOLS5NHUI","created_at":"2026-07-20T02:19:27Z"},{"alias_kind":"pith_short_16","alias_value":"GRKEOLS5NHUI7PFL","created_at":"2026-07-20T02:19:27Z"},{"alias_kind":"pith_short_8","alias_value":"GRKEOLS5","created_at":"2026-07-20T02:19:27Z"}],"graph_snapshots":[{"event_id":"sha256:301eeb3f2c029322e3c9157c5b56ee73c64c8c4adb4ebd16241693c15d2207e6","target":"graph","created_at":"2026-07-20T02:19:27Z","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/2607.16032/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"Let $P^+(n)$ denote the largest prime factor of $n$. One of Erd\\H{o}s and Tur\\'an's conjectures asserts that the asymptotic density of integers $n$ satisfying $P^+(n)<P^+(n+1)$ is 1/2. In this paper, we prove that this density is larger than 0.280, which improves the previous result 0.2017 by L\\\"u and Wang (2025). We also prove that there exists a positive density of $n$ such that $P^+(n)<P^+(n+1)<x^{41/107+\\varepsilon}$. Define $T_c(x):=\\#\\{p\\leq x:P^+(p-1)\\geq p^c\\}$. For $1/2<c<1$, we also show that \\begin{align*}\n  \\mathop{\\lim \\sup}_{x\\rightarrow\\infty}\\frac{T_c(x)}{\\pi(x)}\\leq \\min\\left(","authors_text":"Zhiyuan Yang","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2026-07-17T15:07:16Z","title":"An improvement on the largest prime factors of consecutive integers"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2607.16032","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":"sha256:50496c8da5c12d5cf1e89ba53c214ca086462a7fcdb855646c593eebf2786436","target":"record","created_at":"2026-07-20T02:19:27Z","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":"a516fb3c9aafdb35233c26c7744db85ce891a6a638ab6be9f6584540aa6d0db8","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2026-07-17T15:07:16Z","title_canon_sha256":"88d992ff4f5cf7603a2dda16be29d90d8629f051a637410925564d8a544408cc"},"schema_version":"1.0","source":{"id":"2607.16032","kind":"arxiv","version":1}},"canonical_sha256":"3454472e5d69e88fbcab2d18f976f93025688b7f05ca4684341621e9b166e111","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"3454472e5d69e88fbcab2d18f976f93025688b7f05ca4684341621e9b166e111","first_computed_at":"2026-07-20T02:19:27.349169Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-07-20T02:19:27.349169Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"NW2DMmcrCPdwdmgKxBWPOzUkEPNou7C9N8cVYigmm3rVBijxGCuIrcyNQwLcABj7W8yzWvJLohLUEnYHh9cGAg==","signature_status":"signed_v1","signed_at":"2026-07-20T02:19:27.350031Z","signed_message":"canonical_sha256_bytes"},"source_id":"2607.16032","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:50496c8da5c12d5cf1e89ba53c214ca086462a7fcdb855646c593eebf2786436","sha256:301eeb3f2c029322e3c9157c5b56ee73c64c8c4adb4ebd16241693c15d2207e6"],"state_sha256":"c9e74cecfb15bed7f0c9ecd2b6d81a1cf13858cb768967f4639f9a9588b140e0"}