{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2026:VBVCKEZTZ3YRTJPEEPDESPVPVK","short_pith_number":"pith:VBVCKEZT","schema_version":"1.0","canonical_sha256":"a86a251333cef119a5e423c6493eafaa91ecffae84d47f6a2b02a3dc89265528","source":{"kind":"arxiv","id":"2607.26450","version":1},"attestation_state":"computed","paper":{"title":"Improved Bounds for Distinct Multiples in Intervals","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.CO"],"primary_cat":"math.NT","authors_text":"Kaizhe Chen","submitted_at":"2026-07-29T04:03:50Z","abstract_excerpt":"In this note, we study two functions introduced by Erd\\H{o}s and Pomerance. For any positive integer $n$, let $F(n)$ be the smallest integer $F>0$ such that any $F$ consecutive integers contain a distinct multiple for each positive integer at most $n$, and let $h_{\\mathbb{P}}(n)$ be the smallest integer $H>0$ such that any $H$ consecutive integers contain a distinct multiple for each prime at most $n$. Based on the square-residue digit construction of Green and Ruzsa, we prove \\[\n  F(n)\\ge h_{\\mathbb P}(n)\\ge n\\exp\\!\\left(\\frac{1}{50}\\frac{\\log n}{\\log\\log n}\\right), \\] for sufficiently large "},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"2607.26450","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2026-07-29T04:03:50Z","cross_cats_sorted":["math.CO"],"title_canon_sha256":"71edca6a9defa578c0621f26d5a024abb402ec1933cd6e11f10eea239e402cdf","abstract_canon_sha256":"be26ad790cb7bab201f942383e16f99366e93a37d064cb4e57ace5580473b7bf"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"a86a251333cef119a5e423c6493eafaa91ecffae84d47f6a2b02a3dc89265528","last_reissued_at":"2026-07-30T01:20:32.268506Z","signature_status":"unsigned_v0","first_computed_at":"2026-07-30T01:20:32.268506Z"},"graph_snapshot":{"paper":{"title":"Improved Bounds for Distinct Multiples in Intervals","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.CO"],"primary_cat":"math.NT","authors_text":"Kaizhe Chen","submitted_at":"2026-07-29T04:03:50Z","abstract_excerpt":"In this note, we study two functions introduced by Erd\\H{o}s and Pomerance. For any positive integer $n$, let $F(n)$ be the smallest integer $F>0$ such that any $F$ consecutive integers contain a distinct multiple for each positive integer at most $n$, and let $h_{\\mathbb{P}}(n)$ be the smallest integer $H>0$ such that any $H$ consecutive integers contain a distinct multiple for each prime at most $n$. Based on the square-residue digit construction of Green and Ruzsa, we prove \\[\n  F(n)\\ge h_{\\mathbb P}(n)\\ge n\\exp\\!\\left(\\frac{1}{50}\\frac{\\log n}{\\log\\log n}\\right), \\] for sufficiently large "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2607.26450","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/2607.26450/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"},"aliases":[{"alias_kind":"arxiv","alias_value":"2607.26450","created_at":"2026-07-30T01:20:32.274156+00:00"},{"alias_kind":"arxiv_version","alias_value":"2607.26450v1","created_at":"2026-07-30T01:20:32.274156+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2607.26450","created_at":"2026-07-30T01:20:32.274156+00:00"},{"alias_kind":"pith_short_12","alias_value":"VBVCKEZTZ3YR","created_at":"2026-07-30T01:20:32.274156+00:00"},{"alias_kind":"pith_short_16","alias_value":"VBVCKEZTZ3YRTJPE","created_at":"2026-07-30T01:20:32.274156+00:00"},{"alias_kind":"pith_short_8","alias_value":"VBVCKEZT","created_at":"2026-07-30T01:20:32.274156+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":0,"internal_anchor_count":0,"sample":[]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/VBVCKEZTZ3YRTJPEEPDESPVPVK","json":"https://pith.science/pith/VBVCKEZTZ3YRTJPEEPDESPVPVK.json","graph_json":"https://pith.science/api/pith-number/VBVCKEZTZ3YRTJPEEPDESPVPVK/graph.json","events_json":"https://pith.science/api/pith-number/VBVCKEZTZ3YRTJPEEPDESPVPVK/events.json","paper":"https://pith.science/paper/VBVCKEZT"},"agent_actions":{"view_html":"https://pith.science/pith/VBVCKEZTZ3YRTJPEEPDESPVPVK","download_json":"https://pith.science/pith/VBVCKEZTZ3YRTJPEEPDESPVPVK.json","view_paper":"https://pith.science/paper/VBVCKEZT","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=2607.26450&json=true","fetch_graph":"https://pith.science/api/pith-number/VBVCKEZTZ3YRTJPEEPDESPVPVK/graph.json","fetch_events":"https://pith.science/api/pith-number/VBVCKEZTZ3YRTJPEEPDESPVPVK/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/VBVCKEZTZ3YRTJPEEPDESPVPVK/action/timestamp_anchor","attest_storage":"https://pith.science/pith/VBVCKEZTZ3YRTJPEEPDESPVPVK/action/storage_attestation","attest_author":"https://pith.science/pith/VBVCKEZTZ3YRTJPEEPDESPVPVK/action/author_attestation","sign_citation":"https://pith.science/pith/VBVCKEZTZ3YRTJPEEPDESPVPVK/action/citation_signature","submit_replication":"https://pith.science/pith/VBVCKEZTZ3YRTJPEEPDESPVPVK/action/replication_record"}},"created_at":"2026-07-30T01:20:32.274156+00:00","updated_at":"2026-07-30T01:20:32.274156+00:00"}