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Umans and Wang proposed, as the arithmetic-progression version of their Strong $(\\alpha,\\beta)$-Divisor Conjecture, an $n$-divisor arithmetic progression having at most $n^{2\\beta}$ terms, each of magnitude at most $\\exp(n^\\alpha)$. We prove unconditionally that an $n$-divisor arithmetic progression of height $H$ with $\\log H=o(\\sqrt n)$ must have length \\[\n  L\\ge\n  \\left(\\sqrt{\\frac{8}{27}}-o(1)\\right)\n  \\frac{n^{3/4}}{\\sqrt{\\log n}}. \\] Consequently, the arithmetic-prog"},"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":"2608.06681","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2026-08-07T01:06:46Z","cross_cats_sorted":["math.CO"],"title_canon_sha256":"bde994708553e3bf5cbcd2d72f843893fd9db96bc62ac043714ceefb4cfbc55e","abstract_canon_sha256":"1de0b424e9ca248e6560062c5fce6cc84159ea93af83b7a367c3968480fa339f"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-08-10T01:11:41.485762Z","signature_b64":"o1VjjKHaMnPg6xxLDR98mfs6SUTEx8bDn5CW6ciQqyUsvOuuoFjLlr9xC9on/veraPv3Ydr1fXtPOB6tolc8Cw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"ffab0d82cd401b3bafa559c41b0c896ccf6a8914e4909abe0d9441cb199b90ee","last_reissued_at":"2026-08-10T01:11:41.483338Z","signature_status":"signed_v1","first_computed_at":"2026-08-10T01:11:41.483338Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Refuting a Conjecture of Umans and Wang on Arithmetic-Progression Divisor Covers","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.CO"],"primary_cat":"math.NT","authors_text":"Amit Sahai, Xinjie He","submitted_at":"2026-08-07T01:06:46Z","abstract_excerpt":"An \\emph{$n$-divisor set} is a finite set of positive integers containing a multiple of every integer from $1$ through $n$. Umans and Wang proposed, as the arithmetic-progression version of their Strong $(\\alpha,\\beta)$-Divisor Conjecture, an $n$-divisor arithmetic progression having at most $n^{2\\beta}$ terms, each of magnitude at most $\\exp(n^\\alpha)$. We prove unconditionally that an $n$-divisor arithmetic progression of height $H$ with $\\log H=o(\\sqrt n)$ must have length \\[\n  L\\ge\n  \\left(\\sqrt{\\frac{8}{27}}-o(1)\\right)\n  \\frac{n^{3/4}}{\\sqrt{\\log n}}. \\] Consequently, the arithmetic-prog"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2608.06681","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/2608.06681/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":"2608.06681","created_at":"2026-08-10T01:11:41.484302+00:00"},{"alias_kind":"arxiv_version","alias_value":"2608.06681v1","created_at":"2026-08-10T01:11:41.484302+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2608.06681","created_at":"2026-08-10T01:11:41.484302+00:00"},{"alias_kind":"pith_short_12","alias_value":"76VQ3AWNIANT","created_at":"2026-08-10T01:11:41.484302+00:00"},{"alias_kind":"pith_short_16","alias_value":"76VQ3AWNIANTXL5F","created_at":"2026-08-10T01:11:41.484302+00:00"},{"alias_kind":"pith_short_8","alias_value":"76VQ3AWN","created_at":"2026-08-10T01:11:41.484302+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/76VQ3AWNIANTXL5FLHCBWDEJNT","json":"https://pith.science/pith/76VQ3AWNIANTXL5FLHCBWDEJNT.json","graph_json":"https://pith.science/api/pith-number/76VQ3AWNIANTXL5FLHCBWDEJNT/graph.json","events_json":"https://pith.science/api/pith-number/76VQ3AWNIANTXL5FLHCBWDEJNT/events.json","paper":"https://pith.science/paper/76VQ3AWN"},"agent_actions":{"view_html":"https://pith.science/pith/76VQ3AWNIANTXL5FLHCBWDEJNT","download_json":"https://pith.science/pith/76VQ3AWNIANTXL5FLHCBWDEJNT.json","view_paper":"https://pith.science/paper/76VQ3AWN","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=2608.06681&json=true","fetch_graph":"https://pith.science/api/pith-number/76VQ3AWNIANTXL5FLHCBWDEJNT/graph.json","fetch_events":"https://pith.science/api/pith-number/76VQ3AWNIANTXL5FLHCBWDEJNT/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/76VQ3AWNIANTXL5FLHCBWDEJNT/action/timestamp_anchor","attest_storage":"https://pith.science/pith/76VQ3AWNIANTXL5FLHCBWDEJNT/action/storage_attestation","attest_author":"https://pith.science/pith/76VQ3AWNIANTXL5FLHCBWDEJNT/action/author_attestation","sign_citation":"https://pith.science/pith/76VQ3AWNIANTXL5FLHCBWDEJNT/action/citation_signature","submit_replication":"https://pith.science/pith/76VQ3AWNIANTXL5FLHCBWDEJNT/action/replication_record"}},"created_at":"2026-08-10T01:11:41.484302+00:00","updated_at":"2026-08-10T01:11:41.484302+00:00"}