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Erd\\H{o}s Problem #684 concerns the special threshold $u(n,k)>n^2$ and asks how early this small-prime part can be forced to become large. We prove the density-one analogue for every fixed power threshold. If $f_c(n)$ is the least $k$ for which $u(n,k)>n^c$, then, for each fixed $c>0$, \\[ f_c(n)=\\left(\\frac{c}{1-\\gamma}+o(1)\\right)\\log n \\] for almost all positive integers $n$. In particular, \\[ f_2(n)=\\left(\\frac{2}{1-\\gamma}+o(1)\\right)\\log n =(4.730544237\\ldots+o(1))\\log n \\] for the"},"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":"2606.08216","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2026-06-06T15:10:08Z","cross_cats_sorted":["math.PR"],"title_canon_sha256":"d36ae5d43297e452c0e055a7b81b5dd0b9ba36096f2b0bc3e7e3c92fb8a10f35","abstract_canon_sha256":"294360b8f3d7a5c293beb2ccaedde6a3687a30f0feab732ec71407f0e83128f1"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-06-09T01:05:30.327193Z","signature_b64":"uX9O59ZbG6CZ6WrDNZMqih1xpofotlKF5jvGCTleB2k9z1luEl+W6YI2MskdDpRIrb9gofbm1UWL/NB+c2+vCQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"6ff7aced2c7e990a9ab589fcb576c165452218d26c11cf286a2a692a6f129bcb","last_reissued_at":"2026-06-09T01:05:30.326748Z","signature_status":"signed_v1","first_computed_at":"2026-06-09T01:05:30.326748Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Erd\\H{o}s Problem 684 at Density One: Small-prime Parts of Binomial Coefficients and Gaussian Fluctuations","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.PR"],"primary_cat":"math.NT","authors_text":"Eric Li (Trinity College, United Kingdom), University of Cambridge","submitted_at":"2026-06-06T15:10:08Z","abstract_excerpt":"For $0\\leq k\\leq n$, let $u(n,k)$ be the largest divisor of $\\binom nk$ whose prime factors are at most $k$. Erd\\H{o}s Problem #684 concerns the special threshold $u(n,k)>n^2$ and asks how early this small-prime part can be forced to become large. We prove the density-one analogue for every fixed power threshold. If $f_c(n)$ is the least $k$ for which $u(n,k)>n^c$, then, for each fixed $c>0$, \\[ f_c(n)=\\left(\\frac{c}{1-\\gamma}+o(1)\\right)\\log n \\] for almost all positive integers $n$. In particular, \\[ f_2(n)=\\left(\\frac{2}{1-\\gamma}+o(1)\\right)\\log n =(4.730544237\\ldots+o(1))\\log n \\] for the"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2606.08216","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/2606.08216/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":"2606.08216","created_at":"2026-06-09T01:05:30.326818+00:00"},{"alias_kind":"arxiv_version","alias_value":"2606.08216v1","created_at":"2026-06-09T01:05:30.326818+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2606.08216","created_at":"2026-06-09T01:05:30.326818+00:00"},{"alias_kind":"pith_short_12","alias_value":"N732Z3JMP2MQ","created_at":"2026-06-09T01:05:30.326818+00:00"},{"alias_kind":"pith_short_16","alias_value":"N732Z3JMP2MQVGVV","created_at":"2026-06-09T01:05:30.326818+00:00"},{"alias_kind":"pith_short_8","alias_value":"N732Z3JM","created_at":"2026-06-09T01:05:30.326818+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/N732Z3JMP2MQVGVVRH6LK5WBMV","json":"https://pith.science/pith/N732Z3JMP2MQVGVVRH6LK5WBMV.json","graph_json":"https://pith.science/api/pith-number/N732Z3JMP2MQVGVVRH6LK5WBMV/graph.json","events_json":"https://pith.science/api/pith-number/N732Z3JMP2MQVGVVRH6LK5WBMV/events.json","paper":"https://pith.science/paper/N732Z3JM"},"agent_actions":{"view_html":"https://pith.science/pith/N732Z3JMP2MQVGVVRH6LK5WBMV","download_json":"https://pith.science/pith/N732Z3JMP2MQVGVVRH6LK5WBMV.json","view_paper":"https://pith.science/paper/N732Z3JM","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=2606.08216&json=true","fetch_graph":"https://pith.science/api/pith-number/N732Z3JMP2MQVGVVRH6LK5WBMV/graph.json","fetch_events":"https://pith.science/api/pith-number/N732Z3JMP2MQVGVVRH6LK5WBMV/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/N732Z3JMP2MQVGVVRH6LK5WBMV/action/timestamp_anchor","attest_storage":"https://pith.science/pith/N732Z3JMP2MQVGVVRH6LK5WBMV/action/storage_attestation","attest_author":"https://pith.science/pith/N732Z3JMP2MQVGVVRH6LK5WBMV/action/author_attestation","sign_citation":"https://pith.science/pith/N732Z3JMP2MQVGVVRH6LK5WBMV/action/citation_signature","submit_replication":"https://pith.science/pith/N732Z3JMP2MQVGVVRH6LK5WBMV/action/replication_record"}},"created_at":"2026-06-09T01:05:30.326818+00:00","updated_at":"2026-06-09T01:05:30.326818+00:00"}