{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2026:ZEY3AXHRIJE6I24BNWWME7CQLS","short_pith_number":"pith:ZEY3AXHR","schema_version":"1.0","canonical_sha256":"c931b05cf14249e46b816dacc27c505c939f2c588603c8b8be55710918054b6c","source":{"kind":"arxiv","id":"2606.29062","version":1},"attestation_state":"computed","paper":{"title":"A Resolution of Erd\\H{o}s Problem 731 under Dyadic Regularity","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Eric Li (Trinity College, University of Cambridge)","submitted_at":"2026-06-27T19:39:59Z","abstract_excerpt":"We resolve Erdos Problem 731 under the explicit dyadic-regularity formalization of \"reasonable.\" Let $A(n)$ be the least positive integer not dividing $\\binom{2n}{n}$. On dyadic intervals $X\\le n<2X$, put $L=\\log(2X)$ and ${\\mathcal F}_X=\\sqrt2(\\log2)^{1/4}L^{1/4}\\exp\\sqrt{(\\log2)L}$. Uniformly for $1\\le z\\le Z(X)=o(L^{1/4})$, we prove ${\\mathbb P}_X(A(n)\\le {\\mathcal F}_X\\exp(-z))\\asymp \\exp(-2z)$ and ${\\mathbb P}_X(A(n)>{\\mathcal F}_X\\exp(z))\\ll \\exp(-2z)$. Consequently $\\log A(n)=\\sqrt{(\\log2)\\log n}+\\frac14\\log\\log n+O_{\\rm dens}(1)$. We also prove dyadic nonconcentration: no scalar center"},"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":"2606.29062","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2026-06-27T19:39:59Z","cross_cats_sorted":[],"title_canon_sha256":"5fe111cc6f5c18f59faef1eb77dd6babb070e5d3aef55a45bdae8de45884f548","abstract_canon_sha256":"a53a3f692a6de9572d3edf570f97471b2979cc0fe33e3d227578f101fd63b5b5"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-06-30T01:17:51.254020Z","signature_b64":"Q4XahJ6UdT+Z33lBQzFrHhjgpbNTlj8/iIH1utvJPJhIXTSTx9EKhdDTb/CAw92HkZYmJRAIBSkolV4GIq4+CA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"c931b05cf14249e46b816dacc27c505c939f2c588603c8b8be55710918054b6c","last_reissued_at":"2026-06-30T01:17:51.253533Z","signature_status":"signed_v1","first_computed_at":"2026-06-30T01:17:51.253533Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"A Resolution of Erd\\H{o}s Problem 731 under Dyadic Regularity","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Eric Li (Trinity College, University of Cambridge)","submitted_at":"2026-06-27T19:39:59Z","abstract_excerpt":"We resolve Erdos Problem 731 under the explicit dyadic-regularity formalization of \"reasonable.\" Let $A(n)$ be the least positive integer not dividing $\\binom{2n}{n}$. On dyadic intervals $X\\le n<2X$, put $L=\\log(2X)$ and ${\\mathcal F}_X=\\sqrt2(\\log2)^{1/4}L^{1/4}\\exp\\sqrt{(\\log2)L}$. Uniformly for $1\\le z\\le Z(X)=o(L^{1/4})$, we prove ${\\mathbb P}_X(A(n)\\le {\\mathcal F}_X\\exp(-z))\\asymp \\exp(-2z)$ and ${\\mathbb P}_X(A(n)>{\\mathcal F}_X\\exp(z))\\ll \\exp(-2z)$. Consequently $\\log A(n)=\\sqrt{(\\log2)\\log n}+\\frac14\\log\\log n+O_{\\rm dens}(1)$. We also prove dyadic nonconcentration: no scalar center"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2606.29062","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/2606.29062/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":"2606.29062","created_at":"2026-06-30T01:17:51.253602+00:00"},{"alias_kind":"arxiv_version","alias_value":"2606.29062v1","created_at":"2026-06-30T01:17:51.253602+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2606.29062","created_at":"2026-06-30T01:17:51.253602+00:00"},{"alias_kind":"pith_short_12","alias_value":"ZEY3AXHRIJE6","created_at":"2026-06-30T01:17:51.253602+00:00"},{"alias_kind":"pith_short_16","alias_value":"ZEY3AXHRIJE6I24B","created_at":"2026-06-30T01:17:51.253602+00:00"},{"alias_kind":"pith_short_8","alias_value":"ZEY3AXHR","created_at":"2026-06-30T01:17:51.253602+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/ZEY3AXHRIJE6I24BNWWME7CQLS","json":"https://pith.science/pith/ZEY3AXHRIJE6I24BNWWME7CQLS.json","graph_json":"https://pith.science/api/pith-number/ZEY3AXHRIJE6I24BNWWME7CQLS/graph.json","events_json":"https://pith.science/api/pith-number/ZEY3AXHRIJE6I24BNWWME7CQLS/events.json","paper":"https://pith.science/paper/ZEY3AXHR"},"agent_actions":{"view_html":"https://pith.science/pith/ZEY3AXHRIJE6I24BNWWME7CQLS","download_json":"https://pith.science/pith/ZEY3AXHRIJE6I24BNWWME7CQLS.json","view_paper":"https://pith.science/paper/ZEY3AXHR","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=2606.29062&json=true","fetch_graph":"https://pith.science/api/pith-number/ZEY3AXHRIJE6I24BNWWME7CQLS/graph.json","fetch_events":"https://pith.science/api/pith-number/ZEY3AXHRIJE6I24BNWWME7CQLS/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/ZEY3AXHRIJE6I24BNWWME7CQLS/action/timestamp_anchor","attest_storage":"https://pith.science/pith/ZEY3AXHRIJE6I24BNWWME7CQLS/action/storage_attestation","attest_author":"https://pith.science/pith/ZEY3AXHRIJE6I24BNWWME7CQLS/action/author_attestation","sign_citation":"https://pith.science/pith/ZEY3AXHRIJE6I24BNWWME7CQLS/action/citation_signature","submit_replication":"https://pith.science/pith/ZEY3AXHRIJE6I24BNWWME7CQLS/action/replication_record"}},"created_at":"2026-06-30T01:17:51.253602+00:00","updated_at":"2026-06-30T01:17:51.253602+00:00"}