{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2026:SWJWBTN5SCAPIBOQUO6U7IPIEU","short_pith_number":"pith:SWJWBTN5","schema_version":"1.0","canonical_sha256":"959360cdbd9080f405d0a3bd4fa1e82526d7dbd25a22e19fee02064addd1a4db","source":{"kind":"arxiv","id":"2608.05191","version":1},"attestation_state":"computed","paper":{"title":"Rigidity of Averages over the Two Largest Prime Factors","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Dijia Chen","submitted_at":"2026-08-02T18:31:36Z","abstract_excerpt":"Let \\(P_1(n)\\) and \\(P_2(n)\\) be the largest and second-largest distinct prime factors of \\(n\\), respectively. Alladi and Johnson asked whether there exists a bounded function \\(f\\) on the primes for which both limits \\(\\frac{1}{x}\\sum_{2\\le n\\le x} f(P_1(n)) \\longrightarrow \\kappa_1\\) and \\(\\frac{1}{x}\\sum_{2\\le n\\le x} f(P_2(n)) \\longrightarrow \\kappa_2\\) exist with \\(\\kappa_1\\neq\\kappa_2\\), where we set \\(f(P_2(n))=0\\) when \\(n\\) is a prime power. We prove that this is impossible: convergence of the first average forces convergence of the second to the same limit.\n  On the \\(\\log\\log\\)-scal"},"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":"2608.05191","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2026-08-02T18:31:36Z","cross_cats_sorted":[],"title_canon_sha256":"f4a6c403af83530478a7f059f134769cd381c7d529eb996a6355468c84b01f9d","abstract_canon_sha256":"8734e1c7d2ffed23491814fc130564770d0167d8f491c189c754a2b2f466fb41"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-08-07T00:46:29.930754Z","signature_b64":"nnZFtteqYCXkZsEC8ZwgjP9lRT5XW+U+Aik/nkPn0sTnJWuVIf09JznuV/s4QouYnrRomMmix9DsJzdxc8+mCQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"959360cdbd9080f405d0a3bd4fa1e82526d7dbd25a22e19fee02064addd1a4db","last_reissued_at":"2026-08-07T00:46:29.929392Z","signature_status":"signed_v1","first_computed_at":"2026-08-07T00:46:29.929392Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Rigidity of Averages over the Two Largest Prime Factors","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Dijia Chen","submitted_at":"2026-08-02T18:31:36Z","abstract_excerpt":"Let \\(P_1(n)\\) and \\(P_2(n)\\) be the largest and second-largest distinct prime factors of \\(n\\), respectively. Alladi and Johnson asked whether there exists a bounded function \\(f\\) on the primes for which both limits \\(\\frac{1}{x}\\sum_{2\\le n\\le x} f(P_1(n)) \\longrightarrow \\kappa_1\\) and \\(\\frac{1}{x}\\sum_{2\\le n\\le x} f(P_2(n)) \\longrightarrow \\kappa_2\\) exist with \\(\\kappa_1\\neq\\kappa_2\\), where we set \\(f(P_2(n))=0\\) when \\(n\\) is a prime power. We prove that this is impossible: convergence of the first average forces convergence of the second to the same limit.\n  On the \\(\\log\\log\\)-scal"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2608.05191","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/2608.05191/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":"2608.05191","created_at":"2026-08-07T00:46:29.930649+00:00"},{"alias_kind":"arxiv_version","alias_value":"2608.05191v1","created_at":"2026-08-07T00:46:29.930649+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2608.05191","created_at":"2026-08-07T00:46:29.930649+00:00"},{"alias_kind":"pith_short_12","alias_value":"SWJWBTN5SCAP","created_at":"2026-08-07T00:46:29.930649+00:00"},{"alias_kind":"pith_short_16","alias_value":"SWJWBTN5SCAPIBOQ","created_at":"2026-08-07T00:46:29.930649+00:00"},{"alias_kind":"pith_short_8","alias_value":"SWJWBTN5","created_at":"2026-08-07T00:46:29.930649+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/SWJWBTN5SCAPIBOQUO6U7IPIEU","json":"https://pith.science/pith/SWJWBTN5SCAPIBOQUO6U7IPIEU.json","graph_json":"https://pith.science/api/pith-number/SWJWBTN5SCAPIBOQUO6U7IPIEU/graph.json","events_json":"https://pith.science/api/pith-number/SWJWBTN5SCAPIBOQUO6U7IPIEU/events.json","paper":"https://pith.science/paper/SWJWBTN5"},"agent_actions":{"view_html":"https://pith.science/pith/SWJWBTN5SCAPIBOQUO6U7IPIEU","download_json":"https://pith.science/pith/SWJWBTN5SCAPIBOQUO6U7IPIEU.json","view_paper":"https://pith.science/paper/SWJWBTN5","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=2608.05191&json=true","fetch_graph":"https://pith.science/api/pith-number/SWJWBTN5SCAPIBOQUO6U7IPIEU/graph.json","fetch_events":"https://pith.science/api/pith-number/SWJWBTN5SCAPIBOQUO6U7IPIEU/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/SWJWBTN5SCAPIBOQUO6U7IPIEU/action/timestamp_anchor","attest_storage":"https://pith.science/pith/SWJWBTN5SCAPIBOQUO6U7IPIEU/action/storage_attestation","attest_author":"https://pith.science/pith/SWJWBTN5SCAPIBOQUO6U7IPIEU/action/author_attestation","sign_citation":"https://pith.science/pith/SWJWBTN5SCAPIBOQUO6U7IPIEU/action/citation_signature","submit_replication":"https://pith.science/pith/SWJWBTN5SCAPIBOQUO6U7IPIEU/action/replication_record"}},"created_at":"2026-08-07T00:46:29.930649+00:00","updated_at":"2026-08-07T00:46:29.930649+00:00"}