{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2025:GGPMEBSUKNGYSQCHR5BODEZEEO","short_pith_number":"pith:GGPMEBSU","schema_version":"1.0","canonical_sha256":"319ec20654534d8940478f42e1932423bc412f2b9d6664579ca6732ba35a4e86","source":{"kind":"arxiv","id":"2503.12277","version":4},"attestation_state":"computed","paper":{"title":"On a conjecture of Erd\\H{o}s and Graham about the Sylvester's sequence","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.CA"],"primary_cat":"math.NT","authors_text":"Quanyu Tang, Zheng Li","submitted_at":"2025-03-15T22:22:47Z","abstract_excerpt":"Let $\\{u_n\\}_{n=1}^{\\infty}$ be the Sylvester's sequence (sequence A000058 in the OEIS), and let $ a_1 < a_2 < \\cdots $ be any other positive integer sequence satisfying $ \\sum_{i=1}^\\infty \\frac{1}{a_i} = 1 $. In this paper, we solve a conjecture of Erd\\H{o}s and Graham, which asks whether $$ \\liminf_{n\\to\\infty} a_n^{\\frac{1}{2^n}} < \\lim_{n\\to\\infty} u_n^{\\frac{1}{2^n}} = c_0 = 1.264085\\ldots. $$ We prove this conjecture using a constructive approach. Furthermore, assuming that the unproven claim of Erd\\H{o}s and Graham that \"all rationals have eventually greedy best Egyptian underapproxima"},"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":"2503.12277","kind":"arxiv","version":4},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2025-03-15T22:22:47Z","cross_cats_sorted":["math.CA"],"title_canon_sha256":"cceb46129beb483838f62547bdb5df99f588c814cd56704567b824e3a7a9c994","abstract_canon_sha256":"6387cded000296e019897a52a8f3a3c9603fc44050af9fa2229e566b5d1083b4"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T10:36:38.585403Z","signature_b64":"U10JobUnQIWmGnbn5iZVOBCZKIUJafl6yZHgvmMR6bXHULzCuI2enHP6GfokjmSvGE/H+DhPftRcIUBpYhh+AA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"319ec20654534d8940478f42e1932423bc412f2b9d6664579ca6732ba35a4e86","last_reissued_at":"2026-07-05T10:36:38.584775Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T10:36:38.584775Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"On a conjecture of Erd\\H{o}s and Graham about the Sylvester's sequence","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.CA"],"primary_cat":"math.NT","authors_text":"Quanyu Tang, Zheng Li","submitted_at":"2025-03-15T22:22:47Z","abstract_excerpt":"Let $\\{u_n\\}_{n=1}^{\\infty}$ be the Sylvester's sequence (sequence A000058 in the OEIS), and let $ a_1 < a_2 < \\cdots $ be any other positive integer sequence satisfying $ \\sum_{i=1}^\\infty \\frac{1}{a_i} = 1 $. In this paper, we solve a conjecture of Erd\\H{o}s and Graham, which asks whether $$ \\liminf_{n\\to\\infty} a_n^{\\frac{1}{2^n}} < \\lim_{n\\to\\infty} u_n^{\\frac{1}{2^n}} = c_0 = 1.264085\\ldots. $$ We prove this conjecture using a constructive approach. Furthermore, assuming that the unproven claim of Erd\\H{o}s and Graham that \"all rationals have eventually greedy best Egyptian underapproxima"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2503.12277","kind":"arxiv","version":4},"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/2503.12277/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":"2503.12277","created_at":"2026-07-05T10:36:38.584833+00:00"},{"alias_kind":"arxiv_version","alias_value":"2503.12277v4","created_at":"2026-07-05T10:36:38.584833+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2503.12277","created_at":"2026-07-05T10:36:38.584833+00:00"},{"alias_kind":"pith_short_12","alias_value":"GGPMEBSUKNGY","created_at":"2026-07-05T10:36:38.584833+00:00"},{"alias_kind":"pith_short_16","alias_value":"GGPMEBSUKNGYSQCH","created_at":"2026-07-05T10:36:38.584833+00:00"},{"alias_kind":"pith_short_8","alias_value":"GGPMEBSU","created_at":"2026-07-05T10:36:38.584833+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":1,"internal_anchor_count":1,"sample":[{"citing_arxiv_id":"2607.28387","citing_title":"Eventually greedy best Egyptian underapproximations of rational numbers via optimal control","ref_index":21,"is_internal_anchor":true}]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/GGPMEBSUKNGYSQCHR5BODEZEEO","json":"https://pith.science/pith/GGPMEBSUKNGYSQCHR5BODEZEEO.json","graph_json":"https://pith.science/api/pith-number/GGPMEBSUKNGYSQCHR5BODEZEEO/graph.json","events_json":"https://pith.science/api/pith-number/GGPMEBSUKNGYSQCHR5BODEZEEO/events.json","paper":"https://pith.science/paper/GGPMEBSU"},"agent_actions":{"view_html":"https://pith.science/pith/GGPMEBSUKNGYSQCHR5BODEZEEO","download_json":"https://pith.science/pith/GGPMEBSUKNGYSQCHR5BODEZEEO.json","view_paper":"https://pith.science/paper/GGPMEBSU","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=2503.12277&json=true","fetch_graph":"https://pith.science/api/pith-number/GGPMEBSUKNGYSQCHR5BODEZEEO/graph.json","fetch_events":"https://pith.science/api/pith-number/GGPMEBSUKNGYSQCHR5BODEZEEO/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/GGPMEBSUKNGYSQCHR5BODEZEEO/action/timestamp_anchor","attest_storage":"https://pith.science/pith/GGPMEBSUKNGYSQCHR5BODEZEEO/action/storage_attestation","attest_author":"https://pith.science/pith/GGPMEBSUKNGYSQCHR5BODEZEEO/action/author_attestation","sign_citation":"https://pith.science/pith/GGPMEBSUKNGYSQCHR5BODEZEEO/action/citation_signature","submit_replication":"https://pith.science/pith/GGPMEBSUKNGYSQCHR5BODEZEEO/action/replication_record"}},"created_at":"2026-07-05T10:36:38.584833+00:00","updated_at":"2026-07-05T10:36:38.584833+00:00"}