{"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"}