{"paper":{"title":"Irrationality of rapidly converging series: a problem of Erd\\H{o}s and Graham","license":"http://creativecommons.org/licenses/by-nc-nd/4.0/","headline":"","cross_cats":["math.CA"],"primary_cat":"math.NT","authors_text":"Jiwon Kang, Kevin Barreto, Sang-hyun Kim, Shengtong Zhang, Vjekoslav Kova\\v{c}","submitted_at":"2026-01-29T09:22:08Z","abstract_excerpt":"Answering a question of Erd\\H{o}s and Graham, we show that the double exponential growth condition $\\limsup_{n\\to\\infty}a_n^{1/\\phi^n}=\\infty$ for a strictly increasing sequence of positive integers $\\{a_n\\}_{n=1}^\\infty$ is sufficient for the series $\\sum_{n=1}^\\infty 1/(a_n a_{n+1})$ to have an irrational sum; here $\\phi$ denotes the golden ratio. We also provide a positive generalization to $\\sum_{n=1}^\\infty 1/(a_n^{w_0}\\cdots a_{n+d-1}^{w_{d-1}})$, and a negative result showing that some of its instances are essentially optimal. The original problem was autonomously solved by the AI agent"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2601.21442","kind":"arxiv","version":3},"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/2601.21442/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"}