{"paper":{"title":"An Analytical Exploration of the Erd\\\"os-Moser Equation $ \\sum_{i=1}^{m-1} i^k = m^k $ Using Approximation Methods","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":["math.CO"],"primary_cat":"math.NT","authors_text":"Guillaume Lambard","submitted_at":"2024-11-20T09:20:40Z","abstract_excerpt":"The Erd\\\"{o}s-Moser equation $ \\sum_{i=1}^{m - 1} i^k = m^k $ is a longstanding challenge in number theory, with the only known integer solution being $ (k,m) = (1,3) $. Here, we investigate whether other solutions might exist by using the Euler-MacLaurin formula to approximate the discrete sum $ S(m-1,k) $ with a continuous function $ S_{\\mathbb{R}}(m-1,k) $. We then analyze the resulting approximate polynomial $ P_{\\mathbb{R}}(m) = S_{\\mathbb{R}}(m-1,k) - m^k $ under the rational root theorem to look for integer roots. Our approximation confirms that for $ k=1 $, the only solution is $ m=3 $"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2411.13146","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/2411.13146/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"}