{"about":{"site":"https://codewithpapers.app","non_affiliation":"Code with Papers and Syntology are not affiliated with, endorsed by, or sponsored by Papers with Code, Meta, or the pwc-archive mirror.","licence":"CC BY-SA 4.0","licence_url":"https://creativecommons.org/licenses/by-sa/4.0/legalcode","attribution":"https://codewithpapers.app/attribution","modified":"archive material modified by Syntology; see the attribution page"},"url":"/paper/on-the-elementary-symmetric-functions-of-11-2","title":"On the Elementary Symmetric Functions of $\\{1,1/2,\\dots,1/n\\}\\backslash\\{1/i\\}$","arxiv_id":"2502.18267","date":"2025-02-25","proceeding":null,"authors":["Weilin Zhang","Hongjian Li","Sunben Chiu","Pingzhi Yuan"],"abstract":"In 1946, P. Erd\\H{o}s and I. Niven proved that there are only finitely many positive integers $n$ for which one or more of the elementary symmetric functions of $1,1 / 2$, $\\cdots, 1 / n$ are integers. In 2012, Y. Chen and M. Tang proved that if $n \\geqslant 4$, then none of the elementary symmetric functions of $1,1 / 2, \\cdots, 1 / n$ are integers. In this paper, we prove that if $n \\geqslant 5$, then none of the elementary symmetric functions of $\\{1,1 / 2, \\cdots, 1 / n\\} \\backslash\\{1 / i\\}$ are integers except for $n=i=2$ and $n=i=4$.","url_abs":"https://arxiv.org/abs/2502.18267v1","url_pdf":"https://arxiv.org/pdf/2502.18267v1.pdf","source":{"archive":"pwc-archive (Hugging Face), CC BY-SA 4.0","snapshot":"2025-07-28","licence_url":"https://creativecommons.org/licenses/by-sa/4.0/legalcode","row_kind":"links_only","authors_date_abstract":"arXiv metadata, CC0 1.0 (https://info.arxiv.org/help/license), from the Kaggle arXiv metadata snapshot of 2026-09-12"},"code_links":[{"paper_slug":"on-the-elementary-symmetric-functions-of-11-2","repo_url":"https://github.com/zsben2/esf","is_official":1,"mentioned_in_paper":0,"mentioned_in_github":0,"framework":"none","reach":null}],"tasks":[],"methods":[],"datasets_introduced":[],"methods_introduced":[],"results":[],"syntology":{"syntology_url":null,"atlas_url":null,"mcp":null,"developers":"https://syntology.ai/developers"},"arxiv_metadata":null,"syntology_extracted_results":null}