{"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/the-abc-conjecture-implies-that-only-finitely","title":"The abc Conjecture Implies That Only Finitely Many s-Cullen Numbers Are Repunits","arxiv_id":"2009.04052","date":"2020-09-09","proceeding":null,"authors":["Jon Grantham","Hester Graves"],"abstract":"Assuming the abc conjecture with $\\epsilon=1/6$, we use elementary methods to show that only finitely many $s$-Cullen numbers are repunits, aside from two known infinite families. More precisely, only finitely many positive integers $s$, $n$, $b$, and $q$ with $s,b \\geq 2$ and $n,q \\geq 3$ satisfy \\[C_{s,n} = ns^n + 1 = \\frac{b^q -1}{b-1}.\\]","url_abs":"https://arxiv.org/abs/2009.04052v4","url_pdf":"https://arxiv.org/pdf/2009.04052v4.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":"the-abc-conjecture-implies-that-only-finitely","repo_url":"https://github.com/31and8191/Cullen","is_official":1,"mentioned_in_paper":1,"mentioned_in_github":0,"framework":"none","reach":null}],"tasks":[],"methods":[],"datasets_introduced":[],"methods_introduced":[],"results":[],"syntology":{"atlas_url":null,"mcp":null,"developers":"https://syntology.ai/developers"},"arxiv_metadata":null,"syntology_extracted_results":null}