{"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/mathbb-q-curves-and-the-lebesgue-nagell","title":"$\\mathbb{Q}$-curves and the Lebesgue-Nagell equation","arxiv_id":"2202.09219","date":"2022-02-18","proceeding":null,"authors":["Michael A. Bennett","Philippe Michaud-Jacobs","Samir Siksek"],"abstract":"In this paper, we consider the equation \\[ x^2 - q^{2k+1} = y^n, \\qquad q \\nmid x, \\quad 2 \\mid y, \\] for integers $x,q,k,y$ and $n$, with $k \\geq 0$ and $n \\geq 3$. We extend work of the first and third-named authors by finding all solutions in the cases $q= 41$ and $q = 97$. We do this by constructing a Frey-Hellegouarch $\\mathbb{Q}$-curve defined over the real quadratic field $K=\\mathbb{Q}(\\sqrt{q})$, and using the modular method with multi-Frey techniques.","url_abs":"https://arxiv.org/abs/2202.09219v2","url_pdf":"https://arxiv.org/pdf/2202.09219v2.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":"mathbb-q-curves-and-the-lebesgue-nagell","repo_url":"https://github.com/michaud-jacobs/q-curves","is_official":1,"mentioned_in_paper":1,"mentioned_in_github":1,"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}