{"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-power-values-of-pyramidal-numbers-ii","title":"On power values of pyramidal numbers, II","arxiv_id":"2112.03782","date":"2021-12-07","proceeding":null,"authors":["Andrej Dujella","Kálmán Győry","Philippe Michaud-Jacobs","Ákos Pintér"],"abstract":"For $m \\geq 3$, we define the $m$th order pyramidal number by \\[ \\mathrm{Pyr}_m(x) = \\frac{1}{6} x(x+1)((m-2)x+5-m). \\] In a previous paper, written by the first-, second-, and fourth-named authors, all solutions to the equation $\\mathrm{Pyr}_m(x) = y^2$ are found in positive integers $x$ and $y$, for $6 \\leq m \\leq 100$. In this paper, we consider the question of higher powers, and find all solutions to the equation $\\mathrm{Pyr}_m(x) = y^n$ in positive integers $x$, $y$, and $n$, with $n \\geq 3$, and $5 \\leq m \\leq 50$. We reduce the problem to a study of systems of binomial Thue equations, and use a combination of local arguments, the modular method via Frey curves, and bounds arising from linear forms in logarithms to solve the problem.","url_abs":"https://arxiv.org/abs/2112.03782v2","url_pdf":"https://arxiv.org/pdf/2112.03782v2.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-power-values-of-pyramidal-numbers-ii","repo_url":"https://github.com/michaud-jacobs/pyramidal-2","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}