{"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-differences-of-perfect-powers-and-prime","title":"On differences of perfect powers and prime powers","arxiv_id":"2312.09985","date":"2023-12-15","proceeding":null,"authors":["Pedro-José Cazorla García"],"abstract":"Given a prime number $q$ and a squarefree integer $C_1$, we develop a method to explicitly determine the tuples $(y, n, \\alpha)$ for which the difference $y^n-q^\\alpha$ has squarefree part equal to $C_1$. Our techniques include the combination of the local information provided by Galois representations of Frey-Hellegouarch curves with the effective resolution of Thue-Mahler equations, as well as the use of improved lower bounds for $q-$adic and complex logarithms. As an application of this methodology, we will completely resolve the case when $1 \\le C_1 \\le 20$ and $ 2 \\le q < 25$.","url_abs":"https://arxiv.org/abs/2312.09985v1","url_pdf":"https://arxiv.org/pdf/2312.09985v1.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-differences-of-perfect-powers-and-prime","repo_url":"https://github.com/pjcazorla/differences-between-perfect-and-prime-powers","is_official":1,"mentioned_in_paper":1,"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}