{"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/efficient-resolution-of-thue-mahler-equations","title":"Efficient resolution of Thue-Mahler equations","arxiv_id":"2207.14492","date":"2022-07-29","proceeding":null,"authors":["Adela Gherga","Samir Siksek"],"abstract":"A Thue-Mahler equation is a Diophantine equation of the form $$F(X,Y) = a\\cdot p_1^{z_1}\\cdots p_v^{z_v}, \\qquad \\gcd(X,Y)=1$$ where $F$ be an irreducible homogeneous binary form of degree at least $3$ with integer coefficients, $a$ is a non-zero integer and $p_1, \\dots, p_v$ are rational primes. Existing algorithms for resolving such equations require computations in the number field obtained by adjoining three roots of $F(X,1)=0$. We give a new algorithm that requires computations only in the number field obtained by adjoining one root, making it far more suited for higher degree examples. We also introduce a lattice sieving technique reminiscent of the Mordell--Weil sieve that makes it practical to tackle Thue--Mahler equations of higher degree and with larger sets of primes. We give several examples including one of degree $11$. Let $P(m)$ denote the largest prime divisor of an integer $m \\ge 2$. As an application of our algorithm we determine all pairs $(X,Y)$ of coprime non-negative integers such that $P(X^4-2Y^4) \\le 100$, finding that there are precisely $49$ such pairs.","url_abs":"https://arxiv.org/abs/2207.14492v2","url_pdf":"https://arxiv.org/pdf/2207.14492v2.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":"efficient-resolution-of-thue-mahler-equations","repo_url":"https://github.com/pjcazorla/differences-between-perfect-and-prime-powers","is_official":0,"mentioned_in_paper":0,"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}