{"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-the-sum-of-fourth-powers-in-arithmetic","title":"On the sum of fourth powers in arithmetic progression","arxiv_id":"1907.12351","date":"2019-07-29","proceeding":null,"authors":["Joey M. van Langen"],"abstract":"We prove that the equation ${ (x - y)^4 + x^4 + (x + y)^4 = z^n }$ has no integer solutions ${ x, y, z}$ with ${ \\gcd(x, y) = 1 }$ for all integers ${ n > 1 }$. We mainly use a modular approach with two Frey ${ \\mathbb{Q} }$-curves defined over the field ${ \\mathbb{Q}( \\sqrt{30} ) }$.","url_abs":"http://arxiv.org/abs/1907.12351v3","url_pdf":"http://arxiv.org/pdf/1907.12351v3.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-the-sum-of-fourth-powers-in-arithmetic","repo_url":"https://github.com/jmvlangen/three_fourth_powers_code","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}