{"paper":{"title":"An Ancient Diophantine Equation with applications to Numerical Curios and Geometric Series","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Ajai Choudhry, Jaros{\\l}aw Wr\\'oblewski","submitted_at":"2016-03-20T11:31:09Z","abstract_excerpt":"In this paper we examine the diophantine equation $x^k-y^k=x-y$ where $k$ is a positive integer $\\geq 2$, and consider its applications. While the complete solution of the equation $x^k-y^k=x-y$ in positive rational numbers is already known when $k=2$ or $3$, till now only one numerical solution of the equation in positive rational numbers has been published when $k=4$, and no nontrivial solution is known when $k \\geq 5$. We describe a method of generating infinitely many positive rational solutions of the equation when $k=4$. We use the positive rational solutions of the equation with $k=2,\\,"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1603.06205","kind":"arxiv","version":1},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}