{"paper":{"title":"On certain diophantine equations of diagonal type","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Andrew Bremner, Maciej Ulas","submitted_at":"2013-11-04T14:52:05Z","abstract_excerpt":"In this note we consider Diophantine equations of the form \\begin{equation*} a(x^p-y^q) = b(z^r-w^s), \\quad \\mbox{where}\\quad \\frac{1}{p}+\\frac{1}{q}+\\frac{1}{r}+\\frac{1}{s}=1, \\end{equation*} with even positive integers $p,q,r,s$. We show that in each case the set of rational points on the underlying surface is dense in the Zariski topology. For the surface with $(p,q,r,s)=(2,6,6,6)$ we prove density of rational points in the Euclidean topology. Moreover, in this case we construct infinitely many parametric solutions in coprime polynomials. The same result is true for $(p,q,r,s)\\in\\{(2,4,8,8)"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1311.0717","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"}