{"paper":{"title":"A variety of Euler's conjecture","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Tianxin Cai, Yong Zhang","submitted_at":"2013-09-29T05:12:57Z","abstract_excerpt":"We consider a variety of Euler's conjecture, i.e., whether the Diophantine system \\[\\begin{cases} n=a_{1}+a_{2}+\\cdots+a_{s-1}, a_{1}a_{2}\\cdots a_{s-1}(a_{1}+a_{2}+\\cdots+a_{s-1})=b^{s} \\end{cases}\\] has solutions $n,b,a_i\\in\\mathbb{Z}^+,i=1,2,\\ldots,s-1,s\\geq 3.$ By using the theory of elliptic curves, we prove that it has no solutions $n,b,a_i\\in\\mathbb{Z}^+$ for $s=3$, but for $s=4$ it has infinitely many solutions $n,b,a_i\\in\\mathbb{Z}^+$ and for $s\\geq 5$ there are infinitely many polynomial solutions $n,b,a_i\\in\\mathbb{Z}[t_1,t_2,\\ldots,t_{s-3}]$ with positive value satisfying this Diop"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1309.7537","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"}