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We consider a variety of Euler’s sum of powers conjecture, i.e., whether the Diophantine system $$\left\{\begin{array}{c}n=a_1+a_2+\cdots +a_{s-1},\hfill \\ a_1{a}_2\cdots a_{s-1}(a_1+a_2+\cdots +a_{s-1})=b^s\hfill \end{array}\right.$$ has positive integer or rational solutions $n$, $b$, $a_i$, $i=1,2,\cdots ,s-1$, $s\ge 3.$ Using the theory of elliptic curves, we prove that it has no positive integer solution for $s=3$, but there are infinitely many positive integers $n$ such that it has a positive integer solution for $s\ge 4$. As a corollary, for $s\ge 4$ and any positive integer $n$, the above Diophantine system has a positive rational solution. Meanwhile, we give conditions such that...