A variety of Euler's conjecture

Abstract

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 $n,b,a_i\in {\mathbb{Z}}^{+},i=1,2,\dots ,s-1,s\ge 3.$ 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 $s\ge 5$ there are infinitely many polynomial solutions $n,b,a_i\in \mathbb{Z}[t_1,t_2,\dots ,t_{s-3}]$ with positive value satisfying this Diophantine system.

8 pages

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