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Keywords:
Euler's sum of powers conjecture; elliptic curve; positive integer solution; positive rational solution
Euler's sum of powers conjecture; elliptic curve; positive integer solution; positive rational solution
Summary:
We consider a variety of Euler's sum of powers 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 positive integer or rational solutions $n$, $b$, $a_i$, $i=1,2,\cdots ,s-1$, $s\geq 3.$ 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 $s\geq 4$. As a corollary, for $s\geq 4$ and any positive integer $n$, the above Diophantine system has a positive rational solution. Meanwhile, we give conditions such that it has infinitely many positive rational solutions for $s\geq 4$ and a fixed positive integer $n$.
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