## A variety of Euler’s sum of powers conjecture.(English)Zbl 07442476

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 $$n$$, $$b$$, $$a_i$$, $$i=1,2,\dots,s-1$$, $$s\geq 3$$. 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 $$n$$.

### MSC:

 11D72 Diophantine equations in many variables 11D41 Higher degree equations; Fermat’s equation 11G05 Elliptic curves over global fields
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