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

Cai, Tianxin; Zhang, Yong

Czech. Math. J. 71, No. 4, 1099-1113 (2021).

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

Keywords:

Euler’s sum of powers conjecture; elliptic curve; positive integer solution; positive rational solution