Czechoslovak Mathematical Journal, Vol. 71, No. 4, pp. 1099-1113, 2021
A variety of Euler's sum of powers conjecture
Tianxin Cai, Yong Zhang
Received May 22, 2020. Published online May 10, 2021.
Abstract: We consider a variety of Euler's sum of powers conjecture, i.e., whether the Diophantine system $$\cases n=a_1+a_2+\cdots+a_{s-1}, \\ a_1a_2\cdots a_{s-1}(a_1+a_2+\cdots+a_{s-1})=b^s \endcases$$
has positive integer or rational solutions $n$, $b$, $a_i$, $i=1,2,\cdots,s-1$, $s\geq3.$ 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 $s\geq4$. As a corollary, for $s\geq4$ 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 $s\geq4$ and a fixed positive integer $n$.
Keywords: Euler's sum of powers conjecture; elliptic curve; positive integer solution; positive rational solution
Affiliations: Tianxin Cai, School of Mathematical Sciences, Zhejiang University, 38 Zheda Rd, Hangzhou, 310027, P. R. China, e-mail: txcai@zju.edu.cn; Yong Zhang (corresponding author), School of Mathematics and Statistics, Changsha University of Science and Technology, Hunan Provincial Key Laboratory of Mathematical Modeling and Analysis in Engineering, 960, 2nd Section, Wanjiali RD (S), Changsha 410114, P. R. China, e-mail: zhangyongzju@163.com
