A variety of Euler's conjecture

TL;DR

This paper investigates a specific Diophantine system related to Euler's conjecture, proving the existence or non-existence of solutions for different values of s using elliptic curve theory.

Contribution

It establishes the solvability of the system for s=4 and the existence of polynomial solutions for s≥5, while proving no solutions exist for s=3.

Findings

No solutions for s=3.

Infinitely many solutions for s=4.

Polynomial solutions for s≥5.

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