English

Abstract

We consider a variety of Euler’s sum of powers conjecture, i.e., whether the Diophantine system $$\left\{{\matrix{{n = {a_1} + {a_2} + \ldots + {a_{s - 1}},} \hfill \cr {{a_1}{a_2} \ldots {a_{s - 1}}({a_1} + {a_2} + \ldots + {a_{s - 1}}) = {b^s}} \hfill \cr}} \right.$$ has positive integer or rational solutions n, b, aj, i = 1, 2, …, s − 1, s ⩾ 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 ⩾ 4. As a corollary, for s ⩾ 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 ⩾ 4 and a fixed positive integer n.