A variety of Euler’s sum of powers conjecture

We consider a variety of Euler’s sum of powers conjecture, i.e., whether the Diophantine system { n = a 1 + a 2 + … + a s − 1 , a 1 a 2 … a s − 1 ( a 1 + a 2 + … + a s − 1 ) = b s has positive integer or rational solutions n, b, a j , 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 .