In this paper, we study a Diophantine problem from mathematical physics and prove that for every positive integer k, there exist infinitely many sets of k n-tuples of positive integers with the same sum and the same sum of their cubes. Each set of k n-tuples is “primitive” in the sense that the greatest common divisor of all kn elements is 1. We reduce the corresponding Diophantine system to a family of elliptic curves and apply Nagell’s algorithm, Nagell-Lutz theorem and the theorem of Poincaré and Hurwitz to deal with it. In the end, we raise two open questions about this Diophantine problem.

Diophantine system, Diophantine chains, n-tuples, primitive set, elliptic curves.