On the simultaneous Diophantine equations $m \cdot (x_1^k+x_2^k+ \cdots + x_{t_1}^k)=n \cdot (y_1^k+y_2^k+ \cdots y_{t_2}^k)$; $k=1,3$

Abstract

In this paper, we solve the simultaneous Diophantine equations $m \cdot ( x_{1}^k+ x_{2}^k +\cdots + x_{t_1}^k)=n \cdot (y_{1}^k+ y_{2}^k +\cdots + y_{t_2}^k )$, $k=1,3$, where $t_1, t_2\ge 3$, and m, n are fixed arbitrary and relatively prime positive integers. This is done by choosing two appropriate trivial parametric solutions and obtaining infinitely many nontrivial parametric solutions. Also we work out some examples, in particular the Diophantine systems of $A^k+B^k+C^k=D^k+E^k$, $k=1,3$.