A note on the high power diophantine equations

Abstract

In this paper, we solve the simultaneous diophantine equations $x_{1}^\mu + x_{2}^\mu +\cdots + x_{n}^\mu =k \cdot (y_{1}^\mu + y_{2}^\mu +\cdots + y_{\frac{n}{k}}^\mu )$, $\mu =1,3$, where $n \ge 3$ and $k \ne n$ is a divisor of n ($\frac{n}{k}\ge 2$), and we obtain a nontrivial parametric solution for them. Furthermore, we present a method for producing another solution for the above diophantine equation (DE) for the case $\mu =3$, when a solution is given. We work out some examples and find nontrivial parametric solutions for each case in nonzero integers. Also we prove that the other DE $\sum _{i=1}^n p_{i} \cdot x_{i}^{a_i}=\sum _{j=1}^m q_{j} \cdot y_{j}^{b_j}$, has parametric solution and infinitely many solutions in nonzero integers with the condition that there is an i such that $p_{i}=1$ and ($a_{i},a_{1} \cdot a_{2} \cdots a_{i-1} \cdot a_{i+1} \cdots a_{n} \cdot b_{1} \cdot b_{2} \cdots b_{m})=1$, or there is a j such that $q_{j}=1$ and $(b_{j},a_{1} \cdots a_{n} \cdot b_{1} \cdots b_{j-1} \cdot b_{j+1} \cdots b_{m})=1$. Finally, we study the DE $x^a+y^b=z^c$.