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 $$x^a+y^b=z^c$$ .