# A note on the high power diophantine equations

• Mehdi Baghalaghdam
Article

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

## Keywords

Simultaneous diophantine equations equal sums of the cubes high power diophantine equations

## 2010 Mathematics Subject Classification

Primary: 11D45 Secondary: 11D72 11D25

## Notes

### Acknowledgements

The authors would like to express their hearty thanks to the anonymous referee for a careful reading of the paper and for many useful comments and remarks which improved the quality of the paper.

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