# A note on the high power Diophantine equations

## TL;DR

This paper investigates specific high-power Diophantine equations, providing parametric solutions, methods for generating new solutions, and proving the existence of infinitely many solutions under certain conditions.

## Contribution

It offers new parametric solutions for particular high-power Diophantine equations and introduces methods to generate additional solutions, expanding understanding of their solution spaces.

## Key findings

- Parametric solutions for the equations with u=1,3
- Method to produce new solutions when u=3
- Existence of infinitely many solutions under specified conditions

## Abstract

In this paper, we solve the simultaneous Diophantine equations(SDE) x_1^u+...+x_n^u=k(y_1^u+...+y_{n/k}); u=1,3, where n >3, and k< n, is a divisor of n , and obtain nontrivial parametric solution for them. Furthermore we present a method for producing another solution for the above Diophantine equation (DE) for the case u = 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 p_1x_1^{a_1}+....+p_nx_n^{a_n}=q_1y_1^{b_1}+...+q_my_m^{b_m} , has parametric solution and infinitely many solutions in nonzero integers with the condition that: there is a i such that p_i=1, and (a_i,a_1....a_{i-1}a_{i+1}...b_1...b_m)=1, or there is a j such that q_j=1, and (b_j,a_1...a_nb_1...b_{j-1}b_{j+1}..b_m)=1. Finally we study the DE x^a_y^b=z^c.