# On the simultaneous Diophantine equations m.(x_1^k+....+x_{t_1}^k)=n.(y_1^k+....+y_{t_2}^k); k=1,3

## TL;DR

This paper finds infinitely many solutions to specific simultaneous Diophantine equations involving sums of powers for k=1,3, by constructing parametric solutions and illustrating with examples.

## Contribution

It introduces a method to generate infinitely many solutions for these equations using trivial parametric solutions and applies it to particular cases.

## Key findings

- Established parametric solutions for the equations
- Generated infinitely many solutions for specific cases
- Provided explicit examples of solutions

## Abstract

In this paper, we solve the simultaneous Diophantine equations m.(x_1^k+....+x_{t_1}^k)=n.(y_1^k+....+y_{t_2}^k); k=1,3, where t_1, t_2>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^4; k=1,3.