# On the Diophantine equations $ \sum_{i=1}^n a_ix_{i} ^6+\sum_{i=1}^m b_iy_{i} ^3= \sum_{i=1}^na_iX_{i}^6\pm\sum_{i=1}^m b_iY_{i} ^3 $

## TL;DR

This paper employs elliptic curve theory to find infinitely many positive and parametric solutions to complex Diophantine equations involving sixth and third powers, applicable for arbitrary fixed parameters.

## Contribution

It introduces a novel method using elliptic curves to solve a broad class of Diophantine equations with arbitrary parameters, yielding infinite solutions.

## Key findings

- Infinitely many positive solutions are found.
- Parametric solutions are explicitly constructed.
- Method applies to any fixed nonzero integers $a_i$, $b_i$, $n$, and $m$.

## Abstract

In this paper, the elliptic curves theory is used for solving the Diophantine equations $\sum_{i=1}^n a_ix_{i} ^6+\sum_{i=1}^m b_iy_{i} ^3= \sum_{i=1}^na_iX_{i}^6\pm\sum_{i=1}^m b_iY_{i} ^3$, where $n$, $m$ $\geq 1$ and $a_i$, $b_i$, are fixed arbitrary nonzero integers. By our method, we may find infinitely many nontrivial positive solutions and also obtain infinitely many nontrivial parametric solutions for the Diophantine equations for every arbitrary integers $n$, $m$, $a_i$ and $b_i$.