On the Diophantine equations ${\sum }_{i=1}^n{a}_i{x}_i^6+{\sum }_{i=1}^m{b}_i{y}_i^3={\sum }_{i=1}^n{a}_i{X}_i^6\pm {\sum }_{i=1}^m{b}_i{Y}_i^3$

Abstract

In this paper, the elliptic curves theory is used for solving the Diophantine equations ${\sum }_{i=1}^n{a}_i{x}_i^6+{\sum }_{i=1}^m{b}_i{y}_i^3={\sum }_{i=1}^n{a}_i{X}_i^6\pm {\sum }_{i=1}^m{b}_i{Y}_i^3$, where $n$, $m$ $\ge 1$ 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 $b_i$.

9 pages