On the Diophantine equation $ \sum_{i=1}^n a_ix_{i} ^4= \sum_{j=1}^na_j y_{j}^4 $

Abstract: In this paper, by using the elliptic curves theory, we study the fourth power Diophantine equation ${ \sum_{i=1}^n a_ix_{i} ^4= \sum_{j=1}^na_j y_{j}^4 }$, where $a_i$ and $n\geq3$ are fixed arbitrary integers. We solve the equation for some values of $a_i$ and $n=3,4$, and find nontrivial solutions for each case in natural numbers. By our method, we may find infinitely many nontrivial solutions for the above Diophantine equation and show, among the other things, that how some numbers can be written as sums of three, four, or more biquadrates in two different ways. While our method can be used for solving the equation for every $a_i$ and $n\geq 3$, this paper will be restricted to the examples where $n=3,4$. In the end, we explain how to solve it in general cases without giving concrete examples.