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

TL;DR

This paper uses elliptic curve theory to find solutions to a class of fourth power Diophantine equations, demonstrating the existence of infinitely many solutions and multiple representations of numbers as sums of biquadrates.

Contribution

It introduces a method based on elliptic curves to solve the equation for specific cases and discusses its potential for general application.

Findings

Found nontrivial solutions for specific cases with n=3,4

Established the existence of infinitely many solutions

Showed multiple representations of numbers as sums of biquadrates

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.