# A symmetric diophantine equation involving biquadrates

## TL;DR

This paper derives a multi-parameter explicit solution for a symmetric biquadratic Diophantine equation involving sums of fourth powers, providing smaller solutions than previously known for any number of variables n ≥ 3.

## Contribution

It presents the first explicit parametric solutions for the equation for arbitrary coefficients and any n ≥ 3, expanding the understanding of solutions to such symmetric equations.

## Key findings

- Derived multi-parameter solutions for n=3 and n=4
- Obtained smaller numerical solutions than previous methods
- Provided explicit formulas applicable for any n ≥ 3

## Abstract

This paper is concerned with the diophantine equation $\sum_{i=1}^na_ix_i^4= \sum_{i=1}^na_iy_i^4$ where $n \geq 3$ and $a_i,\,i=1,\,2,\,\ldots,\,n$, are arbitrary integers. While a method of obtaining numerical solutions of such an equation has recently been given, it seems that an explicit parametric of this diophantine equation has not yet been published. We obtain a multi-parameter solution of this equation for arbitrary values of $a_i$ and for any positive integer $n \geq 3$, and deduce specific solutions when $n=3$ and $n=4$. The numerical solutions thus obtained are much smaller than the integer solutions of such equations obtained earlier.