A symmetric diophantine equation involving biquadrates

Abstract

This paper is concerned with the diophantine equation ${\sum }_{i=1}^n{a}_i{x}_i^4={\sum }_{i=1}^n{a}_i{y}_i^4$ where $n\ge 3$ and $a_i,i=1,2,\dots ,n$, 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 $n\ge 3$, 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.

5 pages; typographical errors corrected

References in corpus (1)

• On the Diophantine equation ${\sum }_{i=1}^n{a}_i{x}_i^4={\sum }_{j=1}^n{a}_j{y}_j^4$