On the sums of many biquadrates in two different ways

Abstract

The beautiful quartic Diophantine equation $A^4+hB^4=C^4+hD^4$, where $h$ is a fixed arbitrary positive integer, has been studied by some mathematicians for many years. Although Choudhry, Gerardin and Piezas presented solutions of this equation for many values of $h$, the solutions were not known for arbitrary positive integer values of $h$. In a separate paper (see the arxiv), the authors completely solved the equation for arbitrary values of $h$, and worked out many examples for different values of $h$, in particular for the values which has not already been given a solution. Our method, give rise to infinitely many solutions and also infinitely many parametric solutions for the equation for arbitrary rational values of $h$. In the present paper, we use the above solutions as well as a simple idea to show that how some numbers can be written as the sums of two, three, four, five, or more biquadrates in two different ways. In particular we give examples for the sums of $2$, $3$, $\cdots$, and $10$, biquadrates expressed in two different ways.