A quintic diophantine equation with applications to two diophantine systems concerning fifth powers

Abstract

In this paper we obtain a parametric solution of the quintic diophantine equation $ab(a+b)(a^2+ab+b^2)=cd(c+d)(c^2+cd+d^2)$. We use this solution to obtain parametric solutions of two diophantine systems concerning fifth powers, namely, the system of simultaneous equations $x_1+x_2+x_3 = y_1+y_2+y_3 =0$, $x_1^5+x_2^5+x_3^5 = y_1^5+y_2^5+y_3^5$, and the system of equations given by $\sum_{i=1}^5x_i^k =\sum_{i=1}^5 y_i^k$, $k=1,\,2,\,4,\,5$.