An octic diophantine equation and related families of elliptic curves

Abstract

We obtain two parametric solutions of the diophantine equation [Formula: see text] where [Formula: see text] is the octic form defined by [Formula: see text]. These parametric solutions yield infinitely many examples of two equiareal triangles whose sides are perfect squares of integers. Further, each of the two parametric solutions leads to a family of elliptic curves of rank [Formula: see text] over [Formula: see text]. We study one of the two families in some detail and determine a set of five free generators for the family.