%0 Journal Article
%T On the sums of two cubes
%A Bruce Reznick
%A Jeremy Rouse
%J Mathematics
%D 2010
%I arXiv
%R 10.1142/S1793042111004903
%X We solve the equation $f(x,y)^3 + g(x,y)^3 = x^3 + y^3$ for homogeneous $f, g \in \mathbb C(x,y)$, completing an investigation begun by Vi\`ete in 1591. The usual addition law for elliptic curves and composition give rise to two binary operations on the set of solutions. We show that a particular subset of the set of solutions is ring-isomorphic to $\mathbb Z[e^{2 \pi i / 3}]$.
%U http://arxiv.org/abs/1012.5801v2