On the sums of two cubes

Details On the sums of two cubes On the sums of two cubes

2010-12-28

arxiv.org/abs/1012.5801v2

Int. J. Number Theory, 7 (2011), no. 7, 1863-1882

Mathematics

Number Theory

Revised version, to appear in the International Journal of Number Theory

Scientific paper

10.1142/S1793042111004903

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 $\mathbb Z[e^{2 \pi i / 3}]$.

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