On the sums of two cubes

Bruce Reznick; Jeremy Rouse

doi:10.1142/S1793042111004903

On the sums of two cubes

Bruce Reznick, Jeremy Rouse

Mathematics

Research output: Contribution to journal › Article › peer-review

Abstract

We solve the equation f(x, y)³ + g(x, y)³ = x ³ + y³ for homogeneous f, g ∈ (x, y), completing an investigation begun by Viète 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 [e^2πi/3].

Original language	English (US)
Pages (from-to)	1863-1882
Number of pages	20
Journal	International Journal of Number Theory
Volume	7
Issue number	7
DOIs	https://doi.org/10.1142/S1793042111004903
State	Published - Nov 2011

Keywords

Sums of two cubes
elliptic curves
elliptic surfaces
endomorphism ring

ASJC Scopus subject areas

Algebra and Number Theory

Online availability

10.1142/S1793042111004903

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title = "On the sums of two cubes",

abstract = "We solve the equation f(x, y)3 + g(x, y)3 = x 3 + y3 for homogeneous f, g ∈ (x, y), completing an investigation begun by Vi{\`e}te 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 [e2πi/3].",

keywords = "Sums of two cubes, elliptic curves, elliptic surfaces, endomorphism ring",

author = "Bruce Reznick and Jeremy Rouse",

note = "Funding Information: We happily acknowledge helpful conversations with Bruce Berndt and Ken Ono, and we wish to thank the anonymous referee for helpful comments that improved the exposition. The second author was supported by NSF grant DMS-0901090.",

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N2 - We solve the equation f(x, y)3 + g(x, y)3 = x 3 + y3 for homogeneous f, g ∈ (x, y), completing an investigation begun by Viète 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 [e2πi/3].

AB - We solve the equation f(x, y)3 + g(x, y)3 = x 3 + y3 for homogeneous f, g ∈ (x, y), completing an investigation begun by Viète 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 [e2πi/3].

KW - Sums of two cubes

KW - elliptic curves

KW - elliptic surfaces

KW - endomorphism ring

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On the sums of two cubes

Abstract

Keywords

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