On the sums of two cubes

Bruce Reznick, Jeremy Rouse

Research output: Contribution to journalArticlepeer-review


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].

Original languageEnglish (US)
Pages (from-to)1863-1882
Number of pages20
JournalInternational Journal of Number Theory
Issue number7
StatePublished - Nov 2011


  • Sums of two cubes
  • elliptic curves
  • elliptic surfaces
  • endomorphism ring

ASJC Scopus subject areas

  • Algebra and Number Theory


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