Equal sums of two cubes of quadratic forms

Research output: Chapter in Book/Report/Conference proceedingChapter

Abstract

We give a complete description of all solutions to the equation f31+f32=f33+f34 for quadratic forms fj ∈ ℂ [x, y] and show how Ramanujan’s example can be extended to three equal sums of pairs of cubes. We also give a complete census in counting the number of ways a sextic p ∈ ℂ [x, y] can be written as a sum of two cubes. The extreme example is p(x, y) = xy(x4 − y4), which has six such representations.
Original languageEnglish (US)
Title of host publicationAnalytic and Combinatorial Number Theory
Subtitle of host publicationThe Legacy of Ramanujan
EditorsGeorge E Andrews, Michael Filaseta, Ae Ja Yee
PublisherWorld Scientific
Pages583-608
ISBN (Electronic)9789811277382
ISBN (Print)9789811277368
DOIs
StatePublished - Sep 2024

Publication series

NameMonographs in Number Theory
Volume12
ISSN (Print)1793-8341

Keywords

  • Cubic forms
  • diophantine equations
  • sextic forms

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