An unusual cubic representation problem

Andrew Bremner, Allan Macleod

Research output: Contribution to journalArticlepeer-review

2 Scopus citations

Abstract

For a non-zero integer N, we consider the problem of finding 3 integers (a, b, c) such that (formula presented). We show that the existence of solutions is related to points of infinite order on a family of elliptic curves. We discuss strictly positive solutions and prove the surprising fact that such solutions do not exist for N odd, even though there may exist solutions with one of a, b, c negative. We also show that, where a strictly positive solution does exist, it can be of enormous size (trillions of digits, even in the range we consider).

Original languageEnglish (US)
Pages (from-to)29-41
Number of pages13
JournalAnnales Mathematicae et Informaticae
Volume43
StatePublished - Jan 1 2014

Keywords

  • Cubic representation
  • Elliptic curve
  • Rational points

ASJC Scopus subject areas

  • Computer Science(all)
  • Mathematics(all)

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