Points at rational distances from the vertices of certain geometric objects

Andrew Bremner, Maciej Ulas

Research output: Contribution to journalArticlepeer-review

1 Scopus citations

Abstract

We consider various problems related to finding points in Q2 and in Q3 which lie at rational distance from the vertices of some specified geometric object, for example, a square or rectangle in Q2, and a cube or tetrahedron in Q3. In particular, as one of several results, we prove that the set of positive rational numbers a such that there exist infinitely many rational points in the plane which lie at rational distance from the four vertices of the rectangle with vertices (0, 0), (0, 1), (a, 0), and (a, 1), is dense in R+.

Original languageEnglish (US)
Pages (from-to)104-133
Number of pages30
JournalJournal of Number Theory
Volume158
DOIs
StatePublished - Jan 1 2016

Keywords

  • Elliptic surfaces
  • Rational distances set
  • Rational points

ASJC Scopus subject areas

  • Algebra and Number Theory

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