Hostname: page-component-cd9895bd7-jkksz Total loading time: 0 Render date: 2024-12-26T06:13:07.245Z Has data issue: false hasContentIssue false
  • English
  • Français

On Unit Solutions of the Equation xyz = x + y + z in Not Totally Real Cubic Fields

Published online by Cambridge University Press:  20 November 2018

Liang-Cheng Zhang  and
Jonathan Gordon
Liang-Cheng Zhang
Affiliation:
Department of Mathematics, University of Illinois, 1409 West Green Street, Urbana, Illinois 61801, USA
Jonathan Gordon
Affiliation:
Department of Mathematics, University of Illinois, 1409 West Green Street, Urbana, Illinois 61801, USA
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

It is shown that the equation xyz = x+y+z has unit solutions in only four not totally real cubic fields: two fields which are real and two fields which are imaginary. These fields are then listed.

Keywords

11D2511R2711R16.
Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 34 , Issue 1 , 01 March 1991 , pp. 141 - 144
Copyright
Copyright © Canadian Mathematical Society 1991

References

1. Cassels, J. W. S., On a diophantine equation, Acta. Arith. 6 (1960), 47–52.Google Scholar
2. Mollin, R. A., Small, C., K. Varadarajan and P. G. Walsh. On unit solutions of the equation xyz = x + y + z in the ring of integers of a quadratic field, Acta. Arith. 48 (1987), 341–345.Google Scholar
3. Sierpinski, W., On some unsolved problems of Arithmetics, Scripta Math. 25 (1960), 125–136.Google Scholar
4. Sierpinski, W., Remarques sur le travail de M. J. Cassels, W. S. “On a diophantine equation “, Acta. Arith. 6 (1961), 469–471.Google Scholar
5. Zhang, L. C. and Gordon, J. R., On unit solutions of the equation xyz = x + y + zina number field with unit group of rank 1, to appear.Google Scholar