Explicit solutions to the Oppenheim conjecture forindefinite ternary diagonal forms
Volume 205 / 2022
Acta Arithmetica 205 (2022), 287-307
MSC: Primary 11D75; Secondary 11J25 11J70.
DOI: 10.4064/aa210822-8-9
Published online: 24 October 2022
Abstract
We give a new proof of the Oppenheim conjecture for indefinite ternary diagonal forms of the type $x^2+y^2 -\alpha z^2$ where $\alpha $ is an irrational number. Our method is explicit in the sense that we are able to construct a solution to the problem and we obtain an effective bound on the solution. The method is geometrical and is based on continued fractions.
