Explicit solutions to the Oppenheim conjecture for indefinite ternary diagonal forms

TL;DR

This paper proves the Oppenheim conjecture for a class of indefinite ternary diagonal forms with irrational coefficients, providing explicit solutions and effective bounds using a geometric approach based on continued fractions.

Contribution

It introduces an explicit, constructive method for solving the conjecture for specific forms, with effective bounds, using a geometric approach.

Findings

Proved the Oppenheim conjecture for forms $x^2 + y^2 - eta z^2$ with irrational $eta$.

Constructed explicit solutions with bounds.

Developed a geometric method based on continued fractions.

Abstract

We prove 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.