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Explicit solutions to the Oppenheim conjecture forindefinite ternary diagonal forms

Volume 205 / 2022

Youssef Lazar Acta Arithmetica 205 (2022), 287-307 MSC: Primary 11D75; Secondary 11J25 11J70. DOI: 10.4064/aa210822-8-9 Published online: 24 October 2022


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.


  • Youssef LazarDepartment of Mathematics and Statistics
    College of Science
    Imam Mohammad ibn Saud Islamic University (IMSIU)
    Riyadh, Kingdom of Saudi Arabia

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