All Random Solved Random Open
Let $\alpha \in \mathbb{R}$ be irrational and $\epsilon>0$. Are there positive integers $x,y,z$ such that \[\lvert x^2+y^2-z^2\alpha\rvert <\epsilon?\]
Originally a conjecture due to Oppenheim. Davenport and Heilbronn [DaHe46] solve the analogous problem for quadratic forms in 5 variables.

This is true, and was proved by Margulis [Ma89].

Additional thanks to: Zachary Chase