Problem #496: Let $\alpha \in \mathbb{R}$ be irrational and $\epsilon>0$

Let $\alpha \in \mathbb{R}$ be irrational and $\epsilon>0$. Are there positive integers $x,y,z$ such that\[\lvert x^2+y^2-z^2\alpha\rvert <\epsilon?\]

SOLVED

Problem Statement

Let $\alpha \in \mathbb{R}$ be irrational and $\epsilon>0$. Are there positive integers $x,y,z$ such that\[\lvert x^2+y^2-z^2\alpha\rvert <\epsilon?\]

Categories: Number Theory Diophantine Approximation

Progress

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].

Source: erdosproblems.com/496 | Last verified: January 15, 2026

Problem Statement

Progress

Stay Updated