# On an equality of $x \bmod 1$ (the non-diagonal case)

Let $x \bmod 1\in \Bbb R$ be fractional part of $x \in \Bbb R$.

For what pairs of $(x,y) \in \Bbb R^2$ with $x\neq y$ is $x^2 \bmod 1 = (y \bmod 1)^2$? The case of $x=y$ was answered in On an equality of $x \bmod 1$.

• This question is kind of wierd. You can always choose $y=\sqrt{\{x^2\}}+a$, where $a$ is an integer. Aug 18 '13 at 14:41