On an equality of $x \bmod 1$

Question

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

For what values of $x \in \Bbb R$ is $x^2 \bmod 1 = (x \bmod 1)^2$? Naturally if $|x| < 1$ we have the equality. However, there are some numbers like $x \in 0.5r \pm \Bbb N$ where $r \in \{0, 1\}$ which satisfy this equality. What is the entire solution set?

What about for higher powers?

Answer 1

Written $$x=[x]+\{x\}$$ we have $$\{x^2\}=\{[x]^2+2[x]\{x\}+\{x\}^2\}=\{2[x]\{x\}+\{x\}^2\}$$

So the necessary and sufficient condition for $\{x^2\}=\{x\}^2$ is that $2[x]\{x\}$ is an integer.