elements of order 3 the group $R^2/Z^2$

Question

the group acts on addition is defined by the equivalence of all the reals that differ by squared integers.

I have $3r^2 = M^2-3n^2$ but don't know how to proceed?

Answer 1

The comment is right. $\mathbb{R}^2 / \mathbb{Z}^2$ is the quotient of the plane by its integer lattice points. That is, we have $(x,y) \sim (u,v)$ if and only if there are $n,m \in \mathbb{Z}$ such that $x - u = n$ and $y - v = m$.

Thus, an element $(x,y) \in \mathbb{R}^2 / \mathbb{Z}^2$ has order $k$ if $kx - 0 \in \mathbb{Z}$ and $ky - 0 \in \mathbb{Z}$.

Think you can take it from there?