Equivalence Relation on Integers Modulo 5 induced by Squaring/Number of Equivalence Classes

Theorem
Let $\beta$ denote the relation defined on the integers $\Z$ by:
 * $\forall x, y \in \Z: x \mathrel \beta y \iff x^2 \equiv y^2 \pmod 5$

We have that $\beta$ is an equivalence relation.

The number of $\beta$-equivalence classes is $3$:


 * $\eqclass 0 \beta$, $\eqclass 1 \beta$, $\eqclass 4 \beta$

Proof
That $\beta$ is an equivalence relation is proved in Equivalence Relation on Power Set induced by Intersection with Subset.

The set of residue classes modulo $5$ is:
 * $\set {\eqclass 0 5, \eqclass 1 5, \eqclass 2 5, \eqclass 3 5, \eqclass 4 5}$

Then:

Thus we have that:
 * $\eqclass 1 5^2 = \eqclass 4 5^2 = \eqclass 1 \beta$
 * $\eqclass 2 5^2 = \eqclass 3 5^2 = \eqclass 4 \beta$

Hence the result.