<s>[INST] Into how many equivalences classes does $R$ partition $\mathbb{Z}$? Let $R= \{ (a,b) \in\mathbb{Z}\times\mathbb{Z} \mid a^2\equiv b^2 \bmod 7\}$.
Into how many equivalences classes does $R$ partition $\mathbb{Z}$?
My best guess is that there are $7$ equivalence classes: $a^2-b^2$ has remainder $0 \bmod 7$, $a^2-b^2$ has remainder $1 \bmod 7$, $a^2-b^2$ has remainder $2 \bmod 7, \ldots, a^2-b^2$ has remainder $6 \bmod 7$.
Am I misunderstanding equivalence class partitions?
 [/INST] The fact that $7$ is prime is relevant here.  A number can have more than two square roots modulo a composite number, but modulo a prime number, each number can have only two square roots.  Thus $a\equiv b\bmod 7$ if and only if $a\equiv\pm b\bmod 7$.
\begin{align}
0^2 & \equiv (-0)^2 \equiv 0 \\
1^2 & \equiv (-1)^2 \equiv 1 \\
2^2 & \equiv (-2)^2 \equiv 4 \\
3^2 & \equiv (-3)^2 \equiv 2
\end{align}
Since the only congruence classes modulo $7$ are $0,\pm1,\pm2,\pm3$, this list is complete.  There are $7-1$ non-zero congruence classes and hence $(7-1)/2$ non-zero squares.  Each non-zero square coresponds to one of the equivalence classes you're looking for, $0$ corresponds to another.
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