Is $x^2 \equiv y^3 \pmod 4$ an equivalence relation on the set of all integers?

Question

I know $x \equiv y \pmod n$ is an equivalence relation in general but does squaring or cubing change that?

Answer 1

Yes it does change things for several reasons, one being the fact that there are integers $x$ such that $x^2\not\equiv x^3\pmod 4$.

Answer 2

Define the relation, for $x,y\in\mathbb Z,$ $x\sim y$ if and only if $x^2\equiv y^3\pmod 4.$

One axiom required for an equivalence relation is that for all $x\in\mathbb Z,$ $x\sim x.$ This is certainly not true for our relation, as $3\not\sim3$ ($3^2\equiv 1\not\equiv 3^3\equiv 3\pmod 4$.)