Question

Why does every perfect square no have a moduluar remainder of 2?

Answer 1

Is this the claim for all x, n, it's never the case that

`x^2 \equiv 2 \pmod{n}`

That can't be true, e.g. we know

`3^2 \equiv 2 \pmod{7}`

Try asking again, being more clear about what you're trying to prove.