This question already has an answer here:
Let $p$ be a prime number such that $p \equiv 3 \pmod 4$. Show that $x^2 \equiv -1 \pmod p$ has no solutions.
I noticed that this is equivalent to proving $x^2\equiv 2(2k+1) \pmod p$. I also know that $x^2 \neq 2(2k+1)$. But I still can't prove it. Any help would be greatly appreciated.