<s>[INST] There is no square number that is $3 \mod 4$ Prove that the following cannot be true:
There is no square number that is $3 \mod 4$ 
$x^2 \equiv 3 \mod 4$, I started with examples:
$1^2 \mod 4=1$
$2^2 \mod 4=0$
$3^2 \mod 4=1$
I am more interested in the proof but Im not seeing how to do it.
 [/INST] If $x$ is even then $x=2k$ so the square of $x$ is equal $4k^2$ and leaves no remainder when divided by $4$.
If $x$ is odd then $x=2k-1$ so the square of $x$ is equal $4k^2-4k+1$ and leaves a remainder $1$ when divided by $4$.
So we have either $x^2 \equiv 0 \mod 4$ or $x^2 \equiv 1 \mod 4$.
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