<s>[INST] If $n$ is an integer then $4$ does not divide $n^2 - 3$ Having some trouble proving this. Was going to attack with using the contrapositive of this statement but can't seem to show that $n$ isn't an integer. 
 [/INST] You just need to look at two cases: $n$ is odd and $n$ is even.


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*If $n$ is even, then $n \equiv 0 \textrm{ or } 2 \pmod 4$ but $n^2 \equiv 0 \pmod 4$ regardless, which means that $n^2 - 3 \equiv 1 \pmod 4$, and therefore $$\frac{n^2 - 3}{4} = m + \frac{1}{4},$$ where $m$ is some integer. For example, $n = 10$ gives us $24.25$.

*If $n$ is odd, then $n \equiv 1 \textrm{ or } 3 \pmod 4$ but $n^2 \equiv 1 \pmod 4$ regardless, which means that $n^2 - 3 \equiv 2 \pmod 4$, and therefore $$\frac{n^2 - 3}{4} = m + \frac{1}{2},$$ where $m$ is some integer. For example, $n = 11$ gives us $29.5$.

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