<s>[INST] show that $n + \lfloor \sqrt{n} + \frac{1}{2}\rfloor$ is never a perfect square for all positive integers $n$ show that $n + \lfloor \sqrt{n} + \frac{1}{2}\rfloor$ is never a perfect square for all positive integers $n$
I am thinking of a proof by contradiction by assuming $n + \lfloor \sqrt{n} + \frac{1}{2}\rfloor = k^2 \implies \lfloor \sqrt{n} + \frac{1}{2}\rfloor  = k^2-n $
 [/INST] Let $t=\lfloor\sqrt{n}+\frac{1}{2}\rfloor$, so $t-\frac{1}{2}\le\sqrt{n}<t+\frac{1}{2}$.
If $n+t=k^2$ for some integer $k$, 
then $t^2-t+\frac{1}{4}\le n<t^2+t+\frac{1}{4}\implies t^2+\frac{1}{4}\le n+t<t^2+2t+\frac{1}{4}<(t+1)^2\implies$
$t^2<k^2<(t+1)^2$, which gives a contradiction.
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