Can an integer of a particular form be a perfect square?

Question

Can an integer of the form $27 + 72 n$, where $n \in \mathbb{Z}$, be a perfect square? I just checked the first $100$ squares... would the quad residues be all the numbers relatively prime to $72$? So $27$ would be non residue.

Answer 1

Consider $72n+27\pmod{4}$. First check that(see here or here for proof), every square number can be of the form $4k,4k+1$. Hence, any square number $\pmod{4}$ have to be one of those forms. On the other hand $$72n+27\equiv 3\pmod{4}$$ Hence, there is no integer $n$ for which $72n+27$ is a square number.