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

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.

• What have you tried so far? Do you know what the quadratic residues modulo $72$ are? Aug 30 '18 at 22:16
• 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 Aug 30 '18 at 22:18
• No, the residues are those numbers like $100\equiv 28\mod 72$, the values in the interval $[0,71]$ such that some number squares to that value. Aug 30 '18 at 22:20
• Ok good call so I check and 27 is not a residue . Thank you Aug 30 '18 at 22:22
• A quicker way to check this particular example is to note that $27+72n=9(3+8n)$, giving that $3+8n$ must be square. Aug 30 '18 at 22:25

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.