How to find all naturals $n$ such that $\sqrt{1 {\underbrace{4\cdots4}_{n\text{ times}}}}$ is an integer?
How to find all naturals $n$ such that $\sqrt{1\smash{\underbrace{4\cdots4}_{n\text{ times}}}}$ is an integer?
For $n \geq 4$ your number is equal to $4444$ modulo $10000$, and in particular modulo $16$. If it were a square, then $4444$ would be a square modulo $16$, implying $1111$ is a square modulo $4$. But $1111=3$ mod $4$, contradiction.