Is language of binary representations regular?

Question

Let $bin(n)$ denote binary representation of an integer $n$. Let $L=\left\{bin(n^2):n\in\mathbb{N}\right\}$. Is $L$ a regular language?

Answer 1

In general, if $p$ is any polynomial, then $\{ \operatorname{bin}(p(n)) : n\in\Bbb N \}$ is regular if and only if $p$ is constant or first degree.

It is easy to prove that this case, where $p$ is the second-degree polynomial $n^2$, is irregular, using the Myhill-Nerode theorem.