Yesterday afternoon I met professor Halfbrain at a coffee place. The professor looked very tired. He told me that he hadn't slept for many days, spending his time with writing lots of zeros and with computing square roots.
"Writing lots of zeros?", I asked him surpised. It turned out that the professor had first written down the integer $49$ and then inserted one $0$ in the middle.
"Computing square roots?", I asked him. It turned out that the professor had noticed that his first number $409$ wasn't a perfect square (whereas the number $49=7^2$ is a well-known square). So he had inserted another $0$, and noticed that $4009$ wasn't a perfect square either. He inserted one more $0$, but $40009$ is not square. And so on. After inserting many many further zeros into the number, professor Halfbrain finally managed to detect a square number.
Let us denote by $H(k)$ the integer whose decimal representation is of the form $4000\cdots009$ with exactly $k$ zeros between the digits $4$ and the $9$.
Question: What is the smallest integer $k\ge1$, for which $H(k)$ is a perfect square? Does such a $k$ really exist, or has the professor once again made one of his notorious mathematical blunders?