Here's a hard one!

Suppose that $\begin{array}{@{}c@{\;}c@{}c@{}c} && &P_b \\ & \times & & P_b \\ \ = & & Q & P_b, \\ \end{array}$where $P$, $Q$, and $b$ represent three distinct digits 1-9. If $Q=\frac{P}{2}$ and $P$ is two less than $b$, what is the value of the digit $P$ ?