The density of an electron-hole pair in pure germanium is $3 \times 10^{16} \, \text{m}^{-3}$ at room temperature. On doping with aluminium, the hole density increases to $4.5 \times 10^{22} \, \text{m}^{-3}$. What is the electron density (in $\text{m}^{-3}$) in doped germanium?