Assume that the energy released upon the recombination of a...

The energy released upon the recombination of a hole-electron pair corresponds to the band gap energy $E_g$ of the semiconductor.

The relation between the band gap energy and the maximum emitted wavelength $\lambda$ is given by

$E_g = \dfrac{hc}{\lambda}$

Substituting $hc \approx 1240 \text{ eV} \cdot \text{nm}$ and $\lambda = 820 \text{ nm}$, we get

$E_g = \dfrac{1240}{820} \text{ eV}$

$E_g \approx 1.51 \text{ eV}$

The calculated band gap is approximately $1.5 \text{ eV}$.