How does Band Gap and Refractive index relate to Wavelength?

[Nyfinscyf]

Homework Statement

Nobel prize for physics for blue laser diode using Gallium Nitride with band gap of 3.4 eV and a refractive index of 2.429.
Explain how these parameters determine at what wavelength a Gallium Nitride semiconductor will laser at.

Homework Equations

E=\frac{hc}{\lambda}
Blue wavelength $\approx 445 nm$

The Attempt at a Solution

This equation will give the energy band gap wavelength. But how does the refractive index factor into this? I know it changes the velocity that the light moves through the medium.
I found this article: https://www.quora.com/Whats-the-relation-between-bandgap-the-extinction-coefficient-and-the-index-of-refraction
But I'm still not sure how to answer the question.

 

[Simon Bridge]

When light goes from one medium to another, it's wavelength changes.

 

[Nyfinscyf]

Using the equations I can get:
E=\frac{hc}{\lambda}
\lambda=\frac{hc}{E}=\frac{1240~eV~nm}{3.4~eV} \approx 365~nm
The velocity of the light changes in the medium by v=\frac{c}{n} replacing c in the above by the velocity in the medium gives
\lambda=\frac{hv}{E}=\frac{hc}{nE}=\frac{1240~eV~nm}{(2.429)(3.4)~eV} \approx 150~nm

I don't see how this gives blue light of 445 nm.