Photons have no mass. So, why does $E = pc$ hold?

Question

It's a somewhat theoretical question. In special relativity, The energy of a photon is given by $E = pc$. But, my argument is that, since photons have no mass, how can they have a momentum $p$? The energy $E$ turns out to be 0 always. So, why does this equation hold?

Answer 1

The relativistically correct relation between momentum $p$ and velocity $v$ is $$c^2 p = E v$$ This holds for non-relativistic massive particles ($E = m c^2$ and hence $p=m v$) as well as for massless particles like photons ($v = c$ and therefore $p=E/c$).

Answer 2

Momentum in this case is: $p = h / \lambda$ for a massless particle. The momentum is related to the De Broglie wavelength of the particle with this formula. If you plug it in the equation you have stated you will get back the Energy equation of a massless particle:

$E = hc/\lambda = hf$