Right turn signals flashing rapidly

That usually happens when a bulb or wire is shorted out causing excessive current and heat, thus making the turn signal flasher open and close faster. Are you sure even though no bulbs are burnt out, that they all are actually working, even if flashing fast, make sure at least that they all are working. That includes the indicator on the instrument cluster, also.

I honestly have no idea then why they are flashing fast.

Perhaps you have a bulb that is just bad or weak but still working. Either try swapping them from side to side and see if the problem switches to the left side or just replace all the turn signal bulbs on the right side.