No, the RNG doesn't output successive digits of the prime. That would be period 19937. It's period 2^19937. The random number is 10^6000 digits long.

> 10 digits long[...]100 digits long[...]1000 digits long[...]10^googol digits long I think.

Not 10^googol digits. Googol digits long. Except you don't even need that much, because die rolling is base 6.