they wont ever run out of names...
every name can contain 13 chars max and minimum 3 chars. there are 37 different chars (26 letters, 10 numbers and the _ )
so the the amount of possible names = 37^13+37^12+37^11+37^10+37^9+37^8+37^7+37^6+37^5+37^4+37^3 = 250335035999861340251
correct me if im wrong

im not exactly sure how you would mathmaticaly factor this in buy you havent taken into account that you can use capitals and lowercase letters to increase the possible names