What puzzles me is how we've found the 12th largest number. My naive thinking would be that we slowly increment the highest prime rather than finding one in the middle of other primes.
It's due to the way Mersenne primes are used to simplify the search.
Mersenne primes are a subset of all possible primes, and take the form 2^n - 1. This makes it easy to search for a new higher one because you only need increase n and check the new calculated value of 2^n - 1.
However, increasing n means that you can skip over primes. For example, the first four Mersenne primes are: M2 = (2^2 - 1) = 4-1 = 3; M3 = (2^3 - 1) = 8-1 = 7, M5 = 31 and M7 = 127. We've quite quickly found a 'largest' prime of 127 but in the process we skipped over 11, 13, 17 etc.
The same thing has happened in this story but for a much larger number.