If you're talking about storing it in computer memory during calculation, then no, it's written out. The fast MOD operation you list isn't used, it's an FFT using a weight factor that automatically does the mod operation while squaring a number. You start with S[0] = 4, S[i+1] = Si^2 - 2 MOD N. If S[N-2] = 0 then the number is prime.
So it's 74,207,279 squares using FFT, each likely more than 75 MiB since there will likely be lots of leading zeros (we're probably close to the maximum number of digits using double precision, round off errors dominate unless you start using really small bases and even greater-length FFTs).
That's not counting the two or three compositeness tests that GIMPS uses before starting the Lucas-Lehmer test to prove primality.
For several years I've been working on a program to test for primality of the Generalized Fermat numbers, using NTT instead of FFT. NTT is exact (integer math), and the GFNs likely are more "dense" with primes than the Mersenne Numbers. And they also have a faster form to test, although the primality test has a 50% chance of false negative, so you have to run it a few times to be sure a number isn't prime. That's where Mersenne Numbers are easier to test.