How does GIMPS work and what are these iterations?

Question

I downloaded GIMPS today just out of curiosity and have been running it. On my machine it is checking $M_{52898149}=2^{52898149}-1$.

From what I could find on Wikipedia I suppose that GIMPS uses Lucas-Lehmer primality test which means it will pick $s_{52898147}$ term in Lucas Lehmer sequence and will see whether $M_{52898149}=2^{52898149}-1$ divides it or not?

Now my question is, what does the iterations that my software is doing, means? Here is a pic :

What does iteration $160000/52898149$ means here? I suppose, by the pattern, I must be getting iterations upto 5290 such iterations, and each one of them will take around a couple of weeks, which may increase with higher sequences. Can someone explain what this software is doing to check the primality of this Mersenne prime.

Also, why was I assigned this number, is everyone randomly assigned a prime number for which $M_p$ has not been verified/tested yet as in case of this number it's primality has been tested before as the main page of the site says that all exponents below 78 421 769 have been tested at least once, so I must be just verifying it, meaning most likely it won't come out as a prime, otherwise they must have had it tested on different types of software by now.
Also, can I choose my own prime exponent larger than $78421769$ to check the primality?

P.S.- If this question doesn't fit on this site, please direct me to appropriate place. Also, let me know if I should create a GIMPS tag?

Answer 1

Each one of those iterations is just squaring the number then put it in modulo of whatever your prime exponent is. Iteration 160000/52898149 simply means that your computer system has completed 160000 such iterations. ms/iter counts how many milliseconds are needed on average for the last 10000 iterations, and ETA estimates the time needed to finish the calculations using the current ms/iter.

A prime number exponent is randomly assigned to users using the software for the first time according to the assignment rules. Since M_52898149 is checked once using LL in 2012-12-17 and not checked twice, double-checking is commenced to make sure that the results(residue) of the first test is correct.

If you want to create custom assignments, such as to check the primality of larger prime exponents, you can use the Assignments tab under the Manual Testing column, then choose your number of workers available, number of jobs for a worker, your preferred work type and the (optional) exponent range which you get your exponents.

Answer 2

GIMPS works by doing sucessive layers of checking, to weed out simple cases.

Primes only : It's easily provable that if $d\mid n$ then $2^d-1\mid 2^n-1$ . So any number with factors, is out of the running to be a Mersenne prime exponent.

Special factor form (Trial factoring) : It's easily provable (by Fermat and Lagrange) that if $q\mid 2^p-1$ then $p\mid q-1$ . So any factors of $2^p-1$, are expressible as form $q=2kp+1$ . As 2 is a quadratic residue they are provably congruent to $1,7 \bmod 8$ . Depth checked typically exceeds 80 bits

Gerbicz PRP : This base 3 PRP test is based on a reduced form of Lucas-Lehmer test. A candidate exponent that fails, is likely to fail the Lucas-Lehmer Test.

Lucas-Lehmer Test (LL) is a test devised by Lucas and Lehmer and starts with value $s_0=4$ and iteratively, applies the map $s_i\to s_i^2-2\bmod {2^p-1}$ until $i=p-2$ to check if it divides that iteration. If it does, then it confirms primality of the Mersenne number $2^p-1$