TITLE: How are large prime numbers found?
QUESTION [8 upvotes]: So I've found multiple ways of determining if a number is prime but I was wondering how they pick a number in the first place. For example the GIMPS project I assume is just trying different numbers to square 2 by but what about other methods? Or is it just a process of testing each and every unknown number?

REPLY [4 votes]: By using the power of distributed computing to test potential prime numbers of specific forms.
The most famous is of course the search for Mersenne primes, which are of the form $2^p - 1$, where $p$ is also prime. That right there helps to narrow the search. For example, we immediately know that $2^{91} - 1$ is not prime, because it's divisible by $2^7 - 1$ and $2^{13} - 1$.
But not all primes give primes as Mersenne exponents. For example, $2^{23} - 1 = 47 \times 178481$. There are ways to detect a lot of primes like $23$, further narrowing down the search. According to the Great Internet Mersenne Prime Search (GIMPS), the slower computers in the network are assigned the task of ferreting out primes like $23$, so that the faster computers can concentrate on testing primes likelier to be Mersenne prime exponents.
Once in a while, however, large prime numbers are found almost by accident. For instance, if $a_0 = 20615674205555510$ and $a_1 = 3794765361567513$, the sequence defined by $a_n = a_{n - 2} + a_{n - 1}$ for $n > 1$ contains no primes. However, if you swap $a_0$ and $a_1$, then $a_{138} = 439351292910452432574786963588089477522344721$ is prime.
On November $9$, $2005$, Eric W. Weisstein announced that $a_{156075}$ is also prime. See Mathworld for more details. These are small numbers compared to the largest known Mersenne prime, and even smaller compared to prime numbers large beyond our comprehension.