Prime Numbers

You just never know when knowledge is going to rise up and smack you in the face!
I have been doing research on computers, and this popped up!
I have a headache from trying to wrap my head around this concept......

So, a prime number is a whole number greater than 1, whose only two whole-number factors are 1 and itself.  The first few prime numbers are 2, 3, 5, 7, 11, 13, 17, 19, 23, and 29.

As we proceed in the set of natural numbers N = {1, 2, 3, ...}, the primes become less and less frequent in general.  However, there is no largest prime number.  For every prime number p, there exists a prime number p' such that p' is greater than p.  This was demonstrated in ancient times by the Greek mathematician Euclid.

A computer can be used to test extremely large numbers to see if they are prime.  But, because there is no limit to how large a natural number can be, there is always a point where testing in this manner becomes too great a task even for the most powerful supercomputers.

Various algorithms have been formulated in an attempt to generate ever-larger prime numbers.  These schemes all have limitations.
A Mersenne prime must be reducible to the form 2 ⁿ - 1, where n is a prime number. The first few known values of n that produce Mersenne primes are where n = 2, n = 3, n = 5, n = 7, n = 13, n = 17, n = 19, n = 31, n = 61, and n = 89.
A Fermat prime is a Fermat number that is also a prime number . A Fermat number F _n is of the form 2 ^m + 1, where m is the n th power of 2 (that is, m = 2 ⁿ , where n is an integer).

Does this make ANY sense????
Not to me.....

A Mersenne (also spelled Marsenne) prime is a specific type of prime number. It must be reducible to the form 2 n - 1, where n is a prime number. The term comes from the surname of a French monk who first defined it. The first few known values of n that produce Mersenne primes are where n = 2, n = 3, n = 5, n = 7, n = 13, n = 17, n = 19, n = 31, n = 61, and n = 89.

With the advent of computers to perform number-crunching tasks formerly done by humans, ever-larger Mersenne primes (and primes in general) have been found. The quest to find prime numbers is akin to other numerical searches done by computers. Examples are the decimal expansions of irrational numbers such as pi (the circumference-to-diameter ratio of a circle) or e (the natural logarithm base). But the 'next' prime is more difficult to find than the 'next' digit in the expansion of an irrational number.

It takes the most powerful computer a long time to check a large number to determine if it is prime, and an even longer time to determine if it is a Mersenne prime. For this reason, Mersenne primes are of particular interest to developers of strong encryption methods.

In August 2008, Edson Smith, a system administrator at UCLA, found the largest prime number known to that date. Smith had installed software for the Great Internet Mersenne Prime Search (Gimps), a volunteer-based distributed computing project.  The number (which is a Mersenne prime) is 12,978,189 digits long. It would take nearly two-and-a-half months to write out and, if printed, would stretch out for 30 miles.

Using computers, mathematicians have not yet found any Fermat primes for n greater than 4. So far, Fermat's original hypothesis seems to have been wrong. The search continues for Fermat numbers F n that are prime when n is greater than 4.