Primes are an entire different set of numbers. But even primes numbers have their own categories. Some of these categories are, Fermat primes, Mersenne primes, Twin primes, Palindrome primes and Factorial primes. Soon you will see that they all have their own properties.
Fermat�s Primes
As said earlier, Fermat conjectured that every number of the form 2^N+1 where 2^n=N. Numbers of this form are called Fermat numbers. When a Fermat number is also prime, it�s called a Fermat prime. The only known primes of this sort are the first five Fermat numbers Fn are:
F0=3
F1=5
F2=17
F3=257
F4=65537
It is very liked that there are only finitely many Fermat primes. This is because Euler discovered that every Fermat number Fn with n greater than 2 has the form k(2n+2)+1. This way Euler showed that F5 equaled 128k+1. This way we would try to factor this with 257 and 641 because other numbers 129, 385 and 513 are not prime. This way, you can find that F5 divisible by 641.
Pepin's Test
Another way to check if Fermat numbers are prime is Pepin�s test. In 1877 Pepin proved the theorem:
Fn is prime if and only if:
3(Fn-1)/2= -1 (modulo Fn)
Here once again F4 can be proved to be a prime using this theorem.
F3= 22^3+1
F3=257
3^257-1/2
=11790184577738583171520872861412518665678211592275841109096961
---------------------------
-1 (modulo 257)
=256
11790184577738583171520872861412518665678211592275841109096961 (mod 257)
=256
256=256
Therefore F3 is prime.
Mersenne Numbers
Mersenne numbers are acquired by finding the nth power of 2 in which n is a prime number and subtracting 1 from the result
Mn=2^n-1
These numbers are prime if and only if n is a prime. The reason for that is if m divides n then 2m-1 will divide 2n-1 and it wouldn�t be a prime. Even though in some Mersenne numbers such as M23 where n is a prime, 2^23-1 does not turn out to be prime.
2^23-1
=8388607
8388607 is not a prime (47*178481)
This, just like Fermat numbers proves that not all Mersenne numbers are primes. But unlike Fermat numbers, where there is a restriction after F4, there is no restriction applied to Mersenne numbers. This is why Mersenne are the most widely researched primes in the world.