CHAPTER 15

Definition of the sets of Fermat, Mersenne, perfect prime
numbers by the prime numbers formula "_{}"

_{}^{
}Definition of the set of Fermat's prime numbers

## 15.1. Some general facts concerning Fermat’s numbers (_{})

Fermat’s numbers are in "_{}" form with below formula:

_{}
(1)

First five terms of Fermat’s numbers sequence are:

_{}
(2)

that all of them are prime numbers, Fermat thought that all
of (1) numbers are prime numbers, but Euler in 1732 proved that "_{}" is
composite, because:

_{}

Two "_{}" factors are prime number
and therefore Fermat’s guess disproved it. Till now, prime number except (2)
numbers don’t find in Fermat’s numbers. And most people guess that sequence of
Fermat’s prime number is finite. Some other guesses that rareness of prime
numbers in Fermat’s numbers is because of increasing speed of these numbers.

## 15.2. Definition of the set of Fermat’s prime numbers

by the prime numbers Formula

If we consider _{} as a prime number_{}:

_{} ;

_{}
(3)

According to relation (3), set of Fermat’s prime numbers
define:

(Set of Fermat’s prime numbers)

_{}

It is guessed that set of Fermat’s prime number (IF) is
finite. Considering a kind of factorization of "_{}" by previous
Fermat’s numbers, this guess becomes stronger:

_{}

## 15.3. Some general facts about Mersenne's numbers and

even perfect numbers and the relation between them

Reviewing the Euclid's theorem and Euler’s theorem, existing
infinite even perfect numbers is equal with existing infinite prime numbers in "_{}" form.
These kinds of prime numbers are called as Mersenne’s prime numbers. After this
French monk in 17-th century, searched about these numbers, these numbers are
famous as this name.

### 15.3.1. Definition

Consider that "*n*" be a positive correct
number. n-th Mersenne’s number is correct number _{}. If _{} be a prime number, _{} called as a
Mersenne’s prime number. Trying to factorization the Mersenne's numbers, Fermat
showed that if "*p*" be prime, then every prime divisor of
"_{}"
has residual "1" in dividing by "*p*".

## 15.4. Definition of the sets of Mersenne, even perfect

## prime numbers by the prime numbers formula

If we consider the odd number _{} as a prime number:

_{} ;

_{}
(1)

According to relation (1), set of Mersenne’s prime numbers
is defined as below:

(Set of Mersenne's
prime numbers) _{}

We know that with obtaining a new Mersenne’s prime number,
we obtain a new perfect number. Because general formula of these even perfect
numbers is:

_{}
(2)

According to formula (2) and set of Mersenne’s prime
numbers, set of even perfect number is defined as:

(Set of even perfect numbers)

_{}