CHAPTER 15
Definition of the sets of Fermat, Mersenne, perfect prime
numbers by the prime numbers formula ""
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.
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:
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.
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".
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)