Then We Characterize Primes and Composite Numbers Via Divisibility

In this paper, we show a Theorem which helps us to characterize prime numbers and composite numbers via divisibility; and we use the characterizations of primes and composite numbers to characterize twin primes, Mersenne primes, even perfect numbers, Sophie Germain primes, Fermat primes, Fermat composite numbers and Mersenne composite numbers (we recall that a logic (non recursive) proof of problems posed by twin primes, Mersenne primes, perfect numbers, Sophie Germain primes, Fermat primes, Fermat composite numbers and Mersenne composite numbers, is given in [12].

[[ Prime numbers are well kwown ( see [15] or [19]) and we recall that a composite number is a non prime number. We recall (see [1] or [2] or [3] or [6] or [9] or [10] or [12] or [13] or [14] or [17]) that a {\it{Fermat prime }} is a prime of the form F_{n}=2^{2^{n}}+1, where n is an integer \geq 0; and a {\it{Fermat composite}}  is a non prime number of the form F_{n}=2^{2^{n}}+1, where n is an integer \geq 1; it is known that for every j\in \lbrace 0, 1,2,3,4\rbrace, F_{j} is a Fermat prime,  and it is also known that F_{5} and F_{6} are Fermat composite. We recall (see [11]) that a prime h is called a {\it{Sophie Germain prime}}, if both h and 2h+1 are prime; the first few Sophie Germain primes are 2, 3,5,11,23,29,41, ...; it is easy to check that 233 is a Sophie  Germain prime. A {\it{Mersenne prime }} (see [6] or [10] or [11] or [16] or [19] or [20] or [21]) is a prime of the form M_{m}=2^{m}-1, where m is prime; for example M_{13} and M_{19} are Mersenne prime. A Mersenne composite ( see [7] or [9])  is a non prime number of the form M_{m}=2^{m}-1, where m is prime; it is known that M_{11} and M_{67} are Mersenne composite. We also recall (see [4] or [5] or [7] or [8] or [10] or [11] or [18] or [19] or [20] or [22]) that an integer t is a twin prime,

if t is a prime \geq 3 and if t-2 or t+2 is also a prime \geq 3; for example, it is easy to check that (881,883) is a couple of twin primes. Finally, we recall that Pythagoras saw {\it{perfection}} in any integer that equaled the sum of all the other integers that divided evenly into it (see [7] or [9]). The first perfect number is 6. It's evenly divisible by 1, 2, and 3, and it's also the sum of 1, 2, and 3,

[note 28, 496 and  33550336 are also  perfect numbers (see [7] or [9])]; and perfect numbers are known for some integers  >33550336 ]].