Then We Characterize Primes and Composite Numbers Via Divisibility

Abstract

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 Fermat prime is a prime of the form Fn = 2 2 n + 1, where n is an integer 0; and a Fermat composite is a non prime number of the form Fn = 2 2 n + 1, where n is an integer 1; it is known that for every j 2 f0; 1; 2; 3; 4g, Fj is a Fermat prime, and it is also known that F5 and F6 are Fermat composite. We recall (see [11]) that a prime h is called a Sophie Germain prime, if both h and 2h + 1 are prime; the