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• ### 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 .

[[ Prime numbers are well kwown ( see  or ) and we recall that a composite number is a non prime number. We recall (see  or  or  or  or  or  or  or  or  or ) 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 ) 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  or  or  or  or  or  or ) 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  or ) 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  or  or  or  or  or  or  or  or  or ) 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  or ). 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  or )]; and perfect numbers are known for some integers >33550336 ]].

• ### References

Dickson. Theory of Numbers (History of Numbers. Divisibity and primality) Vol 1. Chelsea Publishing Company. New York , N.Y (1952). Preface.III to Preface.XII. Dickson. Theory of Numbers (History of Numbers. Divisibity and primality) Vol 1. Chelsea Publishing Company. New York , N.Y (1952)
1. G.H Hardy, E.M Wright. An introduction to the theory of numbers. Fith Edition. Clarendon Press. Oxford.
2. Ikorong Anouk Gilbert Nemron An alternative reformulation of the Goldbach conjecture and the twin primes conjecture. Mathematicae Notae. Vol XLIII (2005). 101 - 107.
3. Ikorong Anouk Gilbert Nemron. Around The Twin Primes Conjecture And The Goldbach Conjecture I. Tomul LIII, Analele Stiinti_ce Ale Universitatii "Sectiunea Matematica". (2007). 23 - 34.
4. Ikorong Anouk Gilbert Nemron. An original symposium over the Goldbach conjecture, The Fermat primes, The Fermat composite numbers conjecture, and the Mersenne primes conjecture .Mathematicae Notae. Vol XLV (2008). 31 - 39.
5. Ikorong Anouk Gilbert Nemron. An original abstract over the twin primes, the Goldbach Conjecture, the Friendly numbers, the perfect numbers, the Mersenne composite numbers, and the Sophie Germain primes. Journal of Discrete Mathematical Sciences And Cryptography; Taru Plublications; Vol.11; Number.6, (2008). 715 - 726.
6. Ikorong Anouk Gilbert Nemron. Playing with the twin primes conjecture and the Goldbach conjecture. Alabama Journal of Maths; Spring/Fall 2008. 47 - 54.
7. Ikorong Anouk Gilbert Nemron. A Glance At A Di_erent Kinds Of Numbers. International Journal Of Mathematics And Computer Sciences. Vol.4, No.1, 2009. 43 - 53.
8. Ikorong Anouk Gilbert Nemron. Runing With The Twin Primes, The Goldbach Conjecture, The Fermat Primes Numbers, The Fermat Composite Numbers, And The Mersenne Primes; Far East Journal Of Mathematical Sciences; Volume 40, Issue 2, May2010, 253 - 266.
9. Ikorong Anouk Gilbert Nemron. Speech around the twin primes conjecture, the Mersenne primes conjecture, and the Sophie Germain primes conjecture; Journal of Informatics And Mathematical Sciences; Volume 3, 2011, No 1, 31 - 40.
10. Ikorong Anouk Gilbert Nemron. A Proof Of Eight Famous Number Theory Problems And Their Connection To The Goldbach Conjecture. South Asian Journal of Mathematics; Vol1 (3); 2011, 87 - 105.
11. Ikorong Anouk Gilbert Nemron. Nice Rendez Vous With Primes And Composite Numbers. South Asian Journal Of Mathematics; Vol1
12. (2); 2012, 68 - 80.
13. Ikorong Anouk Gilbert Nemron. Placed Near The Fermat Primes And The Fermat Composite Numbers. International Journal Of Research In Mathematic And Apply Mathematical Sciences; Vol3; 2012, 72 - 82.
14. Ikorong Anouk Gilbert Nemron. Around Prime Numbers And Twin Primes. Theoretical Mathematics And Applications; Vol3, No1; 2013, 211 - 220.
15. Ikorong Anouk Gilbert Nemron. Invited By The Mersenne Primes, The Mersenne Composite Numbers, And The Perfect Numbers. Will appear in Theoretical Mathematics And Applications; Vol3, No2; 2013.
16. Ikorong Anouk Gilbert Nemron. Meeting With Primes And Composite Numbers. Will Appear In Asian Journal of Mathematics and Applications; 2013.  Maria Suzuki. Aternative Formulations of the Twin Primes Problem. The American Mathematical Monthly. Volume 107. Juanuary 2000. 55 - 56.
17. P. Dusart. Inegalits Explicites pour (X), _(X), _(X), et les Nombres Premiers. C.R.Math.Rep.Acad.Sci. Canada. Vol. 21 (1). 1999.53 - 59.
18. Paul Ho_man. Erdos, l'homme qui n'aimait que les nombres. Editions Belin, (2000). 30 - 49.
19. Paul Ho_man. The man who loved only numbers. The story of Paul Erdos and the search for mathematical truth. 1998. 30 - 49.
20. O. Ramare et R. Rumely. Primes in Arithmetic Progressions. Math.Comp. (213). 65. 1996. 397 - 425. View Download Article ID: 1587 DOI: 10.14419/ijams.v2i1.1587