# IJPAM: Volume 57, No. 3 (2009)

**SPECIAL SEMI-GROUPS IN**

ELEMENTARY NUMBER THEORY

ELEMENTARY NUMBER THEORY

Die Eidgenössische Technische Hochschule (ETH) - Zürich

(Swiss Federal Institute of Technology - Zurich)

Automatic Control Laboratory

ETL I 11, Physikstrasse 3, Zurich, 8092, SWITZERLAND

e-mail: mansour@control.ee.ethz.ch

url: www.control.ethz.ch/info/people/mansour

**Abstract.**A new definition of a prime number is given, showing also its relation with the solution of a quadratic equation. Different trivial classifications of odd natural numbers as well as a new grouping of them, in two commutative semi-groups and their cross mapping into a set closed under multiplication, is given. In this grouping the prime numbers are distributed between the two semi-groups. The first semi-group has, as generators the primes which are the sum of two squares, one odd and the other even. The second semi-group has as generators the rest of the primes. The third set is generated by cross multiplication of the first two semi-groups. Combining a trivial classification with this grouping we may get a simplification in checking primness of some natural numbers and some information on the factorization. Also some extensions of Wilson Theorem is presented using these classifications and two new terms are introduced; the single number of sos prime and its multiplication factor.

**Received: **November 14, 2009

**AMS Subject Classification: **11Axx, 11Exx,

**Key Words and Phrases: **number theory, prime number, sos prime

**Source:** International Journal of Pure and Applied Mathematics

**ISSN:** 1311-8080

**Year:** 2009

**Volume:** 57

**Issue:** 3