# IJPAM: Volume 85, No. 5 (2013)

**GENERATION OF PRIME NUMBERS FROM**

ADVANCED SEQUENCE AND DECOMPOSITION METHODS

ADVANCED SEQUENCE AND DECOMPOSITION METHODS

^{1}, Munwar Ali Zardari

^{2}, Youssef A. Y. Rezk

^{3}

^{1}Mathematics Department Science College

Alfaisal University

RIYADH 11533

^{2,3}Department of Computer and Information Sciences

Universirti Teknologi PETRONAS

Bandar Seri Iskandar, 31750, Tronoh, Perak, MALAYSIA

**Abstract. **The generation of prime numbers cause the use of data encryption techniques, as major primal is needed for the generation of pairs of keys. This paper proposes two prime number generation methods which are based on sequence of prime numbers and decomposition of a prime number". In these proposed methods, co-prime and decomposition properties of prime number are used. By considering the co-prime property, any sequence of consecutive primes are coupled together to generate their co-prime numbers. Let n be a number which is co-prime with a sequence of m prime numbers, which can be expressed as:

n=\left ( \prod\limits_{i=1}^{m} p_{i}\right ).k+V_{mod\left ( \prod\limits_{i=1}^{m}p_{i} \right )},

where m is a sequence of prime numbers and p_{i} be the i-th prime number, with p_{1}=1. In the second approach i.e decomposition of prime number, the objective is to generate new prime numbers using decomposition of primes. For all integer numbers represented by X less than p^{2}_{m+1} are prime numbers, it is shown in the following formula.

X\left ( i,w,r \right )=\sum\limits_{j=1}^{i}\left ( \prod\limits_{l\varepsilon I_{i}^{j}} p_{l}^{w_{i}^{d}\left ( l \right )}\left ( -1 \right)^{r_{i}^{d}(l)}\right).

**Received: **August 11, 2012

**AMS Subject Classification: **11XX, 11YXX, 11Y11

**Key Words and Phrases: **prime numbers, twin primes, co-prime number, prime number decomposition.

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**DOI: 10.12732/ijpam.v85i5.2**

International Journal of Pure and Applied Mathematics

**How to cite this paper?****Source:****ISSN printed version:**1311-8080

**ISSN on-line version:**1314-3395

**Year:**2013

**Volume:**85

**Issue:**5