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Session 3265

Analysis of Prime Numbers

Alan Hadad*, Ghaith Hammouri* & Peter Shahdan** *University of Hartford / **North Carolina State University hammouri@hartford.edu

Abstract The problem of determining prime numbers has intrigued and baffled mathematicians throughout the ages. The Sieve of Eratosthenes is perhaps the simplest scheme ever devised to solve the problem. However, this is a mechanical scheme rather than analytical in nature, as is the method of dividing by all possible factors.

The method presented in this paper admittedly becomes cumbersome when dealing with large numbers, but it is analytical rather than mechanical in nature, and it is relatively easy to apply. It is based upon elementary algebraic principles, with special emphasis given to the arithmetic progression formula.

The formulae that yield all possible composite odd numbers and their factors will be derived, within certain convenient bounds. The remaining non-composite numbers within these bounds will therefore be prime numbers. A numerical example will be worked out in some detail to illustrate the process.

Four Divisions of Numbers Consider the set of non-negative integers and separate them into the following four subsets:

P0 = 0, 4, 8, 12, …, 4m

P1 = 1, 5, 9, 13, …, 4m + 1

P2 = 2, 6, 10, 14, …, 4m + 2

P3 = 3, 7, 11, 15,…, 4m + 3

It will be observed that P0 and P2 contain all even numbers; therefore, the only prime number present in these subsets is 2. Since prime numbers are the focus of interest, P0 and P2 will be discarded at this time, and attention focused upon P1 and P3, which contain all the odd numbers, and therefore, all additional primes.

Properties of the Elements of P1 and P3 Theorem 1: The product of any two members of P1 is also a member of P1.

“Proceedings of the 2003 American Society for Engineering Education Annual Conference & Exposition Copyright © 2003, American Society for Engineering Education”

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Hammouri, G., & Hadad, A. (2003, June), The Determination Of Prime Numbers Paper presented at 2003 Annual Conference, Nashville, Tennessee. 10.18260/1-2--11718