Feature Article of Thursday, 20 December 2012

**Columnist: **Sakyi, Kwesi Atta

Overcoming Corruption in Ghana-A New Dimension-Towards a Model-Part 2

Sakyi’s Elimination Sieve Model-Towards the Generation of Twin Primes

By Kwesi Atta Sakyi

16th December 2012

Prime numbers are numbers which have no factors and which are only divisible by 1 and themselves. Most prime numbers can be generated by the term 2n – 1, except those terms which are divisible by 3, 5 and 11.

- Twin primes are prime numbers which follow each other consecutively and are separated by a difference of 2 i.e, (3,5), (5,7), (11,13), (17,19), (29,31), (41,43), (59, 61), (71,73), (89,91) _ _ _ _

Since corruption takes place between two people or two partners, it has been likened to the sex act which involves two people. Hence, the model of twin primes being examined here as sexy primes.

- (1). Consider the arithmetic series 1,7,13,19,25,31,31,37,43, which has a common difference of 6. The general term is 6n – 5. This series has to be sieved by eliminating the series generated by 30n – 5 i.e. 25,55,85,115………..n i.e 5 times the series 5,11,17,23…..

- We also have to sieve it further by eliminating also the squares of the series, 5,11,17,23,29…………n i.e. 25, 121, 289, 529, 841……..n. Also eliminate 11 times the series 5, 11, 17, 23, 29, 35……n. (2). Consider again the series 5, 11, 17, 23, 29, 35…..n. The series has a common difference of 6. The general term is 6n – 1. Sieve the series except 11, by eliminating terms generated by 66n – 55, i.e. 11, 77, 143, 209….. (11 times the series 1, 7, 13, 19…..)

Again except 5, eliminate also the remaining terms of the series generated by the term 30n – 25 i.e, 5, 35, 65, 95, 125 (5 times the series 1,7,13,19….). Eliminate also the squares of the series. The remainder, are prime numbers.

- Combine the remaining numbers under (1) and (2) and you have all the twin prime numbers.

- One set of twin prime numbers can therefore be generated by the terms 30n – 1 and 30n + 1. Another set, by the terms 30n – 19 and 30n – 17. The first 9 twin prime sets are:-

We can discern a pattern on the right where

5, 7 13, 43, 73, 103………..n forming an

11, 13 arithmetic series. We also have 31, 61, 91,

17, 19 121…. forming another series but 121

29, 31 is not prime. On the left, we find the series

41, 43

59, 61 11, 41, 71, 101, 131… and 29, 59,89,119…

71, 73 all with common difference of 30.

89, 91

Let n = 999,999 999

then to generate twin primes, we can use the formula

30n – 1 = 29,999,999,969

30n +1 = 29,999,999,971 Twin primes

30n – 19 = 29,999,999,951

30n – 17 = 29, 999,999,953 Twin primes

This result confirms that there are infinite numbers of twin primes as these four terms are capable of predicting and generating twin primes to infinity. QE.D.

Twin primes are used in cryptology or in encoding messages. This result can be projected for any large number n. Twin primes are also called sexy primes, cousin primes, conjugal or connubial primes, or matrimonial primes (p.Stackel 1892-1919).

Note: Euclid (325-265 B.C) was the first Greek living in Alexandria who produced a sieve for generating prime numbers. It is also said Eratosthenes and Euler (1737) also did. Other mathematicians who have come up with the twin primes conjecture include Gauss, Polignac, Goldston, Hardy – Littlewood (1923), Brun (1919), Marcus du Sautoy, among others. My thesis proves that there is infinite number of twin primes and this resolves one of the unresolved issues in mathematics, namely the twin primes conjecture. Try the number 10, 999, 999,999 x 10, 999, 999,999 by representing it as n in the terms which I have proposed, namely 30n -1, 30n+ 1, 30n – 19 and 30n – 17 and you can generate any sets of twin primes to infinity. You will observe that these terms do not generate three twin prime sets (3,5), (5,7), (17,19) because 17 and 19 are used in the general terms. Also the primes 3,5 and 7 are primary primes, or rudimentary primes.

The limitation of the terms I have discovered is that they cannot generate three of the twin prime sets below 20. These sets are (3,5), (5,7) and (17,19). That notwithstanding, the terms I have discovered or derived can generate infinite sets of prime pairs or twin primes up to infinity. Thus, the formula for generating twin primes are:-

(30n – 1) and (30n + 1) on the one hand, and (30n – 19) and (30n – 17) on the other hand. Assume n= ?, therefore 30?- 1 <30? + 1 by 2.

and

30? - 19 < 30? - 17 by 2.

Reference: www.sjsu.edu//twinprimes.pdf

Presented by Kwesi Atta Sakyi

P.O. Box 50121

International School of Lusaka

Zambia

Email: [email protected]

Cell: +260973 790 152

© 2012

We will need a computer programme that can be used to generate twin primes, using the terms which I have arrived at by using my sieve, (30n-1), (30n+1): (30n -19) and (30n – 17). Since the natural numbers are infinite and twin primes are part of the natural numbers, they are also infinite by deduction.

References

mathworld.wolfram.com

www.math.sjsu.edu/..//twinprimes.pdf

http://listserv.nodak.edu/scripts/wa.exe....