CHAPTER 17

On the history of attempts for proving the conjecture of existence of infinity many twin prime numbers

 


 


17.1. Twin prime numbers

17.1.1. A. De Polignac's conjecture

The problem of twin prime numbers depends on a conjecture, famous as "A. De Polignacís conjecture" that has 100 years record and it is that if "k" be an arbitrary natural number, in sequence of prime numbers , infinite successive pairs like "" and "" exist so that "".

I. It is clear that three prime numbers in form of p, †and †donít exist only for "". But three numbers of prime numbers exist in "p", †and †form, like prime numbers 5, 7, 11. Such three prime numbers are called a prime triplet. One of the greatest known triplets is:

10014491,†††††† 10014493,†††††† 10014497

Four prime numbers in p,, and †form are called as prime quartet. The first six prime quartets are obtained for:

p = 5, 11, 101, 191, 821, 1481

Some great prime quartets are as follow():

†††††† 294311, 294313, 294317, 294319 ;

†††† 299471, 299473, 299477, 299479

We donít have too much information about distribution of prime triplets and quartets.

II. In a quartet," p , " and also " , " are twin, but some paris of twin exist that no prime number separates them from each other but they donít form quartet like twin pairs (419, 421) and (431, 433).

Three twins also exist that no prime number separates them like (179,181),
(191, 193) and (197, 199). There isnít any prime number between 181 and 191 and also between 193 and 197. At last, we know below quartet from prime twins and with the same property:

(9419, 9421), (9431, 9433), (9437, 9439), (9461, 9463)

17.2. Clementís theorem (1949)

We suppose that "n" be a non-one natural number. If and only if "n" and "" be a prime twin, then:

.

17.3. Approaching to the solution of problem of infinity many twin prime numbers

In 1737, Euler proved that sum of †that is added on prime numbers, is infinite. An interesting method for approaching to problem of many twin prime numbers is following sum:

17.4. The distances of prime numbers

†Different questions are propounded about distances of prime numbers. It is clear that except 2 and 3, any two successive natural numbers do not exist so that both of them are prime number. More over, some sequence of natural numbers exists as long as we want that none of its term is prime.

17.4.1. Remark

Twin prime numbers namely pairs of prime numbers that their difference is "2", exist (Like (3,5) , (5,7) , (11,13) , (17,19)).

It is clear with calculating that the number of prime twins less than 105, 106 and 3.107, are respectively 1224, 8164, 152892.

In 1972, apparently the pair 76.3139 Ė1 and 76.3139+1 was the greatest known prime twin, but even in some references of 1974, 1012 +9649 and 1012+9651 is known as† the greatest prime twin. There are only "20" prime twins between 1012 and 1012+10000. From this, some bodies have estimated that prime twins become rare as they become greater.

According to new researches and calculations, it is guessed that set of prime twins is infinite, but this guess is not proved or disproved till now.

17.5. Definitions and notes

It has been conjectured (but never proven) that there are infinitely many twin primes. If the probability of a random integer "m" and the integer being prime were statistically independent events, then it would follow from the prime number theorem that there are about †twin primes less than or equal to "m". These probabilities are not independent, so Hardy and Littlewood conjectured that the correct estimate should be the following:

Here the infinite product is the twin prime constant (estimated by Wrench and others to be approximately 0.6601618158...) and we introduce an integral to improve the quality of the estimate. This estimate works quite well! For example,

the number of twin primes less than N:

 

N

actual

estimate

8169

8248

440312

440368

27412679

27411417

 

There is a longer table by Kutnib and Richstein available online.

In 1919 Brun showed that the sum of the reciprocals of the twin primes converges to a sum now called Brun's Constant (Recall that the sum of the reciprocals of all primes diverges). By calculating the twin primes up to (and discovering the infamous pentium bug along the way). Thomas Nicely heuristically estimates Brun's constant to be "1.902160578".

 

 

 


Twin primes table

rank

†††††† Twin Primes

digits

who

when

1

33218925.

51090

g259

2002

2

60194061.

34533

g294

2002

3

1765199373.

32376

g182

2002

4

318032361.

32220

p100

2001

5

1807318575.

29603

g216

2001

6

665551035.

24099

g216

2000

7

1046886225.

21082

p146

2004

8

1940734185.

20012

g336

2004

9

781134345.

20011

p53

2001

10

1693965.

20008

g183

2000

11

83475759.

19562

g144

2000

12

37831341.

18605

g277

2003

13

1014561345.

18434

g326

2004

14

291889803.

18098

g191

2001

15

4648619711505.

18075

IJW

2000

16

2409110779845.

18075

IJW

2000

17

47280435.

16361

g389

2005

18

488162409.

16064

p135

2004

19

332494785.

15395

g344

2004

20

168030135.

15395

g344

2004

 


17.6. Problems concerning twin prime numbers

1. Prove that in a prime twin different from the pair 3 and 5, the smaller number is congruent with "5" with module "6".

2. Prime twin 5 and 7 have this property that numeral average of its numbers, namely "5" and "7", is a triangular number. Is there another prime twin with the same property?

3. Prime twin of previous question has is property that numeral average of its members is a perfect number. Is there another prime twin with the same property?