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 Polignac’s 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 don’t 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 don’t 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 don’t 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 isn’t 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. Clement’s 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 –1 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?