CHAPTER 17
On the history of attempts for proving the conjecture
of existence of infinity many twin prime numbers
_{}
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)
We suppose that "n" be a non-one natural number. If and only if "n" and "_{}" be a prime twin, then:
_{}.
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:
_{}
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.
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 10^{5}, 10^{6} and 3.10^{7}, are respectively 1224, 8164, 152892.
In 1972, apparently the pair 76.3^{139} –1 and 76.3^{139}+1 was the greatest known prime twin, but even in some references of 1974, 10^{12} +9649 and 10^{12}+9651 is known as the greatest prime twin. There are only "20" prime twins between 10^{12} and 10^{12}+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.
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".
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 |
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?