Transcript Historical Problem Presentation

Meng Li
10-05-2012

Content:
There are infinitely many primes p such
that p + 2 is also prime.

History:
“The twin prime conjecture” was came up by a
French Mathematician Polignac in 1894. And the
Hardy–Littlewood conjecture is the strong form of
the twin prime conjecture which was came up by
Hardy and Littlewood in1923.
Question: Do you believe in the conjecture?

A prime number (or a prime) is a natural number
greater than 1 that has no positive divisors other
than 1 and itself.


2, 3, 5, 7, 11, 13, 17, 19, 23, 29,
31, 37, 41, 43, 47, 53, 59, 61, 67, 71,
73, 79, 83, 89, 97, 101, 103, 107, 109, 113,
127, 131, 137, 139, 149, 151, 157, 163, 167,
179, 181, 191, 193, 197, 199, 211, 223, 227,
233, 239, 241, 251, 257, 263, 269, 271, 277,
283, 293, 307, 311, 313, 317, 331, 337, 347,
353, 359, 367, 373, 379, 383, 389, 397, 401,
419, 421, 431, 433, 439, 443, 449, 457, 461,
467, 479, 487, 491, 499
173,
229,
281,
349,
409,
463,
Question: Can you find all the twin primes within 100?



By the definition of the twin prime conjecture we
can get all twin primes from 0 to 100:
(3,5);(5,7);(11,13);(17,19);(29,31);(41,43);(59,61);
and (71,73). So there are 8 pairs in total within
100.
It is obvious that with the number becomes
greater and greater the distribution of twin
primes will become sparser and sparser, so it will
also be harder and harder to find twin primes.
Question: Is there exists a certain threshold such that after it
the twin prime will no longer exists?

As we all know that Euclid has already proved
that there exist infinite prime numbers, so it
is unnecessary for us to worry about it. Since
based on the property of prime numbers,
people think that there must exist infinite
twin prime, which is the topic that I am going
to talk about today. As I mentioned before,
there are 2 forms of the twin prime
conjecture. Let’s start with the Polignac's
conjecture.

Polignac's conjecture from 1849 states that
for every positive even natural number k,
there are infinitely many consecutive prime
pairs p and p′ such that p′ − p = k. The case
k = 2 is the twin prime conjecture. When k=4
is the cousin prime. The case k=6 is the sexy
prime. (ps: sex means six in Latin.)While the
conjecture has not been proved or disproved
for any value of k.

The Hardy–Littlewood conjecture is a
generalization of the twin prime conjecture. It
is concerned with the distribution of prime
constellations, including twin primes, in
analogy to the prime number theorem. Let
π2(x) denote the number of primes p ≤ x
such that p + 2 is also prime.
Define the twin prime constant C2:
C2=
P( P  2)
 (P -1)^2  0.660161815846869573927812110014...
P 3


And also it gives us the approximation:
n
n
dt
π2(n)~2C2 (ln n)^2 ~2C2 2 (ln t )^2
There is form that can show how exact the
conjecture is.
n
The number of twin prime
Hardy-Littlewood conjecture
100,000
1,224
1,249
1,000,000
8,169
8,248
10,000,000
58,980
58,754
100,000,000
440,312
440,368
27,412,679
27,411,417
10,000,000,000


1.
After knowing lots of information about the
twin prime conjecture, now we are going to
see some proof of it.
Generally speaking, there are two methods
to proof it:
Non-estimate results: It was came up in
1966 by a Chinese Mathematician jingrun
Chen, and he prove that there are infintely
many primes p such that p + 2 is either a
prime or a product of two primes by using
sieve method.
2. estimate results:
 The results achieved by Goldston and Yildirim are in
the category. Such results estimate that the minimum
spacing between adjacent primes, more precisely:
Δ := limn→∞inf[(pn+1-pn)/ln(pn)]
Obviously, if the twin prime conjecture established, then Δ=0.
Because the twin prime conjecture that pn+1-pn=2
should be true for infinitely many n, while ln(pn) →∞
then the minimum value of the ratio of the two sets ( and thus
for the entire set of prime numbers also) tends to zero.
 In 2005, Goldston, János Pintz and Yıldırım
established that Δ can be chosen to be arbitrarily
small, Δ=0.


Though Δ = 0 has been proved by Goldston
and Yildirim, it is only a necessary condition,
but not sufficient condition. Since that it is
still very far away to prove the twin prime
conjucture compeletly, but it certainly is the
most striking results.




http://en.wikipedia.org/wiki/Twin_prime
http://www.changhai.org/articles/science/m
athematics/twin_prime_conjecture.php
http://en.wikipedia.org/wiki/prime
http://wenku.baidu.com/view/3b105c0203d
8ce2f0066239c.html