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Historical Problem Presentation

Historical Problem Presentation. Meng Li 10-05-2012. The Twin Prime Conjecture . Content : There are infinitely many primes p such that p + 2 is also prime . History:

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Historical Problem Presentation

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  1. Historical Problem Presentation Meng Li 10-05-2012

  2. The Twin Prime Conjecture • 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?

  3. Before we start… • A prime number (or a prime) is a natural number greater than 1 that has no positive divisors other than 1 and itself.

  4. All Prime numbers within 500 • 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, 173, 179, 181, 191, 193, 197, 199, 211, 223, 227, 229, 233, 239, 241, 251, 257, 263, 269, 271, 277, 281, 283, 293, 307, 311, 313, 317, 331, 337, 347, 349, 353, 359, 367, 373, 379, 383, 389, 397, 401, 409, 419, 421, 431, 433, 439, 443, 449, 457, 461, 463, 467, 479, 487, 491, 499 • Question: Can you find all the twin primes within 100?

  5. The Twin prime conjecture • 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?

  6. The Twin Prime Conjecture • 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.

  7. 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.

  8. Hardy–Littlewood conjecture • 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=

  9. Continue • And also it gives us the approximation: π2(n)~2C2 ~2C2 • There is form that can show how exact the conjecture is.

  10. Proof • 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.

  11. Continue • 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.

  12. While… • 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.

  13. Reference • http://en.wikipedia.org/wiki/Twin_prime • http://www.changhai.org/articles/science/mathematics/twin_prime_conjecture.php • http://en.wikipedia.org/wiki/prime • http://wenku.baidu.com/view/3b105c0203d8ce2f0066239c.html

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