If the prime number is finite, then the reciprocal sum is naturally finite. But Euler proved that this sum is divergent, that is, infinite. This shows that there are infinite prime numbers. 19 19, the Norwegian mathematician Bloom imitated Euler's method to find the reciprocal sum of all twin prime numbers:
If it can be proved that this sum is greater than any number, it proves that there are infinitely many twin prime numbers. It's a good idea, but the fact goes against Bloom's wishes. He proved that the reciprocal sum is a finite number, and this constant is called Bloom's constant: b = 1.902 16054 ... Bloom also found that for any given integer m, m adjacent prime numbers can be found, and none of them are twin prime numbers.
1920s, Vigo Bulun of Norway proved that 2 can be expressed as the difference between two numbers with at most nine prime factors by using the famous sieve theory. This conclusion is somewhat similar to the twin prime conjecture. It can be seen that the twin prime conjecture can be proved as long as the number with at most 9 prime factors in this proof is improved to the number with at most 1 prime factors.
1966 was obtained by screening method by the late China mathematician Chen Jingrun. Chen Jingrun proved that there are infinitely many prime numbers P, so p+2 is either a prime number or the product of two prime numbers. This result is very similar to his conjecture about Goldbach. It is generally believed that due to the limitations of the screening method itself, this result is difficult to be surpassed within the scope of the screening method.
On May 4th, 20 13, Nature magazine reported online that Zhang proved that "there are infinite prime pairs with differences less than 70 million". This study was immediately regarded as a major breakthrough in the limit number theory of twin prime conjecture, and some even thought that its influence on academic circles would exceed Chen Jingrun's "1+2" proof. In the latest research, Zhang did not rely on unconfirmed inference, and found that there are infinitely many prime pairs with a difference of less than 70 million, thus taking a big step forward on the road of the important problem of twin prime conjecture.
The conjecture of twin prime numbers can be weakened to "whether a positive number can be found so that the difference between infinite pairs of prime numbers is less than this given positive number". In the twin prime conjecture, this positive number is 2. The positive number found by Zhang is "70 million". Although it is a long distance from 20 million to 70 million, the report in Nature still calls it an "important milestone". As Dan GoldSi Tong, a professor of number theory at San Jose State University, said, "The distance from 70 million to 2 (referring to the unfinished work in the conjecture) is insignificant compared with the distance from infinity to 70 million (referring to Zhang's work)."
On May 20 1313, Zhang delivered a keynote speech at Harvard University in the United States, introducing his own research progress. According to a report in Nature magazine, if this result holds, it is the first time that someone has officially proved the existence of infinite pairs of prime numbers whose distance is less than a fixed value. In other words, Zhang will guess a real "head" for twin prime numbers. The Annals of Mathematics, the world's top mathematical journal, is going to accept this certified article, and the reviewer also commented that it is "proved to be correct and a first-class mathematical work".
Zhang's paper was published on May 6th, 2004 at 5438+04. Two weeks later, on May 28th, the constant dropped to 60 million. Only two days later, on May 3 1, it dropped to 42 million. Three days later, June 2nd,130,000. The next day, 5 million. On June 5, 400 thousand. In the "Polymath" project initiated by British mathematician Tim Gowers and others, the twin prime number problem has become a typical example of mathematicians around the world using the network to cooperate. People continue to improve Zhang's proof, further narrowing the distance to finally solve the twin prime conjecture. As of 20 14, 10,9 (2014-10-09) [update], the difference between prime numbers has been reduced to ≤ 246. From 246 to 2, although it is close to the crown of dual prime numbers, the road is getting harder and harder. It is unknown who can win the championship and when. Another result of proving the conjecture of twin prime numbers is an estimation result. This result estimates the minimum interval δ between adjacent prime numbers, more precisely:
Translated into vernacular Chinese, this expression defines the minimum value of the ratio between the interval of two adjacent prime numbers and the logarithm value of the smaller prime number in the whole prime set. Obviously, if the twin prime conjecture holds, then δ must be equal to 0. Because the conjecture of twin prime numbers shows that pn+ 1-pn=2 holds for an infinite number of n, and when ln(pn)→∞, for the set of twin prime numbers (and thus for the whole set of prime numbers), the minimum value of the ratio between them tends to zero. However, it should be noted that δ = 0 is only a necessary condition, but not a sufficient condition, for the twin prime conjecture to be established. In other words, if Δ ≠ 0 can be proved, the conjecture of twin prime numbers does not hold; But proving that δ = 0 does not mean that the twin prime conjecture must be established.
The simplest estimate of δ comes from the prime number theorem. According to the prime number theorem, for a sufficiently large X, the probability that a prime number appears near X is 0, indicating that the average interval between prime numbers is ln(x) (which is why ln(pn) appears in the expression of δ), thus giving the ratio of the interval between adjacent prime numbers to the average interval, and the average value is obviously 1. The average value is 1, and the minimum value is obviously less than or equal to 1, so the prime number theorem gives δ≤ 1.
Further estimates of δ began with Hardy and Littlewood. 1926, they proved by circle method that if the generalized Riemann conjecture holds, δ ≤ 2/3. This result was later improved by Rankin to δ≤ 3/5. But both of these results depend on the unproven generalized Riemann conjecture, so they can only be regarded as conditional results. 1940, Erd? S first gives an unconditional result by screening method: δ.
In 2003, Goldston and Yildirim published a paper claiming to prove that δ = 0. But on April 23rd, 2003, Andrew glanville (University of Montreal) and Cannan Sodalarn (University of Michigan) discovered an error in the proof of GoldSi Tong and Yildirim. In 2005, they cooperated with Janos Pintz to complete the proof. In addition, if Elliott-Halberstam conjecture holds, then when δ = 0, the weakened version of the twin prime conjecture-there are infinite pairs of prime numbers separated by16-will also hold.
After δ = 0 was proved, people's attention naturally turned to the study of the way δ tends to 0. The conjecture of twin prime numbers needs δ ~ [log (pn)] (because pn+ 1-pn=2 holds for the infinite number of n). Goldston's and Yildirim's proofs give δ ~ [log (pn)], and there is a considerable distance between them. However, some mathematicians who have read GoldSi Tong and Yildirim's manuscripts think that the methods used by GoldSi Tong and Yildirim have room for improvement. In other words, their method may be able to make a stronger estimate of the way δ tends to 0. Therefore, the value of GoldSi Tong and Yildirim's proof lies not only in the result itself, but also in the fact that it is likely to become the starting point of a series of future studies. In 1849, Alfonso de Polignac put forward a more general conjecture: for all natural numbers k, there are infinite prime pairs (p, p+2k). The case of k= 1 is the twin prime conjecture. Therefore, Polinak is sometimes regarded as the initiator of the conjecture of twin prime numbers.
192 1 year, British mathematicians Hardy and Littlewood put forward the following enhanced conjecture: Let it be the number of twin prime numbers in the first n natural numbers. therefore
Where the constant is the so-called twin prime constant, where p stands for prime number.
Hardy and Littlewood conjecture is actually an enhanced version of the long-standing twin prime conjecture. The conjecture of twin prime numbers refers to "there are infinite twin prime numbers". This conjecture has not been proved. However, Hardy's and Littlewood's guesses don't need to be based on the hypothesis of twin prime numbers.