Collective Intelligence in Number Theory
In May, an article was published on Habré, which tells about the work of Yitang Zhang in the field of the theory of prime numbers, which caused a great resonance in the scientific community. This article provides a brief summary of the results that the scientific community has achieved nine months after the publication of this paper.
Recall that one of the important, but still unproved, hypotheses in number theory is the twin prime conjecture hypothesis, which assumes that there are infinitely many pairs of prime numbers that differ by two, for example: (3, five); (11, 13), and so on. In this case, it is assumed, of course, that there are infinitely many primes - a fact that was proved by Euclid. In a more rigorous form, this conjecture about twin numbers can be formulated as follows.
')
Where
denotes the n-th prime number. The proof of the existence of this limit leads to the proof of the original hypothesis. Zhang was the first to present evidence that there is a finite estimate of the interval for an infinite number of pairs of prime numbers. Using the record above, the result of Zhang can be written as follows.
It is important to make a digression and say that:
1. This does not mean that all prime numbers are separated from each other at a distance of less than 70,000,000. In fact, the greater the n, the greater the average distance between two prime numbers and the less likely it is to meet a pair of prime numbers that are small distance from each other. See related article in wikipedia on this topic.
2. This does not mean that by taking any natural number, you are guaranteed to find a prime number in the range plus or minus 70,000,000.
3. The proof is that no matter how far you go and whatever large natural number you take, there are always a couple of prime numbers located further along the axis, such that the interval between these prime numbers will be less than 70,000,000.
This work caused a great resonance and pushed many scientists to further improve this boundary. One of the most talented mathematicians of our time, Terence Tao, with a group of colleagues, organized the collective project polymath8a , whose goal was to lower the proven upper limit. In just a few months, jointly, we managed to significantly improve the result of Zhang and prove that
The results of this work are being prepared for publication, although a draft (attention, 177 pages) can be read now.
November 19, 2013 followed by the next breakthrough. James Maynard proved that
The brilliant Terence Tao did not lose his head and organized a new collective project, polymath8b , aimed at improving the newly acquired border. The last result is that
Thus, it is known that there is an infinite set of pairs of primes, the interval between them does not exceed 270. Potentially, this limit can be reduced to 6 if you can prove another hypothesis (in mathematics it often happens that the proof of one hypothesis follows from the proof / refutation of another - so it was, for example, with the Fermat theorem).
Even in this case, however, this does not immediately lead to the ultimate goal - the proof of the hypothesis of twin twin numbers, although it will significantly approximate the mathematicians to it.
Recall that one of the important, but still unproved, hypotheses in number theory is the twin prime conjecture hypothesis, which assumes that there are infinitely many pairs of prime numbers that differ by two, for example: (3, five); (11, 13), and so on. In this case, it is assumed, of course, that there are infinitely many primes - a fact that was proved by Euclid. In a more rigorous form, this conjecture about twin numbers can be formulated as follows.
')
Where
It is important to make a digression and say that:
1. This does not mean that all prime numbers are separated from each other at a distance of less than 70,000,000. In fact, the greater the n, the greater the average distance between two prime numbers and the less likely it is to meet a pair of prime numbers that are small distance from each other. See related article in wikipedia on this topic.
2. This does not mean that by taking any natural number, you are guaranteed to find a prime number in the range plus or minus 70,000,000.
3. The proof is that no matter how far you go and whatever large natural number you take, there are always a couple of prime numbers located further along the axis, such that the interval between these prime numbers will be less than 70,000,000.
This work caused a great resonance and pushed many scientists to further improve this boundary. One of the most talented mathematicians of our time, Terence Tao, with a group of colleagues, organized the collective project polymath8a , whose goal was to lower the proven upper limit. In just a few months, jointly, we managed to significantly improve the result of Zhang and prove that
The results of this work are being prepared for publication, although a draft (attention, 177 pages) can be read now.
November 19, 2013 followed by the next breakthrough. James Maynard proved that
The brilliant Terence Tao did not lose his head and organized a new collective project, polymath8b , aimed at improving the newly acquired border. The last result is that
Thus, it is known that there is an infinite set of pairs of primes, the interval between them does not exceed 270. Potentially, this limit can be reduced to 6 if you can prove another hypothesis (in mathematics it often happens that the proof of one hypothesis follows from the proof / refutation of another - so it was, for example, with the Fermat theorem).
Even in this case, however, this does not immediately lead to the ultimate goal - the proof of the hypothesis of twin twin numbers, although it will significantly approximate the mathematicians to it.
Source: https://habr.com/ru/post/211120/
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