Unknown mathematician made a breakthrough in the theory of twin prime numbers

In mathematics it is extremely rare for a scientist older than 40 to publish the first serious scientific work. More rarely it happens that this work has a great scientific value. Such a rare case is the assistant professor of the University of New Hampshire, Ethan Zhang (Yitang Zhang), who still does not have either a professor or a web page with a list of scientific works. Nevertheless, he managed to take a serious step towards solving one of the oldest mathematical problems, the hypothesis of twin numbers.

When the “Annals of Mathematics” magazine received Zhang's scientific work on April 17, 2013, they were skeptical. An application for a breakthrough research from an unknown scientist? This is too trite and often meets to be true. To the surprise of the editorial board, several scientific experts studied in detail Zhang's work — and found evidence of the hypothesis about the distance between paired prime numbers extremely clear, precise, and indisputable.

As a result, the magazine approved the work for publication in an extremely short time - three weeks after it was received.

At 50+, Ethan Zhang teaches algebraic geometry at a university, but number theory was his hobby. As usual, mathematicians are often addicted to prime numbers as one of the most interesting puzzles in this field of science. Zhang's attention was attracted by the twin prime number theorem.
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The sieve of Eratosthenes is a simple algorithm for finding all primes up to some integer number n, by crossing out all the numbers that are divided by the prime divisor: 2, 3, 5, 7, etc.

Mathematicians have long noticed that the distribution of primes in an infinite number space has certain regularities. In particular, a strange phenomenon are the prime numbers-twins, which differ from each other by 2. The greater the number of characters, the less often there are numbers-twins, but still they continue to occur again and again.

In the original version of the hypothesis states that there are an infinite number of prime numbers of twins. This assumption has not yet been proven or disproved. The largest found prime twins known to science are 3756801695685 × 2 ⁶⁶⁶⁶⁶⁹ - 1 and 3756801695685 × 2 ⁶⁶⁶⁶⁶⁹ + 1.

Ethan Zhang proved that there are an infinitely large number of prime numbers, the distance between which does not exceed 70 million. These pairs will occur less and less, but they will never disappear, despite the effect of the theorem on the average distance between prime numbers of 2.3 × N, where N is the number of digits.

In other words, the average distance between the numbers will approach infinity as the number of digits grows, but it will always contain prime numbers that are no more than 70 million apart, which is simply amazing.

“This job will change the rules of the game,” said Andrew Granville, a number theorist at the University of Montreal. “Sometimes after the appearance of a new evidence, what seemed difficult to prove earlier becomes just a small extension.” Now we need to study the work and understand what's what. " But there is no question on the quality of the proof: “He worked through every detail, so no one would put his work in doubt,” Granville added.

UPD. Zhang's article itself was not published in the public domain, but it was possible to find excerpts from his speech in Gervard on May 13, 2013 (thanks, EvgeshaS ).