An unknown mathematician has made a breakthrough in the theory of twin primes

In mathematics, it is extremely rare for a scientist older than 40 to publish his first serious scientific work. It is even rarer that this work has great scientific value. It is such a rare case that the associate professor of the University of New Hampshire, Ethan Zhang (Yitang Zhang), who still does not have a professorship or a web page with a list of scientific papers. Nevertheless, he managed to take a serious step towards solving one of the oldest mathematical problems - the hypothesis of twin primes.

When the magazine “Annals of Mathematics”received on April 17, 2013 Zhang's scientific work, they were skeptical about it. An application for breakthrough research from an unknown scientist? It is too commonplace and often found to be true. To the surprise of the editorial board, several scientific experts studied Zhang's work in detail - and found a proof of the hypothesis about the distance between paired primes as extremely clear, clear and undeniable.

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

In his 50+ years, Ethan Zhang has been teaching algebraic geometry at the university, but number theory was his hobby. As usual, mathematicians are often addicted to primes as one of the most interesting puzzles in this field of science. Zhang’s attention was drawn to the twin prime theorem.


The sieve of Eratosthenes is a simple algorithm for finding all primes up to a certain integer n, by deleting all numbers that are divided by a prime divisor: 2, 3, 5, 7, etc.

Mathematicians have long noticed that the distribution of primes in infinite number space has certain patterns. In particular, the twin primes, which differ from each other by 2, are a strange phenomenon. The larger the number of characters, the less common are the twin numbers, but still they continue to meet again and again.

In the original version, the hypothesis states that there are an infinite number of twin primes. No one has yet proven or disproved this assumption. The largest twin primes found in science are 3756801695685 × 2 ⁶⁶⁶⁶⁶⁹  - 1 and 3756801695685 × 2 ⁶⁶⁶⁶⁶⁹  + 1.

Ethan Zhang proved that there is infinitea large number of primes, the distance between which does not exceed 70 million. These pairs will meet less and less, but will never disappear, despite the action of the theorem on the average distance between primes 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 there will always be prime numbers that are no more than 70 million distant from each other, which is simply amazing.

“This work will change the rules of the game,” says Andrew Granville, a theoretician in number theory from the University of Montreal. - Sometimes after the appearance of new evidence, what previously seemed difficult to prove becomes just a small extension. Now we need to study the work and understand what's what. ” But there is no question about the quality of the evidence: “He worked out every detail, so no one would question his work,” Granville added.

UPD. Zhang's article itself was not publicly available, but managed to find excerpts from his speech at Hervard on May 13, 2013 (thanks, EvgeshaS ).