An important finding about prime numbers

Mathematicians made the first progress in 76 years in the question: How many consecutive prime numbers can be separated by the largest distance?

In May 2013, mathematician Zhang Yitang (Zhang Yishang, a Chinese-American) at the University of New Hampshire demonstrated for the first time that even the greater the number of prime elements, the more pairs of numbers we will always find. the element is separated by a blocked distance - he has proven that this is within 70 million. After Zhang's announcement, many mathematicians joined the study to improve the results, and lowered this limit to 246, substantially closer to assuming twin primes - assuming that there are infinitely many pairs of primes separated by two units.

Paul Erdos (left) and Terence Tao in 1985

Now mathematicians have made the first significant progress in 76 years with the question in the opposite direction: "What is the largest distance between two consecutive prime numbers? So far? Who can answer this issue ".

"This is a very obvious question, one of the first questions about prime numbers , " said Andrew Granville, an arithmetic theorist at Montreal University (Canada). 'But we still have almost no further steps in nearly 80 years.'

Last August, two groups of mathematicians - the first group of four members including Terrence Tao of the University of California and the second group James Maynard of Oxford University - published their studies, proving a fake. Paul Erdos's long-standing theory of the magnitude of the distance between two primes. After that, these two groups came together to improve their results. That result was recently published in December 2014 (available at: http://arxiv.org/abs/1412.5029).

Erdos, one of the most accomplished mathematicians of the twentieth century, thought of hundreds of different mathematical problems; Whoever wins will be rewarded, usually at 25 USD. As for bonuses for the question of the distance between primes, Erdos raised up to 10,000 USD.

Erdos's hypothesis is based on a lower bound (oddly looking), discovered in 1938 by Robert Alexander Rankin, a Scottish mathematician. With a sufficiently large number X, Rankin demonstrates that the maximum distance between two consecutive prime numbers (hereinafter referred to as the prime distance) less than X is always greater than or equal to

Terence Tao once said: The number theory formula is famous because there are many "log" (short for natural logarithms). Even digital theory is joking: "How will the number-drowning theorist call?" - 'Log log log log .'.

Terence Tao thinks Rankin's result is: 'a funny formula, you don't think it can appear naturally. Everyone believes that this formula can be quickly improved. ' But aside from some minor improvements, no progress has been made with Rankin's formula for more than seven decades.

Many mathematicians believe that the actual size of the elemental distance can be much larger - up to (log X) 2, as suggested by Swedish mathematician Harald Cramer in 1936. This gap occurs If we assume that the set of primes is like a set of random numbers, in fact these two sets have many similarities. But no one can prove Cramer's theory. Terence Tao concludes 'We still don't have a good understanding of prime numbers'.

Erdos's hypothesis is more modest: It is possible to replace 1/3 of Rankin's formula with any number as long as we increase X to be large enough. That means that the elemental spacing received may be much larger in Rankin's formula, though not as big as in Cramer's formula. The two proofs for Erdos's hypothesis mentioned above are based on a simple construction of large elemental distances like a long series of numbers.

For example, here's a way to build a sequence of 100 consecutive numbers: Get 100 numbers from 2 to 101 and add 101! (factorial of 101). This series becomes 101! + 2, 101! + 3, 101! + 4, ., 101! + 101. For 101! Divide by numbers from 2 to 101, each number in the upper row is definitely a combination of numbers: 101! + 2 divided by 2, 101! + 3 divided by 3, .

James Maynard said: 'Big elemental proofs all use small variations of this construction method.' The numbers in the upper sequence are all very large because 101! Has 160 digits. of Rankin, mathematicians have to prove the existence of much smaller number of arrays - can add a smaller number of more than 101 to the range 2, 3, ., 101 and still create a series of numbers Both groups achieved this by using recent findings - each group uses a different finding - about how the primes are distributed, in addition, Maynard's work uses a number of development tools. Last year involved small elemental distances.

Now, the five researchers of the two groups are focusing on making a better general project that, according to Tao, will push Rankin's method as far as possible within the limits of current techniques.

This work does not have immediate applications although understanding the large elemental distance can affect encryption algorithms. If the elemental distance found is greater than Cramer's hypothesis, it is possible that encryption algorithms based on finding large primes will have problems. Maynard said: 'If the unfortunate algorithm starts to look at the beginning of a very large prime distance, the program will take a lot of time.'

Tao has a more personal motivation when studying elemental distances.'After a while, these things start making you uncomfortable ,' he said, 'You are supposed to be a prime expert, but you can't answer the basic questions like that, even if people have thought about them for centuries ".

Erdos died in 1996 but Ronald Graham of the University of California, San Diego, who collaborated extensively with Erdos, decided to sponsor the $ 10,000 prize that Erdos committed.

In 1985, 10-year-old prodigy Tao first solved an Erdos math problem and he met Erdos at a mathematical event . "He treated me equally, he told me serious math problems," Tao, who received the 2006 Fields Medal, recalls.

Recent advances in small and large elemental distances have created a generation of digital theorists who think nothing is impossible, Granville said: 'When I was studying, we thought there were The problem will not have an answer until a new mathematical period . But I think in the past few years, the attitude has changed. Many young mathematicians have much bigger ambitions because they see that we can make great progress in mathematics. "