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Numbers with a rare property, that's what a mathematician loves

Numbers with a rare property, that’s what a mathematician loves

Is there a sequence of one million consecutive numbers, and none of them are prime? Is there a prime number for every number between this number and its multiple? Is there a prime number more than a billion digits without the number 7? Are there infinitely many double primes, pairs of primes that differ only by two?

It is questions like these that pique the curiosity of Britain’s James Maynard (35) of Oxford University. But even more than the questions themselves, he is interested in the basic techniques needed to answer them. This has given a huge boost in recent years. Maynard continues to find new ways, even when things seem hopeless. Mathematician Frits Bokers said, In the year 2020 about in Norwegian Refugee Council† Maynard is now the recipient of the Fields Medal, the highest award for young mathematicians.

Read about the work of a Fields Medal winner: How do you stack oranges in eight dimensions?

The above questions have in common that almost everyone can understand. Of course, you need to know what primes are (numbers that are only divisible by 1 and by themselves), but other than that there is no experience. The answers to the questions are different. The first question is still easy: multiply the first natural number from a million and one and add 2, and you will have the first number of the desired sequence. To find out, you do not have to perform the calculations, but consider the structure of this complex number.

Question 2 is more difficult. The answer (yes) was given in 1850 by the Russian Pavnuty Chepsgov. Question 3 – about large primes without the number 7 – was an open problem until a few years ago, until Maynard proved that a non-seven prime with more than a billion digits actually existed. In fact, there are an infinite number of them.

The last of the problems presented is a pressing issue of number theory, which no one has yet been able to solve. However, Maynard came close: at the end of 2013, he proved that there are an infinite number of pairs of primes that differ at most 600 from each other. This is not an axiom, because primes get rarer as numbers increase. The fact that the distance between two consecutive primes is always greater than 600 from a given moment is not a priori impossible. Maynard’s proof was startling, although Chinese mathematician Yitang Zhang gave a very similar result a few months ago.

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jeans and shirt

In 2016, Maynard – who always wore jeans, a T-shirt, and two top buttons – gave a lecture at the Dutch Sports Conference in Amsterdam. He was walking around without a doubt, chatting to everyone during his coffee break. This newspaper did Report multiple times of its results. Maynard was always on hand to answer emails full of questions about his abstract work. His most recent work is about the “zero points of the Riemann zeta function”, which is related to the Riemann hypothesis, the most widely followed puzzle in mathematics. Preliminary version Put online last month.

Read about the Riemann hypothesis: How coincidental is the distribution of prime numbers?

for youtube channel numberphile Maynard has made several short films in recent years to explain his work to a wide audience. One of those about the aforementioned Seven Prime Numbers† There are still too many twenty-digit primes without 7, eg 49135,832,685.980. But with very high numbers, the absence of sevens is exceptional. The vast majority of numbers, for example, a million numbers, contain all numbers from 0 to 9. It is by no means clear that the set of infinite natural numbers contains an infinite number of primes. By the way, there is nothing special about the number 7. You can ask the same question if you replace the number 7 with a different number.

Maynard proved that there are always primes missing a certain number, no matter how high the number ladder goes. The difficulty, of course, is how to prove such a thing? Maynard needs ninety pages. his article from 2019 in math inventions It contains a mixture of ideas from analysis, synthesis and algebra.

You have to watch how far you go with trimming the numbers

The obvious follow-up question is: Are there also infinitely many primes missing two finite numbers, eg 4 and 7? “This is an open problem, we don’t know,” Maynard said. numberphile-video. doubts that the answer is yes. “But with not just a single number, we’re really at the limit of current proof techniques,” he adds. So Maynard doesn’t expect a follow-up question to be answered any time soon. “Furthermore, you have to be careful how far you go with trimming the numbers,” he says. Simple example: even though there are an infinite number of even-numbered numbers only, there are no primes with this property except for 2.

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In less than ten years, Maynard has risen from a doctoral student to one of the leading figures in his field. His work gradually reveals the mysterious world of prime numbers. The hope, of course, is that the tools it develops will be useful in the long run for finding answers to big questions, like the double major problem.