Mathematicians Uncover a New Way to Count Prime Numbers

A new proof brings mathematicians one step closer to understanding the hidden ordering of these “arithmetic atoms” – prime numbers.

Prime numbers—numbers that are only divisible by themselves and 1—are the most fundamental building blocks of mathematics. They are also the most mysterious. At first glance, they appear to be randomly scattered across the number line. But of course, prime numbers are not random. They are completely deterministic, and a closer look at them reveals all sorts of strange patterns that mathematicians have spent centuries trying to unravel. A better understanding of how prime numbers are distributed will illuminate vast areas of the mathematical universe.

But while mathematicians have formulas that give them a rough idea of ​​where the primes are, they can’t pinpoint them. Instead, they had to take a more indirect approach.

Around 300 BC, Euclid proved that there are infinitely many prime numbers. Since then, mathematicians have used his theorem as a basis to prove the same statement for prime numbers that satisfy additional conditions. (A simple example: Are there infinitely many prime numbers that do not contain the number 7?) Over time, mathematicians have made these criteria more and more stringent. By showing that there are still infinitely many primes that satisfy this increasingly tight constraint, they have been able to learn more about the locations of the primes.

But such claims are difficult to prove. “There aren’t many results like this,” said Jonny Teravainen Professor at the University of Turku, Finland.

Now, two mathematicians— Ben Green Oxford University and Mehtab Soni Columbia University professor – has demonstrated such a claim for a particularly challenging type of prime numbers. Their proof is Publish online October not only deepened mathematicians’ understanding of prime numbers. It also draws on a set of tools from very different areas of mathematics, suggesting that these tools are much more powerful than mathematicians imagined and may be ripe for application elsewhere.

“That’s great,” said John Friedlander from the University of Toronto. “It really surprised me that they did that.”

Try the set

Mathematicians tend to study families of prime numbers that are complex enough to be interesting but simple enough to make progress. For example, they might try to prove that there are infinitely many prime numbers that are 500 units apart. Or we can construct an infinite number of prime numbers by adding the squares of other numbers.

This last restriction was particularly useful, guiding centuries of mathematical progress. In 1640, Pierre de Fermat conjectured that an infinite number of prime numbers could be found by squaring two integers and adding them. (For example, the prime number 13 can be written as 2² + 3².) Leonhard Euler later proved this. But tweaking the problem a little – insisting that one of the numbers you want to square might be an odd number, or a perfect square – makes the problem much more difficult. “The more restricted a set is, the harder it is to find prime numbers in it,” Green said.

In the 19th century, the study of such statements led to the development of much of modern number theory. In the 20th century, it inspired one of the most ambitious mathematical efforts to date: Langlands Plan. Entering the 21st century, the study of such prime numbers continues to produce new technologies and insights.

In 2018, Friedlander and Henrik Izhanik A Rutgers professor asked whether there are infinitely many prime numbers of the form p² + 4q²two of which p and q It must also be a prime number. (For example, 41 = 5² + 4 × 2².) Resolving this limitation proved particularly challenging. But if mathematicians can solve this problem, they could succeed in gaining a new degree of control over prime numbers—exactly what they’ve always wanted to do.

A fruitful visit

Neither Green nor Sony had played this type of prime number counting game before. But they all had experience studying strange patterns caused by prime numbers.

In July, the two mathematicians met at a conference in Edinburgh. Sony, who just finished graduate school, has always admired Green. Sony said a pioneering result Green demonstrated 20 years ago was “one of the things that got me into this subject.” “I was like, ‘Oh, man, how can you do this?'” Greene was also impressed by the young mathematician. “Metab was an outstanding mathematician,” he said. “He somehow knows everything.”

The two decided to cooperate. They just need to find the right problem to solve. After some discussion, they finally confirmed Friedlander and Ivanik’s conjecture.

Green invited Sony to come to Oxford for a week. They knew that to prove similar conjectures, mathematicians often rely on a specific set of counting techniques. But because the prime numbers in the problem were so tightly defined, Green and Soni couldn’t find a way to make this traditional toolkit work.

Instead, they hoped to prove the conjecture in a more roundabout way—through mathematical moves like playing chess. But first, they must prove they were allowed to take the action in the first place.

By the end of the Sony visit, he and Green had figured out how to do this, allowing them to prove the conjecture. To do so, they ended up making a surprising connection to another area of ​​mathematics.

try another set

It was not possible for Green and Soni to directly calculate the number of primes obtained by squaring two other primes and adding them together. But what if they loosened the reins a little? They realized they could solve a slightly weaker version of the problem – one in which the squared numbers only needed to be “roughly” prime.

Rough primes are easier to find than prime numbers. Suppose you want to calculate all rough prime numbers between 1 and 200. These numbers are roughly prime. In this case, you end up with 50 rough primes: 46 of them are actually primes, while the remaining four (121, 143, 169, and 187) are not. Since rough prime numbers have a much lower random distribution than prime numbers, they are easier to work with. “We have a much better understanding of crude primes,” Soni said.

Green and Soni showed that an infinite number of primes can be obtained by squaring two crude primes and adding them. They now only had to prove that this statement implied the problem they really wanted to solve: There are infinitely many primes that can be written as sums of squares of actual primes.

But it’s not obvious. They must analyze a special set of functions, called Type I and Type II sums, for each version of the problem and then prove that these sums are equivalent regardless of which constraints are used. Only then would Green and Soni learn that they could plug rough prime numbers into their proof without losing information.

They soon realized: they could prove that these sums were equivalent using tools that each of them had encountered independently in previous work. The tool is called the Gaussian norm and was developed by the mathematician decades ago Gowers A measure of how random or structured a function or set of numbers is. On the surface, the Gauss norm seems to belong to an entirely different area of ​​mathematics. “As an outsider, it’s almost impossible to tell whether these things are related,” Soni said.

But using landmark results demonstrated by mathematicians in 2018 Terence Tao and Tamar ZieglerGreen and Sawhney found a way to relate the Gowers norm to the sum of Type I and Type II. Essentially, they needed to use the Gaussian norm to prove that their two sets of primes—the set built using crude primes and the set built using real primes—were similar enough.

It turns out Sony knows how to do it. Earlier this year, to solve an unrelated problem, he developed a technique for comparing sets using the Gaussian norm. To his surprise, the technique was enough to show that both groups had the same sum of Type I and Type II.

With this, Green and Soni proved Friedlander and Ivanik’s conjecture: there are infinitely many prime numbers that can be written as p² + 4q². Eventually, they were able to extend their results to show that there are infinitely many primes belonging to other types of families as well. The results mark a major breakthrough on a class of problems where progress is often difficult to achieve.

More importantly, this work shows that the Gaussian norm can serve as a powerful tool in new fields. “Because it’s so new, at least in this part of number theory, there’s potential to do a lot of other things with it,” Friedlander said. Mathematicians are now hoping to expand the scope of the Gaussian norm even further – trying to use it to solve other problems in number theory besides counting prime numbers.

“It’s fun for me to see unexpected new applications for things I thought of a while ago,” Ziegler said. “It’s like as a parent, when you let your children be free, they grow up and do mysterious, unexpected things.”