To make progress on one of number theory’s most elementary questions, two mathematicians turned to an unlikely source.
Mathematicians discovered how to pick out a particular form of prime number from the garden of all primes.
A new proof has brought mathematicians one step closer to understanding the hidden order of those “atoms of arithmetic,” the prime numbers.
The primes — numbers that are only divisible by themselves and 1 — are the most fundamental building blocks in math. They’re also the most mysterious. At first glance, they seem to be scattered at random across the number line. But of course, the primes aren’t random. They’re completely determined, and a closer look at them reveals all sorts of strange patterns, which mathematicians have spent centuries trying to unravel. A better understanding of how the primes are distributed would illuminate vast swaths of the mathematical universe.
But while mathematicians have formulas that give an approximate sense of where the primes are located, they can’t pinpoint them exactly. Instead, they’ve had to take a more indirect approach.
Around 300 BCE, Euclid proved that there are infinitely many prime numbers. Mathematicians have since built on his theorem, proving the same statement for primes that meet additional criteria. (A simple example: Are there an infinite number of primes that don’t contain the number 7?) Over time, mathematicians have made these criteria stricter and stricter. By showing that there are still infinitely many primes that satisfy such increasingly rigid constraints, they’ve been able to learn more about where the primes live.
But these kinds of statements are very difficult to prove. “There are not many results like that out there,” said Joni Teräväinen of the University of Turku in Finland.
Now, two mathematicians — Ben Green of the University of Oxford and Mehtaab Sawhney of Columbia University — have proved just such a statement for a particularly challenging type of prime number. Their proof, which was posted online in October, doesn’t just sharpen mathematicians’ understanding of the primes. It also makes use of a set of tools from a very different area of mathematics, suggesting that those tools are far more powerful than mathematicians imagined, and potentially ripe for applications elsewhere.
“It’s terrific,” said John Friedlander of the University of Toronto. “It really surprised me that they did this.”
A Trying Set
Mathematicians tend to study families of primes that are just complicated enough to be interesting but still simple enough to make progress on. They might try to prove, for instance, that there are infinitely many primes that are 500 units apart. Or that we can build infinitely many primes by adding up the squares of other numbers.
Mehtaab Sawhney realized, to his surprise, that work he’d done earlier this year was key to resolving a major, seemingly unrelated problem in number theory.
This last constraint has been particularly useful, guiding centuries of mathematical progress. In 1640, Pierre de Fermat conjectured that there are infinitely many primes that can be formulated by squaring two whole numbers and adding them together. (The prime number 13, for instance, can be written as 22 + 32.) Leonhard Euler later proved it. But tweaking the question just a little bit — by insisting that one of the numbers you’re squaring be odd, perhaps, or a perfect square — makes the problem much harder. “The more you constrain a set, the harder it is to find primes in it,” Green said.
In the 19th century, research on these kinds of statements led to the development of much of modern number theory. In the 20th century, it helped inspire one of the most ambitious mathematical efforts to date, the Langlands program. And in the 21st, work on these sorts of primes has continued to yield new techniques and insights.
In 2018, Friedlander and Henryk Iwaniec of Rutgers University asked if there are infinitely many primes of the form p2 + 4q2, where both p and q must also be prime. (For example, 41 = 52 + 4 × 22.) The constraint turned out to be particularly challenging to deal with. But if mathematicians could solve the problem, they’d succeed in exerting a new level of control over the primes — exactly what they’d hoped to do all along.
A Fruitful Visit
Neither Green nor Sawhney had played this type of prime-counting game before. But they both had experience in studying the strange patterns the primes give rise to.
In July, the two mathematicians met at a conference in Edinburgh. Sawhney, who had just finished graduate school, had always admired Green. A seminal result Green had proved 20 years earlier was “one of the things that brought me into the subject,” Sawhney said. “I was like ‘Oh man, how could you do this?’” Green was impressed by the younger mathematician, too. “Mehtaab is an exceptional, exceptional mathematician,” he said. “He knows everything somehow.”
The two decided to collaborate. They just had to find the right problem to work on. After some discussion, they settled on Friedlander and Iwaniec’s conjecture.
Ben Green, a mathematician at the University of Oxford, is fascinated by the mysterious patterns that characterize the prime numbers.
Green invited Sawhney to Oxford for a week. They knew that to prove similar conjectures, mathematicians usually relied on a particular suite of counting techniques. But because the primes in their problem were so strictly defined, Green and Sawhney couldn’t figure out a way to get this traditional tool kit to work.
Instead, they hoped to prove the conjecture in a more roundabout way — by making a mathematical chess move of sorts. But first, they’d have to prove that they were allowed to make the move in the first place.
By the end of Sawhney’s visit, he and Green had figured out how to do that — which allowed them to prove the conjecture. To do so, they ended up making a surprising connection to another area of math.
Try Another Set
It wasn’t possible for Green and Sawhney to directly count the number of primes made by squaring two other primes and adding them together. But what if they loosened their constraint just a bit? They realized they could solve a slightly weaker version of their problem — one where the numbers getting squared only had to be “roughly” prime.
Rough primes are much easier to find than primes. Say you want to count all the rough primes between 1 and 200. First, consider a handful of the smallest primes — like 2, 3, 5 and 7. Then list all the numbers that aren’t divisible by those primes. These numbers are the rough primes. In this case, you end up with 50 rough primes: 46 of them are actually prime, while the remaining four (121, 143, 169 and 187) are not. Because the rough primes are much less randomly distributed than the primes are, they’re significantly easier to work with. “The rough primes are a set that we understand much, much better,” Sawhney said.
Tamar Ziegler’s seminal work on prime numbers allowed researchers to port a mathematical technique called a Gowers norm into a new field.
Green and Sawhney proved that there are infinitely many primes that can be made by squaring two rough primes and adding them together. They now just had to show that this statement would imply the problem they actually wanted to solve: There are also infinitely many primes that can be written as the sum of squares of actual primes.
But that wasn’t obvious. They’d have to analyze a special set of functions, called Type I and Type II sums, for each version of their problem, then show that the sums were equivalent no matter which constraint they used. Only then would Green and Sawhney know they could substitute rough primes into their proof without losing information.
They soon came to a realization: They could show that the sums were equivalent using a tool that each of them had independently encountered in previous work. The tool, known as a Gowers norm, was developed decades earlier by the mathematician Timothy Gowers to measure how random or structured a function or set of numbers is. On its face, the Gowers norm seemed to belong to a completely different realm of mathematics. “It’s almost impossible to tell as an outsider that these things are related,” Sawhney said.
But using a landmark result proved in 2018 by the mathematicians Terence Tao and Tamar Ziegler, Green and Sawhney found a way to make the connection between Gowers norms and Type I and II sums. Essentially, they needed to use Gowers norms to show that their two sets of primes — the set built using rough primes, and the set built using real primes — were sufficiently similar.
As it turned out, Sawhney knew how to do this. Earlier this year, in order to solve an unrelated problem, he had developed a technique for comparing sets using Gowers norms. To his surprise, the technique was just good enough to show that the two sets had the same Type I and II sums.
With this in hand, Green and Sawhney proved Friedlander and Iwaniec’s conjecture: There are infinitely many primes that can be written as p2 + 4q2. Ultimately, they were able to extend their result to prove that there are infinitely many primes belonging to other kinds of families as well. The result marks a significant breakthrough on a type of problem where progress is usually very rare.
Even more important, the work demonstrates that the Gowers norm can act as a powerful tool in a new domain. “Because it’s so new, at least in this part of number theory, there is potential to do a bunch of other things with it,” Friedlander said. Mathematicians now hope to broaden the scope of the Gowers norm even further — to try using it to solve other problems in number theory beyond counting primes.
“It’s a lot of fun for me to see things I thought about some time ago have unexpected new applications,” Ziegler said. “It’s like as a parent, when you set your kid free and they grow up and do mysterious, unexpected things.”