A conjecture of John Friedlander and Henryk Iwaniec has inspired a recent breakthrough in number theory. John discusses the circle of ideas in this article:
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.”
Read more in Quanta Magazine