Two mathematicians have solved a prime number puzzle that seemed impossible to decipher

Prime numbers have a lot of charm. They have it, at least, for many mathematicians who find them indisputably attractive. After all, they are very special. A very brief review in case anyone is a little confused: prime numbers are integers greater than 1 that only are divisible by 1 and by themselves. That’s all. 2, 3, 5, 7 or 11 are prime numbers, among many others. Said like that, they don’t seem like much, but in reality, they are very important. And they are because they have relatively exotic properties.

The most obvious of all of them is that they are infinite, but, in addition, some are twins because they are separated by 2 (such as 3 and 5 or 11 and 13); Its distribution seems irregular, although it is not; or they can take special forms, such as, for example, the Mersenne or Fermat primes (we are not going to delve into them so as not to unnecessarily complicate this article). Be that as it may, these numbers are the true protagonists of a story that has captivated us. We hope that you, our readers, find it as interesting as we do.

Researchers now know a little better where prime numbers “live”

Mathematicians have flirted with these numbers for centuries. Even the greatest have been seduced by them. About 300 years BC. C. the Greek mathematician Euclid demonstrated that there are infinitely many prime numbers, and since then an infinite number of mathematicians have set out to find out with the greatest possible precision how they are distributed along the number line. More than two millennia have passed, and yes, we now know better than ever how to find them and what their properties are, but this does not mean that all the work is done.

For mathematicians, the search for a rule that allows us to describe how they are distributed has become a real puzzle. A millennia-old puzzle. Fortunately, Ben Green, from the University of Oxford (England), and Mehtaab Sawhney, from Columbia University (USA), have managed to demonstrate that there are a very challenging type of prime numbers that satisfies a rigorous constraint. These types of demonstrations are not at all frequent, and this one in particular has established itself as a very important contribution that allows mathematicians to better understand prime numbers.

Green and Sawhney have published their proof on the open access repository arXiv, and in some ways it is a work of art. A work of art in the field of mathematics. Their starting point when they decided to embark on the adventure that concluded with the publication of their scientific article was Friedlander and Iwaniec’s conjecture. These two mathematicians from Rutgers University, in New Jersey (USA), asked themselves in 2018 if there are infinitely many prime numbers that can be described by the equation p² + 4q², where both p and q must also be prime.

Green and Sawhney rightly thought that directly counting all the prime numbers that satisfy that condition was unfeasible, but they realized that by relaxing that requirement a little they could solve this problem. And they did so assuming that squared numbers only They had to be approximately cousins. Approximate primes are easier to locate than regular primes. And they are not prime numbers in the traditional sense because it is enough that they are not divisible by the smallest primes to be able to work with them. Even though they aren’t actually cousins.

Mathematicians often work with these types of simplifications to be able to address challenges that seem unsolvable, and there is no problem as long as they explain well what they have done. “Approximate cousins ​​are a set that we understand much, much better,” says Sawhney. And so much. Taking them as a starting point, Ben Green and Mehtaab Sawhney have managed to solve Friedlander and Iwaniec’s conjecture.

In this article we are interested in knowing the history of these two mathematicians and reviewing why prime numbers have so much charmand not so much the details of its demonstration. If you are not intimidated by mathematics, I suggest you take a look at Green and Sawhney’s article, although, fortunately, it is not necessary to understand it to understand what makes prime numbers so special.

Image | Pixabay

More information | arXiv | Quanta Magazine

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