A new method for counting prime numbers emerges.
To advance in one of the most fundamental questions of number theory, two mathematicians turned to an unexpected source.
A new breakthrough in the understanding of prime numbers has been achieved by mathematicians, who have presented a proof focusing on the hidden order of these “atoms of arithmetic.” Prime numbers, which are only divisible by themselves and one, are the cornerstone of mathematics but also its greatest mystery. At first glance, they appear to be randomly scattered along the number line; however, they are not random, as their distribution is entirely determined and reveals fascinating patterns that mathematics has been attempting to decipher for centuries.
Although there are formulas that provide an approximation of the location of primes, mathematicians have had to rely on indirect methods to study them. Since Euclid demonstrated in 300 BC that there are infinitely many prime numbers, more specific theorems for certain types of prime numbers have been developed. Over time, the restrictions in these theorems have become stricter, allowing mathematicians to learn more about the distribution of prime numbers. However, such claims are complicated to prove.
Recently, two mathematicians, Ben Green from the University of Oxford and Mehtaab Sawhney from Columbia University, managed to prove a statement for a particularly complex category of prime numbers. Their proof, published in October, not only deepens the mathematical understanding of primes but also employs tools from a very different area of mathematics, suggesting that these tools may be more useful than initially thought.
Throughout the history of mathematics, researchers have explored sets of primes that are interesting yet manageable. Many of these studies have led to significant advancements in number theory. For example, Pierre de Fermat conjectured in 1640 that there are infinitely many prime numbers that can be expressed as the sum of the squares of two integers. This conjecture was later proven by Leonhard Euler, but by imposing new restrictions on the question, mathematicians have encountered much more challenging problems.
In July, Green and Sawhney met at a conference in Edinburgh. Although both were mathematicians with different backgrounds, they decided to collaborate on the Friedlander and Iwaniec conjecture. Inviting Sawhney to Oxford turned out to be key to their progress, as they implemented an innovative approach to solve the problem, sidestepping the difficulties posed by the strict way that defined their problem.
While they could not directly count the primes generated by summing the squares of other primes, they chose to relax the restriction slightly and work with what they called "imprecise primes.” These are easier to find and allow for better mathematical manipulation. Ultimately, Green and Sawhney demonstrated that there are infinitely many primes that can be expressed in the form of p² + 4q², where p and q are primes. This discovery is an important milestone in the realm of prime numbers and has opened the door to new applications of Gowers norms in number theory.
The collaboration between both mathematicians not only resulted in this achievement but also highlighted the potential for future applications in other areas of mathematics, opening a range of opportunities to explore unresolved problems in number theory.