Two mathematicians took another perspective: approximate primes are much easier to find than primes.
Os numbers Primes are a fascination – or a problem – for anyone who deals with mathematics on a daily basis.
These numbers, which are divisible only by themselves and 1, are considered one of the fundamental building blocks of mathematics. And also one of the most mysterious.
They seem random, but mathematicians have spent centuries trying to identify strange patterns, with much still to be discovered today, remembers.
But now two mathematicians have adopted another perspective: approximate cousins They are much easier to find than their cousins.
Ben Green of Oxford University and Mehtaab Sawhney of Columbia University have achieved a significant advance in number theory by solving a long-standing problem related to a specific type of prime numbers.
Their discovery clarifies intricate patterns of prime numbers and introduces a versatile mathematical tool – the Gowers norm – to a new domain.
The challenge faced by the duo is based on a conjecture presented in 2018 by John Friedlander and Henryk Iwaniec. The conjecture asks whether there are infinitely many prime numbers of the form \( p^2 + 4q^2 \), where \( p \) and \( q \) are also primes. The restrictive conditions of this problem made it particularly difficult to solve.
With experience in studying prime number patterns, the two mathematicians noticed that traditional counting techniques proved to be inadequate, leading them to resort to an innovative alternative solution: approximate primes.
The approximate cousins are numbers that are not divisible by a small set of primes, functioning as an approximation to real primes.
Using approximate primes, mathematicians proved a less restrictive version of the problem and then faced the more complex task of linking this result to the real primes.
This required the analysis of special mathematical functions and the demonstration of equivalence between results obtained with approximate primes and real primes.
A Gowers norma tool created by Timothy Gowers to measure randomness and structure, played a crucial role in this.
Although originally unrelated to number theory, Green and Sawhney adapted the norm to fill the gap between their results with approximate and real primes.
Their efforts culminated in the proof of Friedlander and Iwaniec’s conjecture.
Furthermore, they extended the work to demonstrate that there are infinitely many prime numbers in other restricted forms.
“This result is spectacular,” said Friedlander.
