For the first time in more than 25 years, mathematicians have proven a new way to identify prime numbers, and in doing so developed a toolkit that could allow further advances in number theory.
Prime numbers, which can only be divided by themselves or one, are the mathematical building blocks of whole numbers, and mathematicians have explored how they can be found and combined for hundreds of years. 鈥淣ew results about primes don鈥檛 come along that often, so whenever we get some new development, it feels worthwhile,鈥� says at the University of Oxford.
One of the most famous problems involving primes is Fermat鈥檚 last theorem, a statement first proposed by mathematician Pierre de Fermat in around 1640. It says that there are no integers, or whole numbers, a, b, and c that can fit in the equation an + bn = cn for any integer n greater than 2.
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Because whole numbers can be divided into their constituent primes, mathematicians quickly realised that proving Fermat鈥檚 idea would only require doing so for cases where n is a prime number, says at Rutgers University in New Jersey. 鈥淚t turns out that there are many questions that, if you answer them about the prime numbers, then you actually answer them about all numbers.鈥�
It took until 1993 for Andrew Wiles, then at Princeton University, to finally publish his stunning proof of Fermat鈥檚 last theorem, which also led to breakthroughs in other areas of mathematics related to prime numbers.
A few years later in 1998, mathematicians Henryk Iwaniec and John Friedlander proved a related concept, showing that you could make primes by adding whole numbers in the form x虏 + y4, where one of the numbers themselves was prime, but they were unable to solve a variant of their equation, known as the Gaussian primes conjecture, which says that are an infinite number of pairs of prime numbers x and y such that when combined in the form x虏 + (2y)虏 they will also give a prime number.
Now, Green and at Columbia University in New York have done just that, in the first new result about combining numbers to form primes since Iwaniec and Friedlander.
鈥淧eople were waiting 25 years and they didn鈥檛 know what kinds of techniques would be necessary to get a result of this quality, but [Green and Sawhney] managed to push it through. It鈥檚 a fantastic achievement,鈥� says Kontorovich. 鈥淸I] am greatly impressed,鈥� says Friedlander.
To prove the statement, Green and Sawhney deployed a mathematical toolkit of cutting-edge techniques such as Type II sums, which can help work out the distribution of prime numbers, and Gowers norms, an approach invented by mathematician Timothy Gowers that can reveal the degree to which a set of numbers are chaotic or more structured.
These two techniques are from distant mathematical realms, namely number theory and combinatorics, says Green. 鈥淧robably the most interesting thing about the work is the fact that these two somewhat different types of areas can be combined.鈥�
As well as being an important result in its own right, the tools that Green and Sawhney used to prove the conjecture could help other mathematicians make breakthroughs in other areas. 鈥淭he most exciting thing about it is actually the guts of the proof, which has all kinds of new ideas. Who knows what other breakthroughs these ideas will lead to?鈥� says Kontorovich.
arXiv
Article amended on 22 October 2024
We have corrected the definition of the Gaussian primes conjecture
