For centuries, prime numbers have captured the imagination of mathematicians, who continue to seek new models that help identify them and how they are distributed among other figures. The prime numbers are whole numbers greater than 1 and are only divisible by 1 and themselves. The three smallest prime numbers are 2, 3 and 5. It is easy to know if small numbers are privileged – you just have to check which figures can take them into account. When mathematicians, however, consider a large number, the task of discernment which are privileged Mushrooms quickly in difficulty. Although it can be practical to check whether, for example, figures 10 or 1000 have more than two factors, this strategy is unfavorable or even untenable to check whether the gigantic numbers are first or composite. For example, the The biggest known first order numberwhich is 2136279841 – 1, measure 41,024,320 figures long. At first, this number may seem greater in the mind. Since there are infinity of positive integers of all different sizes, this number is tiny compared to even larger prime numbers.
In addition, mathematicians want to be more than simply trying to such Factor numbers one by one To determine if a given integer is essential. “We are interested in prime numbers because there are much of them, but it is very difficult to identify all the models,” explains Ken Ono, a mathematician at the University of Virginia. However, a main objective is to determine how prime numbers are distributed in larger numbers.
Recently, Ono and two of his colleagues – William Craig, a mathematician at US Naval Academy, and Jan -Willem Van Ittersum, mathematician at the University of Cologne in Germany – identified a brand new approach to find prime numbers. “We have infinitely described new types of criteria to determine exactly the set of prime numbers, which are all very different from ‘if you cannot take it into account, it must be first’ ‘, says Ono. He and his colleagues’ paper, published in the Proceedings of the National Academy of Sciences USA,, was a finalist for a prize in physical sciences which recognizes scientific excellence and originality. In a certain sense, the discovery offers an infinite number of new definitions so that the numbers are privileged, notes Ono.
At the heart of the team’s strategy is a notion called Integer partitions. “The theory of scores is very old,” explains Ono. He dates back to the Swiss mathematician of the 18th century Leonhard Euler, and he continued to be extended and refined by mathematicians over time. “The scores, at first glance, seem to be the fabric of the child’s game,” said Ono. “How many ways can you add numbers to get other numbers?” For example, the number 5 has seven partitions: 4 + 1, 3 + 2, 3 + 1 + 1, 2 + 2 + 1, 2 + 1 + 1 + 1 and 1 + 1 + 1 + 1.
However, the concept is powerful as a hidden key which unlocks new ways of detecting prime numbers. “It is remarkable that such a classic combinatorial object – the score function – can be used to detect prime numbers in this new way,” explains Kathrin Bringmann, mathematician at the University of Cologne. (Bringmann has already worked with Ono and Craig, and she is currently the postdoctoral advisor of Van Ittersum, but she was not involved in this research.) Ono notes that the idea of this approach is from a question posed by one of his former students, Robert Schneider, who is now a mathematician at Michigan Technological University.
Ono, Craig and Van Ittersum have proven that prime numbers are the solutions of an infinite number of a particular type of polynomial equation in partition functions. Appointed Diophantine equations After the mathematician of the third century Diophantus of Alexandria (and studied long before him), these expressions can have whole or rational solutions (which means that they can be written as a fraction). In other words, the discovery shows that “the whole partitions detect the prime numbers in an infinitely”, wrote the researchers in their PNA paper.
George Andrews, mathematician at the Pennsylvania State University, who edited the PNA Document but was not involved in research, describes the observation as “something brand new” and “not something that was expected”, which makes it difficult to predict “where it will lead”.
In relation: What is the greatest known main number?
The discovery goes beyond probe the distribution of prime numbers. “We actually nail all the prime numbers on the nose,” said Ono. In this method, you can connect an integer which is 2 or more in particular equations, and if they are true, then the whole is first. One of these equations is (3n3 – 13n2 + 18n – 8)M1(n) + (12n2 – 120n + 212)M2(n) – 960M3(n) = 0, where M1(n),, M2(n) And M3(n) are well -studied partition functions. “More generally”, for a particular type of partition function “, we prove that there are many detection equations of those of this type with constant coefficients”, wrote the researchers in their PNA paper. In terms more simply, “it’s almost as if our work gives you new definitions for bonus,” says Ono. “It’s a bit breathtaking.”
The team’s conclusions could lead to many new discoveries, Bringmann notes. “Beyond her intrinsic mathematical interest, this work can inspire additional investigations on the surprising algebraic or analytical properties hidden in combinatorial functions,” she says. In combination – Counting mathematics – Combinatorial functions are used to describe the number of ways whose elements in the sets can be chosen or arranged. “More broadly, it shows the richness of mathematics connections,” she adds. “These types of results often stimulate a new reflection on sub-champs.”
Bringmann suggests potential means that mathematicians could rely on research. For example, they could explore which other types of mathematical structures could be found using partition functions or search for means for the main results to study different types of numbers. “Are there generalizations of the main result to other sequences, such as composite numbers or the values of arithmetic functions?” she asked.
“Ken Ono is, in my opinion, one of the most exciting mathematicians today,” said Andrews. “It is not the first time that he has seen a classic problem and has put really new things.”
There remains a superabundance of Open questions about prime numbersmany of which are for a long time. Two examples are the twin conjecture And Goldbach conjecture. The Twin Prime conjecture indicates that there is an infinity of two numbers with two numbers which are separated by a value of two. Numbers 5 and 7 are prime numbers, as is 11 and 13. The Goldbach conjecture indicates that “each uniform number of more than 2 is a sum of two prime numbers in a way at least in a way,” says Ono. But no one has proven that this conjecture was true.
“Problems like this have confused mathematicians and theorists of numbers for generations, almost throughout the history of the theory of numbers”, explains Ono. Although the recent discovery of his team does not solve these problems, he says, this is a deep example of the way mathematicians push the limits to better understand the mysterious nature of prime numbers.
This article was published for the first time at American scientist. © Scienticamer.com. All rights reserved. Follow Tiktok and Instagram,, X And Facebook.
