Mathematicians found a completely new way to detect prime numbers
Kyiv • UNN
Scientists have discovered a new way to find prime numbers using number partition theory. This revolutionary discovery could change the approach to studying prime numbers.
Mathematicians have discovered an entirely new way of finding prime numbers, reports Live Science, writes UNN.
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For centuries, prime numbers have captivated the imagination of mathematicians, who continue to search for new patterns that help identify them and how they are distributed among other numbers.
Prime numbers are whole numbers greater than 1 that are only divisible by 1 and themselves. The three smallest prime numbers are 2, 3, and 5. It is easy to find out if small numbers are prime - you just need to check which numbers can factor them.
However, when mathematicians consider large numbers, the task of distinguishing which are prime quickly becomes complicated. While it may be practical to check whether, say, 10 or 1000 have more than two factors, this strategy is disadvantageous or even impossible for checking whether giant numbers are prime or composite. For example, the largest known prime number, which is 2^136,279,841 - 1, has a length of 41,024,320 digits. At first, this number may seem overwhelmingly large. However, given that there are infinitely many positive integers of all different sizes, this number is minuscule compared to even larger prime numbers.
Moreover, mathematicians want to do more than just tediously try to factor numbers one by one to determine if a number is prime. "We are interested in prime numbers because there are infinitely many of them, but it is very difficult to detect any patterns in them," says Ken Ono, a mathematician at the University of Virginia. However, one of the main goals is to determine how prime numbers are distributed in larger sets of numbers.
Recently, Ono and two of his colleagues - William Craig, a mathematician at the U.S. Naval Academy, and Jan-Willem van Ittersum, a mathematician at the University of Cologne in Germany - identified a completely new approach to finding prime numbers.
"We described a host of new kinds of criteria to precisely define the set of prime numbers, and they are all very different from 'If you can't factor it, it must be prime,'" says Ono.
His and his colleagues' article, published in the Proceedings of the National Academy of Sciences USA, took second place at the Physical Science Award, which recognizes scientific excellence and originality. In a sense, this discovery offers a host of new definitions of what it means for numbers to be prime, Ono notes.
At the heart of the team's strategy is a concept called number partitioning. "Partition theory is very old," says Ono. It originates from the 18th-century Swiss mathematician Leonhard Euler, and over time, mathematicians continue to expand and refine it. "Partitions, at first glance, seem like child's play," says Ono. "How many ways can you combine numbers to get other numbers?" For example, the number 5 has seven partitioning options: 4+1, 3+2, 3+1+1, 2+2+1, 2+1+1+1, and 1+1+1+1+1.
However, this concept proves to be powerful like a hidden key that unlocks new ways to detect prime numbers. "It is remarkable that such a classic combinatorial object - the partition function - can be used to detect prime numbers in such a new way," says Kathrin Bringmann, a mathematician at the University of Cologne.
Ono, Craig, and van Ittersum proved that prime numbers are solutions to an infinite number of a certain type of polynomial equations in partition functions. In other words, the discovery shows that "integer partitions reveal prime numbers in infinitely many natural ways," the researchers wrote in their PNAS article.
George Andrews, a mathematician at Pennsylvania State University who edited the PNAS article but was not involved in the research, describes the discovery as "something new" and "not what was expected," which makes it difficult to predict "where this will lead."
The team's findings could lead to many new discoveries, Bringmann notes. "Besides its intrinsic mathematical interest, this work can inspire further research into the amazing algebraic or analytic properties hidden in combinatorial functions," she says. In combinatorics - the mathematics of counting - combinatorial functions are used to describe the number of ways in which elements in sets can be chosen or ordered. "More broadly, it shows the richness of connections in mathematics," she adds. Such results often stimulate fresh thinking in subfields.
"Ken Ono is, in my opinion, one of the most interesting mathematicians of our time," says Andrews. "This is not the first time he has delved into a classic problem and brought truly new things to light."
