For centuries, prime numbers have captured the imaginations of mathematicians, who continue to search for new patterns that help identify them and the way they’re distributed among other numbers. Primes are whole numbers that are greater than 1 and are divisible by only 1 and themselves. The three smallest prime numbers are 2, 3 and 5. It's easy to find out if small numbers are prime—one simply needs to check what numbers can factor them. When mathematicians consider large numbers, however, the task of discerning which ones are prime quickly mushrooms in difficulty. Although it might be practical to check if, say, the numbers 10 or 1,000 have more than two factors, that strategy is unfavorable or even untenable for checking if gigantic numbers are prime or composite. For instance, the largest known prime number, which is 2136279841 − 1, is 41,024,320 digits long. At first, that number may seem mind-bogglingly large. Given that there are infinitely many positive integers of all different sizes, however, this number is minuscule compared with even larger primes.
Furthermore, mathematicians want to do more than just tediously attempt to factor numbers one by one to determine if any given integer is prime. “We’re interested in the prime numbers because there are infinitely many of them, but it’s very difficult to identify any patterns in them,” says Ken Ono, a mathematician at the University of Virginia. Still, one main goal is to determine how prime numbers are distributed within 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 whole new approach for finding prime numbers. “We have described infinitely many new kinds of criteria for exactly determining the set of prime numbers, all of which are very different from ‘If you can’t factor it, it must be prime,’” Ono says. He and his colleagues’ paper, published in the Proceedings of the National Academy of Sciences USA, was runner-up for a physical science prize that recognizes scientific excellence and originality. In some sense, the finding offers an infinite number of new definitions for what it means for numbers to be prime, Ono notes.
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At the heart of the team’s strategy is a notion called integer partitions. “The theory of partitions is very old,” Ono says. It dates back to the 18th-century Swiss mathematician Leonhard Euler, and it has continued to be expanded and refined by mathematicians over time. “Partitions, at first glance, seem to be the stuff of child’s play,” Ono says. “How many ways can you add up numbers to get other numbers?” For instance, 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 + 1.
Yet the concept turns out to be powerful as a hidden key that unlocks new ways of detecting primes. “It is remarkable that such a classical combinatorial object—the partition function—can be used to detect primes in this novel way,” says Kathrin Bringmann, a mathematician at the University of Cologne. (Bringmann has worked with Ono and Craig before, and she’s currently van Ittersum’s postdoctoral adviser, but she wasn’t involved with this research.) Ono notes that the idea for this approach originated in a question posed by one of his former students, Robert Schneider, who’s now a mathematician at Michigan Technological University.
Ono, Craig and van Ittersum proved that prime numbers are the solutions of an infinite number of a particular type of polynomial equation in partition functions. Named Diophantine equations after third-century mathematician Diophantus of Alexandria (and studied long before him), these expressions can have integer solutions or rational ones (meaning they can be written as a fraction). In other words, the finding shows that “integer partitions detect the primes in infinitely many natural ways,” the researchers wrote in their PNAS paper.
George Andrews, a mathematician at Pennsylvania State University, who edited the PNAS paper but wasn’t involved with the research, describes the finding as “something that's brand new” and “not something that was anticipated,” making it difficult to predict “where it will lead.”
The discovery goes beyond probing the distribution of prime numbers. “We’re actually nailing all the prime numbers on the nose,” Ono says. In this method, you can plug an integer that is 2 or larger into particular equations, and if they are true, then the integer is prime. One such equation 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 infinitely many such prime detecting equations with constant coefficients,” the researchers wrote in their PNAS paper. Put more simply, “it’s almost like our work gives you infinitely many new definitions for prime,” Ono says. “That’s kind of mind-blowing.”
The team’s findings could lead to many new discoveries, Bringmann notes. “Beyond its intrinsic mathematical interest, this work may inspire further investigations into the surprising 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 that items in sets can be chosen or arranged. “More broadly, it shows the richness of connections in mathematics,” she adds. “These kinds of results often stimulate fresh thinking across subfields.”
Bringmann suggests some potential ways that mathematicians could build on the research. For instance, they could explore what other types of mathematical structures could be found using partition functions or look for ways that the main result could be expanded to study different types of numbers. “Are there generalizations of the main result to other sequences, such as composite numbers or values of arithmetic functions?” she asks.
“Ken Ono is, in my opinion, one of the most exciting mathematicians around today,” Andrews says. "This isn’t the first time that he has seen into a classic problem and brought really new things to light.”
There remains a glut of open questions about prime numbers, many of which are long-standing. Two examples are the twin prime conjecture and Goldbach’s conjecture. The twin prime conjecture states that there are infinitely many twin primes—prime numbers that are separated by a value of two. The numbers 5 and 7 are twin primes, as are 11 and 13. Goldbach’s conjecture states that “every even number bigger than 2 is a sum of two primes in at least one way,” Ono says. But no one has proven this conjecture to be true.
“Problems like that have befuddled mathematicians and number theorists for generations, almost throughout the entire history of number theory,” Ono says. Although his team’s recent finding doesn’t solve those problems, he says, it’s a profound example of how mathematicians are pushing boundaries to better understand the mysterious nature of prime numbers.
