For hundreds of years, mathematicians have been fascinated by prime numbers, continually seeking new patterns that can help identify them and understand how they are spread out among other numbers. Prime numbers are defined as whole numbers greater than 1 that can only be divided evenly by 1 and themselves. The smallest three prime numbers are 2, 3, and 5. Recognizing if smaller numbers are prime is straightforward — one just needs to verify their factors. However, determining the primality of larger numbers becomes increasingly complex. While testing numbers like 10 or 1,000 for factors is manageable, applying this to extremely large numbers is impractical. For example, the largest known prime number, which is 2136279841 − 1, spans an astonishing 41,024,320 digits. Although this might seem enormous, it pales in comparison to even more substantial prime numbers.
Moreover, mathematicians aspire to find more efficient methods than simply trying to factor numbers individually to assess if a number is prime. “We are interested in prime numbers because there are infinitely many of them, yet identifying patterns within them is challenging,” states Ken Ono, a mathematician at the University of Virginia. A primary goal remains to understand how prime numbers are distributed among larger groups of numbers.
Recently, Ono and his colleagues — William Craig from the U.S. Naval Academy and Jan-Willem van Ittersum of the University of Cologne in Germany — discovered a completely new method for identifying prime numbers. “We have defined infinitely many new criteria for precisely determining the set of prime numbers, all distinctly different from the idea that ‘if you can’t factor it, it must be prime,'” says Ono. Their paper, published in the Proceedings of the National Academy of Sciences USA, was awarded a runner-up title for a prize recognizing scientific excellence and innovation. In essence, this finding presents an infinite array of new definitions for prime numbers, according to Ono.
Central to the team’s approach is a concept known as integer partitions. “The theory of partitions has a long history,” Ono says, tracing back to the 18th-century Swiss mathematician Leonhard Euler, with further developments by mathematicians over time. “At first glance, partitions may seem simplistic, like a child’s game — how can you combine numbers to yield other numbers?” For instance, the number 5 can be partitioned in seven different ways: 4 + 1, 3 + 2, 3 + 1 + 1, 2 + 2 + 1, 2 + 1 + 1 + 1, and 1 + 1 + 1 + 1 + 1.
However, this concept serves as a powerful tool that unveils new methods for identifying primes. “It is quite striking that such a classical combinatorial idea — the partition function — can be utilized to detect primes in this innovative manner,” states Kathrin Bringmann, a mathematician at the University of Cologne. (Bringmann has collaborated with Ono and Craig in the past, and she currently mentors van Ittersum, but she was not involved in this particular research.) The inspiration for this method arose from a question posed by one of Ono’s former students, Robert Schneider, now a mathematician at Michigan Technological University.
Ono, Craig, and van Ittersum demonstrated that prime numbers are solutions to an infinite number of specific polynomial equations related to partition functions. These equations, known as Diophantine equations, named after the ancient mathematician Diophantus of Alexandria, can yield integer or rational solutions (expressible as fractions). Their finding indicates that “integer partitions reveal the primes in infinitely many natural ways,” as noted in their PNAS paper.
George Andrews, a mathematician at Pennsylvania State University who edited the PNAS paper but didn’t take part in the research, characterizes the discovery as “completely new” and “unexpected,” making its future implications hard to foresee.
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This new discovery goes beyond just looking into the distribution of prime numbers. “We are accurately pinpointing every prime number,” Ono notes. Through this new method, inputting any integer 2 or higher into specific equations confirms its primality if the equations hold true. 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) represent well-studied partition functions. More generally, for a special category of partition function, “we demonstrate that infinitely many prime-detecting equations with constant coefficients exist,” as detailed in their PNAS paper. Simply put, “our work offers an infinite number of new definitions for primes,” states Ono. “That’s truly astonishing.”
The implications of this research could lead to many new findings, according to Bringmann. “Aside from its intrinsic mathematical significance, this work may ignite further exploration into the intriguing algebraic or analytic properties found in combinatorial functions,” she remarks. In combinatorics — the study of counting — combinatorial functions describe the various ways items can be selected or arranged. “More broadly, it highlights the rich interconnections within mathematics,” she adds. “Such findings often inspire new ideas across different subfields.”
Bringmann proposes potential avenues for further exploration based on the research. Mathematicians could investigate what other mathematical structures might be uncovered using partition functions or how the main results could be adapted to explore other kinds of numbers. “Can we broaden the main findings to include other sequences, like composite numbers or values of arithmetic functions?” she queries.
“Ken Ono is, in my view, one of the most exciting mathematicians today,” Andrews states. “This isn’t the first time he has reevaluated a classic problem and revealed truly new insights.”
Numerous open questions about prime numbers remain, many of which have puzzled mathematicians for ages. Two notable examples are the twin prime conjecture and Goldbach’s conjecture. The twin prime conjecture claims that there are infinitely many twin primes — pairs of prime numbers that differ by two. For instance, 5 and 7 are twin primes, as are 11 and 13. Meanwhile, Goldbach’s conjecture asserts that “every even number greater than 2 can be expressed as the sum of two primes in at least one way,” states Ono. However, this conjecture has yet to be proven true.
Matters like these have baffled mathematicians and number theorists for generations, spanning nearly the entire history of number theory,” Ono remarks. Although his team’s recent discovery does not resolve those questions, it exemplifies how mathematicians are pushing the limits to gain a better understanding of the enigmatic world of prime numbers.
This article was first published at Scientific American. © ScientificAmerican.com. All rights reserved. Follow on TikTok and Instagram, X and Facebook.
