For centuries, prime numbers have fascinated mathematicians, who continuously seek new patterns to recognize and analyze their distribution among other integers. A prime number is defined as an integer greater than 1 that can only be divided evenly by 1 and itself. The smallest prime numbers are 2, 3, and 5. Identifying these initial primes is straightforward; one simply checks which numbers can divide them. However, as mathematicians expand their focus, identifying primes within larger sets of numbers becomes increasingly challenging. For instance, it may be feasible to check smaller groups of numbers, such as those up to 10 or 1,000, but impractical when addressing significantly larger candidates. A notable example is the largest known prime number, which is 2136279841 – 1, consisting of 41,024,320 digits. While it appears overwhelmingly large, it is rather conservative compared to the infinite array of positive integers that exist.
Moreover, mathematicians aim to broaden their investigation beyond tedious factor analysis that typically defines whether a specific integer is prime. “We are intrigued by prime numbers because there are infinitely many of them, yet identifying their patterns proves to be extremely difficult,” states Ken Ono, a mathematician at the University of Virginia. A primary objective remains to understand how prime numbers are distributed through a larger numerical continuum.
Recently, Ono along with his colleagues—William Craig from the US Naval Academy and Jan Willem van Ittersum from the University of Cologne—have unveiled a novel approach to discovering prime numbers. “We have articulated various new criteria to pinpoint sets of prime numbers, drastically different from the conventional method of addressing divisibility,” remarks Ono. Their research, detailed in a paper published in the Proceedings of the National Academy of Sciences, highlights that their findings redefine what it means for a number to be prime.
Central to their strategy is the concept of integer partitions. “This theory, which dates back to the Swiss mathematician Leonhard Euler in the 18th century, continues to evolve through mathematical research,” according to Ono. He explains, “At first glance, partitions may appear simplistic—how many ways can I combine numbers to reach another number?” For example, the number 5 can be partitioned in seven distinct ways: 4+1, 3+2, 3+1+1, 2+2+1, 2+1+1+1, and 1+1+1+1+1. However, this concept is proving to be a pivotal key for unearthing new methods for detecting prime numbers. “It’s remarkable how we can apply these classic combination tools (partition functions) to discover primes in this innovative manner,” adds Kathrin Bringmann, a mathematician from the University of Cologne. Ono mentions that the inspiration for this approach stemmed from an inquiry posed by one of his former students, Robert Schneider, who is currently a mathematician at the Michigan University of Technology.
Ono, Craig, and Van Ittersum have demonstrated that a prime number is a solution to an infinite class of polynomial equations involving partition functions, specifically known as the Ziphantin equation, named after the mathematician Diophantus of Alexandria from the 3rd century. These expressions can yield both integer and rational solutions (the latter being expressible as fractions). Essentially, the researchers concluded that “integer partitions reveal primes in infinitely numerous natural manners,” as stated in their pnas paper.
Edited by George Andrews, a mathematician at Penn State University. The pnas paper was not a part of the research but comments on its findings as “brand new” or “unexpected,” complicating prediction of their potential implications.
Related: What is the largest known prime number?
Their discoveries extend beyond simply analyzing the distribution of prime numbers. “We are accurately identifying every significant number,” says Ono. This methodology allows for the insertion of two or more integers into a specific equation. If validated, the integer is designated as prime. One example of such an 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 appropriately studied partition functions. “More broadly,” the researchers affirm, “this indicates that there exists an infinite number of such prime detection equations with constant coefficients” as discussed in their pnas paper. Put more succinctly, “our work provides endless new definitions for what characterizes a prime,” shares Ono. “It’s truly fascinating.”
The findings of the team may pave the way for numerous new discoveries, according to Bringmann. “Beyond essential mathematical interests, this work could inspire further inquiries into the unexpected algebraic or analytical properties masked within the combined function,” she remarks. Combinations can illustrate the variety of ways items can be selected or arranged based on combinatorial mathematics. “More broadly, it illustrates the richness of connections within mathematics,” she adds. “Such results frequently inspire innovative thinking across multiple subfields.”
Bringmann suggests several avenues for further exploration by mathematicians. For instance, partition functions could be employed to uncover other mathematical structures or examine various number types that could extend the main findings. “Could there be generalizations of the primary results concerning other sequences, such as compound numbers or arithmetic functions?” she inquires.
“In my view, Ken Ono stands as one of the most captivating mathematicians today,” remarks Andrews. “This is not the first time he has tackled classical issues and produced something genuinely novel.”
Significant unresolved public inquiries surrounding prime numbers continue to exist, some dating back many years. Two notable examples include twin prime conjecture and Goldbach’s conjecture. The twin prime conjecture asserts that there are infinitely many primes that differ by two values, like 11 and 13 or 5 and 7. Goldbach’s conjecture states that every even integer greater than two can be expressed as the sum of two primes in at least one way. However, no proof exists for this conjecture.
“Such challenges have perplexed mathematicians and number theorists for generations, throughout the evolution of number theory,” states Ono. While the recent discoveries of his team do not resolve these issues, they exemplify how mathematicians consistently stretch the boundaries of their understanding to unravel the enigmatic nature of prime numbers.
This article originally appeared in Scientific American. © ScientificAmerican.com. Unauthorized reproduction is forbidden. Follow me on TikTok and Instagram, Twitter, and Facebook.
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