The Horizons of Mathematics, with Ken Ono (University of Virginia, USA)

Ken Ono is Marvin Rosenblum Professor of Mathematics and Chancellor’s STEM Advisor at the University of Virginia (USA). His research areas include data science, mathematics and statistics. Image courtesy of the author.

A new approach based on partition theory allows the construction of modified Diophantine equations whose only integer solutions are prime numbers. Ken Ono (University of Virginia, USA) will present his results at the upcoming Joint Mathematics Colloquium ICMAT-UAM-UCM-UAM on May 9 at 12:30.

Ágata Timón García-Longoria (ICMAT-CSIC)

125 years ago, the mathematician David Hilbert brought attention to the following question: Is there a method to decide whether a Diophantine equation— that is, an equation where both the coefficients and the solutions sought are integers —has a solution or not? This problem, number 10 on Hilbert’s list, was answred negatively in the 1970s by Yuri Matiyasevich. There is no general method or algorithm that always works.

“One very curious feature of this theorem is the crazy corollary that it implies: there is a polynomial whose positive values are the prime numbers,” says Ken Ono, professor at the University of Virginia (USA). This polynomial was discovered in 1976 by James P. Jones, Daihachiro Sato, Hideo Wada, and Douglas Wiens (pictured above).

Despite the fame Hilbert’s list once held within the mathematical community, many of the problems it includes—especially those already solved—have fallen into obscurity. Last year, Ono, was attending a conference when he realized that today’s graduate students were unfamiliar with the famous work from the 1970s on Hilbert’s tenth problem. “A few days later, I started to wonder if there are other similar prime detectors out there, ones that are not as crazy as the polynomial of Wada and his collaborators. A few weeks later, I attended another conference in Cologne (Germany), where I discussed the problem with my former student Will Craig and postdoc Jan Willem van Ittersum” he recalls.

Within minutes, they had an idea that turned out to be the key to creating more manageable analogs of that strange polynomial. Their work, published in the Proceedings of the National Academy of Sciences of the United States of America (PNAS), was selected as a finalist for the 2025 Cozzarelli Prize, which honors the best papers published each year in the journal across different disciplines. Ono will present these results next Friday at 12:30 in the joint mathematics colloquium, under the title Partitions Detect Primes, to be held in Room 520 of the Mathematics Department at the Universidad Autónoma de Madrid (UAM).

Instead of considering classical Diophantine equations, Ono and his collaborators use special Diophantine equations involving so-called partition functions. Partitions are ways of dividing an object. For example, the partitions of a positive integer n are the ways to write n as a sum of positive integers (regardless of the order of the summands). Thus, the five partitions of the number 4 are: 1+1+1+1, 2+1+1, 3+1, 2+2, and 4.

The Indian mathematician Ramanujan studied the partition function in depth and discovered several surprising properties.

“At first glance, partitions may seem like child’s play,” explains Ono. “However, as is often the case in number theory, a simple concept quickly leads to beautiful and fascinating problems. Partitions appear in diverse areas of mathematics, such as combinatorics, Lie theory, representation theory, mathematical physics, or the theory of special functions. The process of breaking numbers into sums is fundamental when studying permutations—that is, rearrangements of sets—or the symmetries of objects,” he elaborates.

Partitions also hold the key to these new special Diophantine equations that detect prime numbers. “Prime numbers, the fundamental building blocks of the natural numbers, turn out to be the solutions to infinitely many special Diophantine equations in well-known partition functions.” In their work, they provide several such expressions (see image below), whose integer solutions greater than one are exactly the prime numbers.

“This approach can be used to locate primes in several ways—beyond simply finding values of n that make the expression equal to zero and then checking that n is prime. These equations can also be encoded as Cauchy-type contour integrals that depend on n. Such integrals vanish only if n is prime,” Ono explains. As a method for tracking primes, these new expressions are much more efficient than the 1976 polynomial, although not as fast as other primality tests or the Sieve of Eratosthenes.

Their advances also have other implications, particularly in the study of symmetric groups. “In fact, our main contribution—which leads to these questions about primes—does not initially seem related to them at all. We obtained a new set of generators for the space of so-called quasimodular forms. Our generators are designed to capture partitions, which allowed us to show that all quasimodular forms are explicit expressions in partitions,” says Ono. “This new perspective is, for many, the main contribution of our work; the consequence regarding primes would be its first application,” he admits.