Like physics, mathematics has its own “fundamental particles,” or prime numbers, which cannot be decomposed into smaller natural numbers. They can only be divided by themselves and by 1.
In a new development, it turns out that these mathematical “particles” offer a new way to tackle physics’ deepest mysteries. Over the past year, researchers have discovered that a mathematical formula based on prime numbers can explain the characteristics of black holes. Number theorists have spent hundreds of years deriving theorems and conjectures based on prime numbers. These new relationships suggest that the mathematical truths that govern prime numbers may also govern some fundamental laws of the universe. So, can physics be expressed in terms of prime numbers?
A black hole is the place where the strongest gravitational force in the universe acts. At their center is a single point called a singularity, where classical physics predicts gravity to be infinite and our understanding of space and time collapses. But in the 1960s, physicists discovered that a kind of chaos emerges immediately around singularities. And it’s very similar to the kind of chaos we found recently with prime numbers.
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Physicists hope to exploit this connection. “I think a lot of high-energy physicists don’t really know much about that side of number theory,” says Eric Perlmutter of the Thackray Institute for Theoretical Physics.
The conjecture underlying number theory regarding prime numbers is the Riemann conjecture of 1859. German mathematician Bernhard Riemann presented a formula containing two principal terms in a handwritten paper. The first provided a surprisingly accurate estimate of how many prime numbers exist that are smaller than a given number. The second term is the zeta function, whose zeros (where the function equals zero) adjust the original estimate. The mysterious way in which zeta zero always improves estimates is the subject of the Riemann conjecture. This hypothesis is so important to number theory that anyone who can prove it will receive a $1 million Clay Mathematics Institute prize.
In the late 1980s, physicists began to wonder if there were physical systems in which energy levels could be based on prime numbers. Bernard Julia, a physicist at France’s École Normale Supérieur, was challenged by a colleague to find an analogue to the physics described by the zeta function. His solution was to propose a hypothetical kind of particle with energy levels given by the logarithm of a prime number. Julia called these particles “primons” and groups of them “primons”. This gas partition function (a survey of the possible states of the system) is just the Riemann zeta function.
At the time, Julia’s concept was a thought experiment. Most scientists doubted whether Primon actually existed. But deep inside the black hole, a mathematical connection was waiting to be discovered. More than 20 years later, physicists Jan Fedorov of King’s College London, Geis Heary of Ohio State University, and John Keating of Oxford University discovered hints that fractal chaos emerges from zero fluctuations in the zeta function, an idea that was definitively proven in 2025.
Einstein’s theory of general relativity shows that the same chaos occurs near singularities.
In a February 2025 preprint, University of Cambridge physicist Sean Hartnall and graduate student Ming Yang brought Julia’s research to the real world. They found that “conformal” symmetry appears inside the chaos near the singularity. Hartnoll likens conformal symmetry to Dutch artist MC Escher’s famous bat paintings. The same structure is repeated at different scales. This scaling symmetry and a bit of mathematics revealed a quantum system near the singularity where the spectrum organizes into prime numbers: a conformal main gas cloud.
Five months later, they uploaded a preprint with new developments. The research team, which also included physicist Marine de Klerk from the University of Cambridge, extended their analysis to a five-dimensional universe rather than the usual four-dimensional universe. They discovered that the additional dimensions required new capabilities. To track the dynamics of the singularity, we now needed a “complex” prime number, known as a Gaussian prime, with an imaginary component (a number multiplied by the square root of -1). Gaussian primes cannot be further divided by other complex numbers. The authors named this system “complex master gas.”
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“It remains to be seen whether the appearance of randomness in prime numbers near the singularity has a deeper meaning,” Hartnoll says. “But in my opinion, it is very interesting that the relevance extends to higher-dimensional theories of gravity, including some candidates for a fully quantum mechanical theory of gravity.”
And in a late 2025 preprint, Perlmutter proposed a new framework that includes Zeta Zero. He relaxed the restrictions on the zeta function, allowing it to depend not only on integers but also on all real numbers, including irrational numbers. In doing so, it opened up an even more powerful zeta function approach to understanding quantum gravity. John Keating, a physicist at the University of Oxford who was not involved in the study, said such a broader perspective could reveal new ways to tackle long-standing problems. “It’s only when you step back and look at the whole mountain that you think, ‘Oh, there’s a better way to get up there,'” he says.
Perlmutter is cautiously hopeful that the wave of major physics will hasten new discoveries, but his approach is one of many fighting for acceptance. “The kinds of things we’re trying to understand, quantum gravitational black holes, are certainly dominated by some beautiful structures,” he says. “And number theory is like a natural language.”
This article first appeared in Scientific American. ©ScientificAmerican.com. All rights reserved. Follow us on TikTok, Instagram, X, and Facebook.
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