Like physics, mathematics has its own set of “fundamental particles“- the one prime numberwhich cannot be broken down into smaller natural numbers. They can only be divided between themselves and 1.
And in a new development, it turns out that these mathematical “particles” offer new ways to tackle some of physics’ deepest mysteries. Over the past year, researchers have found that formulas based on the prime numbers can describe features of black holes. Number theorists have spent hundreds of years deriving theorems and assumptions based on prime numbers. These new connections 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?
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Physicists hope to make use of the connection. “I would say that many high-energy physicists actually don’t know much about that side of number theory,” says Eric Perlmutter of the Institute of Theoretical Physics, Saclay.
Number theory’s basic conjecture about prime numbers is the Riemann hypothesis from 1859. In a handwritten paper, the German mathematician Bernhard Riemann gave a formula with two main terms. The first gave a startlingly close estimate of how many prime numbers there are that are smaller than a given number. The second term is the zeta function, whose zeros (the places where the function equals zero) adjust the original estimate. The mysterious way the zeta zeros always improve the estimate is the subject of the Riemann hypothesis. The hypothesis is so crucial to number theory that anyone who can prove it will earn a $1 million Clay Mathematics Institute prize.
In the late 1980s, physicists began to wonder if there was a physical system whose energy levels might be based on the prime numbers. Physicist Bernard Julia at the École Normale Supérieure in France was challenged by a colleague to find a physics analogue described by the zeta function. His solution was to propose a hypothetical type of particle with energy levels given by the logarithms of prime numbers. Julia called these particles “primones” and a group of them a “primone gas”. The partition function—a count of a system’s possible states—for this gas is exactly the Riemann zeta function.
At the time, Julia’s concept was a thought experiment—most scientists doubted that primones actually existed. But deep inside black holes, a mathematical link was waiting to be discovered. A little more than two decades later, physicists Yan Fyodorov of King’s College London, Ghaith Hiary of Ohio State University, and Jon Keating of the University of Oxford saw hints that fractal chaos emerges from the fluctuations in the zeros of the zeta function, an idea that was crucial. proven in 2025.
Einstein’s general theory of relativity shows that the same chaos also occurs near a singularity.
In a February 2025 preprint, University of Cambridge physicist Sean Hartnoll and doctoral student Ming Yang brought Julia’s work into the real world. Inside the chaos near a singularity, they found that a “conformal” symmetry arises. Hartnoll compares conformal symmetry to Dutch artist MC Escher’s famous drawings of bats — the same structure repeats itself on different scales. This scaling symmetry, along with some mathematics, revealed a quantum system near the singularity whose spectrum is organized into primes—a conformal primon gas cloud.
Five months later, they uploaded a preprint with a new twist. The team, which now included University of Cambridge University physicist Marine De Clerck, extended the analysis to a five-dimensional universe instead of the usual four. They found that extra dimension forced a new function: keeping track of the singularity’s dynamics now required a “complex” prime, known as a Gaussian prime, which includes an imaginary component (a number multiplied by the square root of –1). Gaussian primes cannot be further divided by other complex numbers. The authors called this system a “complex primon gas.”
“We don’t yet know whether the appearance of randomness of prime numbers near a singularity has a deeper meaning,” says Hartnoll. “But in my opinion, it’s very exciting that the connection extends to higher dimensional theories of gravity,” including some candidates for a full quantum mechanical theory of gravity.
Perlmutter cautiously hopes that the influx of first-rate physics will accelerate new discoveries, but the approach is one of many fighting for acceptance. “The kinds of things we’re trying to understand, black holes in quantum gravity, are certainly governed by some beautiful structures,” he says. “And number theory seems to be a natural language.”
This article was first published at Scientific American. © ScientificAmerican.com. All rights reserved. Follow along TikTok and Instagram, X and Facebook.
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