Exotic prime numbers could be hiding inside black holes
Like physics, mathematics has its own set of “fundamental particles” – THE prime numberswhich cannot be decomposed into smaller natural numbers. They can only be divided by themselves and 1.
And in a new development, it turns out that these mathematical “particles” offer new ways to approach some of physics’ deepest mysteries. Over the past year, researchers have discovered that formulas based on prime numbers can describe the characteristics of black holes. Number theorists have spent hundreds of years developing theorems and guesses based on prime numbers. These new connections suggest that the mathematical truths that govern prime numbers might 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 exploit this connection. “I would say that many high-energy physicists don’t know much about this aspect of number theory,” says Eric Perlmutter of the Institute of Theoretical Physics at Saclay.
The fundamental conjecture of number theory about prime numbers is the Riemann hypothesis of 1859. In a handwritten paper, German mathematician Bernhard Riemann provided a formula consisting of two main terms. The first offered a surprisingly close estimate of the number of prime numbers that are smaller than a given number. The second term is the zeta function, whose zeros (places where the function equals zero) adjust the original estimate. The mysterious way that 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 win a million-dollar prize from the Clay Mathematics Institute.
In the late 1980s, physicists began to wonder whether there was a physical system whose energy levels could be based on prime numbers. Physicist Bernard Julia of the École Normale Supérieure in France was challenged by a colleague to find a physical analogue described by the zeta function. His solution was to propose a hypothetical type of particle whose energy levels were given by the logarithms of the prime numbers. Julia called these particles “primons” and a group of them a “primon gas.” The partition function – a census of the possible states of a system – of this gas is exactly the Riemann zeta function.
At the time, Julia’s concept was a thought experiment: most scientists doubted the real existence of primons. But deep within the black holes, a mathematical link was waiting to be discovered. Just over two decades later, physicists Yan Fyodorov of King’s College London, Ghaith Hiary of Ohio State University, and Jon Keating of Oxford University saw evidence that fractal chaos emerged from fluctuations in the zeros of the zeta function, an idea that proved successful. proven in 2025.
Einstein’s general theory of relativity shows that the same chaos also appears near a singularity.
In a February 2025 preprint, University of Cambridge physicist Sean Hartnoll and graduate student Ming Yang brought Julia’s work into the real world. Inside the chaos close to a singularity, they found that a “conformal” symmetry emerges. Hartnoll compares conformal symmetry to Dutch artist MC Escher’s famous bat drawings — the same structure is repeated at different scales. This scale symmetry, combined with a bit of mathematics, revealed a near-singularity quantum system whose spectrum is organized into prime numbers – a cloud of conformal primon gas.
Five months later, they posted a preprint online with a new twist. The team, which now included physicist Marine De Clerck from the University of Cambridge, extended their analysis to a universe with five dimensions instead of the usual four. They found that the an extra dimension forced a new feature: tracking the dynamics of the singularity now required a “complex” prime number, called a Gaussian prime, which includes an imaginary component (a number multiplied by the square root of –1). Gaussian primes can no longer be divided by other complex numbers. The authors dubbed this system a “complex primon gas.”
“We don’t yet know whether the appearance of a random prime number near a singularity has a deeper meaning,” says Hartnoll. “However, in my opinion, it is very intriguing that the link extends to higher-dimensional theories of gravity,” including some candidates for a fully quantum-mechanical theory of gravity.
Perlmutter cautiously hopes that this wave of primordial physics will accelerate new discoveries, but this approach is just one of many struggling to gain acceptance. “The kinds of things we’re trying to understand, black holes in quantum gravity, are surely governed by beautiful structures,” he says. “And number theory seems to be a natural language.”
This article was first published on Scientific American. © ScientificAmerican.com. All rights reserved. Follow TikTok and Instagram, X And Facebook.
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