What could prime numbers possibly have to do with a black hole? Prime numbers look like the definition of “pure math,” but new theoretical work argues they may also be the right language for one of nature’s strangest places.
In two papers published in 2025, researchers link chaotic spacetime dynamics near a spacelike singularity to mathematical objects built from primes and the Riemann zeta function.
This does not mean primes physically sit inside a black hole. The claim is subtler and, in a way, more interesting. When gravity collapses toward a singularity and the geometry starts acting like a pinball machine, the best way to write down that chaos may be in the same “alphabet” number theorists use to study primes.
A new bridge between gravity and number theory
The first step came from University of Cambridge physicist Sean A. Hartnoll and graduate student Ming Yang. Their February 2025 arXiv preprint later became a peer-reviewed Journal of High Energy Physics paper published July 29, 2025.
They start with the BKL picture, a long-studied model of what spacetime does extremely close to a spacelike singularity. In their formulation, the dynamics at each point can be mapped to a simple “bouncing particle” problem in an abstract space with powerful symmetries.
Here is the punchline the authors emphasize. When they quantize that motion, they find wavefunctions tied to automorphic L functions and, along a particular mathematical direction, an L function can be rewritten like the partition function of a gas of non-interacting oscillators labeled by prime numbers. That is the weird part.
Why singularities can look like “cosmic billiards”
Black holes are famous for their event horizons, but the interior is where theory runs out of road. Classical general relativity predicts a singularity where the equations break down, which is why physicists expect a quantum description of gravity is needed.
In the BKL scenario, the approach to a singularity is not smooth. Instead, spacetime cycles through a jagged sequence that behaves chaotically, and nearby points can effectively decouple so each region follows its own violent rhythm.
That behavior can be represented with a metaphor that is surprisingly easy to picture. Imagine a billiard ball ricocheting off invisible walls, except the “table” is a curved mathematical space and each bounce encodes how the geometry stretches and squeezes as time runs out.
Prime numbers and the million-dollar question behind them
Prime numbers are the whole numbers greater than 1 that cannot be factored into smaller whole numbers, and they sit behind every multiplication table you ever memorized. Yet their pattern among all integers is famously irregular.
Riemann’s 1859 insight was that this apparent randomness is linked to the behavior of a function now called the Riemann zeta function. The Clay Mathematics Institute explains the Riemann hypothesis as the claim that all the “non-obvious” zeros of this function lie on a line where the real part is one half.
It also notes that the hypothesis has been checked for the first 10,000,000,000,000 such zeros, without a general proof.
The stakes are not symbolic. Clay’s Millennium Prize Problems set aside a $7 million fund with $1 million for each solution, and the Riemann hypothesis is still on that list.
The primon gas thought experiment that would not go away
In the late 1980s, physicists started asking if primes could show up as the energy levels of a quantum system. Bernard Julia proposed a hypothetical particle whose energy depends on the logarithms of prime numbers and called it a “primon.”
In that setup, a collection of non-interacting primons forms a “primon gas” whose partition function is exactly the zeta function. Reviews of the idea describe it as a clean mathematical bridge between statistical physics and number theory, even if “primons” were never expected to be real particles.
Hartnoll and Yang’s work revives that bridge in a new setting. The point is not that black holes contain primons, but that the quantum bookkeeping near a singularity can be reorganized so it behaves like a prime-labeled gas constrained by symmetry.
A five dimensional twist introduces “complex primes”
The second paper adds a twist that sounds like science fiction but is really mathematics. A team led by Marine De Clerck, Hartnoll, and Yang extended the billiard analysis to five dimensional gravity, and the article was published November 25, 2025, in the Journal of High Energy Physics.
With an extra dimension, the natural “prime” objects in the equations are no longer just ordinary primes. The authors describe L functions whose factorization runs over complex primes such as Gaussian primes, which live in number systems that include an imaginary component.
They also stress how open the interpretation remains. “We don’t know yet whether the appearance of prime number randomness close to a singularity has a deeper meaning,” Hartnoll told Scientific American, while calling the higher-dimensional extension “very intriguing.”
Why an ecology site should pay attention anyway
So why should anyone who worries about climate, clean energy, or the electric grid care about primes in a black hole? There is no direct line from these papers to cleaner air, cheaper solar panels, or a lower electric bill, at least not today.
But the broader point is worth noticing. Deep number theory already supports real world tools, including cryptography and secure computing, which sit behind everything from research data sharing to the digital systems that help run modern power grids.
Nature is full of “organized mess,” from turbulent oceans to the noisy ups and downs of weather, so any new way to describe structured chaos expands the shared toolbox. For now, this prime number and black hole link is theory first, but it hints at how far math can travel across disciplines.
The study was published in Journal of High Energy Physics.
