Prime numbers are sometimes called math’s “atoms” because they can be divided by only themselves and 1. For two millennia, mathematicians have wondered if the prime numbers are truly random, or if some unknown pattern underlies their ordering. Recently number theorists have proposed several surprising conjectures on prime patterns—in particular, probabilistic patterns that show up in large groups of the mathematical atoms.
The patterns in the primes trace back to an 1859 hypothesis involving the legendary Riemann zeta function. Mathematician Bernhard Riemann derived a function that counts the number of primes up to a number x. It includes three main ingredients: a smooth estimate, a set of corrective terms coming from the Riemann zeta function, and a small error term.
Much has been written about the Riemann zeta function, but the most important thing to know is that it provides a correction to the smooth estimate. To do so, it takes on a wavy pattern, sometimes raising the count, sometimes lowering it. These corrective oscillations are determined by the locations of the zeros of the Riemann zeta function. In fact, the celebrated Riemann hypothesis claims that all such zeros lie on a “critical line” where the real part equals 1⁄2.
On supporting science journalism
If you’re enjoying this article, consider supporting our award-winning journalism by subscribing. By purchasing a subscription you are helping to ensure the future of impactful stories about the discoveries and ideas shaping our world today.
The zeros intrigue mathematicians for two reasons. First, they imply that the zeta function is encoding as-yet-unknown information about the primes. Second, they suggest that the spacing of the primes, despite irregularities, is as orderly as possible; smaller fluctuations would contradict the density of the primes.
Taken together, this means the error in Riemann’s prime counting formula is as minimal as possible.
The hypothesis has been verified all the way into the trillions—but never proven. It would only take a single counterexample to upend much of modern number theory, so proving the hypothesis has been a priority in mathematics for decades.
For the century following Riemann’s discovery, however, mathematicians were stymied by the seemingly random structure of the prime numbers. The problem was so difficult and so important that in 2000 the Clay Mathematics Institute set up a million-dollar bounty for anyone who could prove Riemann’s hypothesis.
Prime Numbers and the Probability Oracle
In particular, the prime numbers were shown to obey certain random measures. In math, a measure is concerned with the statistical behavior of a large number of things. For instance, a single particle of gas might be easy to model, but to predict the behavior of a large cloud of billions of particles would be beyond today’s computational power. Instead the overall statistics of the cloud’s movements can be captured as a particular type of random measure.
Northwestern University mathematician Maksym Radziwill calls the technique a probability oracle. “I can quickly get the truth out of probability,” he says. “I can find the right model, and then I can figure out what is the right answer for pretty much any question.” But the oracle fails to explain the deeper meaning behind that answer, leaving mathematicians with few insights for how to prove their new discoveries.
To be clear, the primes are not random numbers; they are completely deterministic. But if you choose a large number of primes, their distribution—theway they are spread across the number line—behaves statistically like certain types of random sequences. But what kinds?
The first measure of the primes was found in the 1970s during a chance discussion between University of Cambridge Ph.D. student Hugh Montgomery and famed physicist Freeman Dyson of the Institute for Advanced Study. Montgomery was wary of bothering the venerable Dyson but diffidently told him about his work, says Jon Keating, a mathematical physicist at the University of Oxford familiar with the story. Dyson reacted with extreme excitement, realizing that Montgomery’s ideas tied into projects he was already working on.
Dyson was well versed with random measures because of a collaboration with Nobel Prize–winning physicist Eugene Wigner to understand the mathematics of the nuclei of heavy atoms. Directly calculating the allowed energies of such heavily populated nuclei was too complex, so Wigner statistically predicted the energy levels. The results showed energies that fell on “regularly” irregular spacings; they weren’t clumped tightly together or extremely far apart.
Montgomery happened to find strikingly similar behavior in the prime numbers— specifically, the correlations between the positions of the notorious zeros of the Riemann zeta function. They weren’t evenly spaced, but neither were they completely uncorrelated.
In a discovery as shocking as it was beautiful, the spacings between the zeros of the Riemann zeta function were shown to match the same type of random measure that described quantum systems. For the prime numbers, it hinted at subtle patterns woven into otherwise murky statistics.
Prime Numbers and Chaos
Since then, close to a dozen random measures have been linked to the primes, but many of the findings amount to conjectures. “A lot of these results really build your intuition,” Radziwill says. “They tell you what a typical object looks like, but they don’t actually prove results by themselves.”
At a September 2025 conference, Adam Harper, a number theorist at the University of Warwick in England, presented a proof of a different random measure’s suitability in the quest to find prime patterns. Gaussian multiplicative chaos captures highly fluctuating, scale-invariant randomness, which describes various chaotic systems, from turbulence to quantum gravity and even financial markets. Because fractals are scale-invariant, it is sometimes also referred to as a “random fractal measure.” Surprisingly, Harper’s proof showed that statistics associated with the zeros of the zeta function could also be captured by random fractal measures.
Furthermore, Harper, Max Wenqiang Xu of New York University and Kannan Soundararajan of Stanford University found a way to predict when this chaotic behavior emerged in the primes. Random measures describe large collections of prime numbers. But as you consider smaller and smaller collections, the statistics change, losing their probabilistic patterns and reverting to pure, unstructured randomness. The group announced during a 2025 summer conference that if random fractal measures described the numbers up to x, then for all the intervals in a transition period (x to x + y, where y is small) they could calculate the exact mix of randomness and chaos. Following this interval, the statistics reverted to random fractal measures.
When mathematicians tried to look at the short interval (x to x + √x), they were thrust into deeper mathematical waters dubbed “beyond the square root barrier.” Inside this small stretch, Harper conjectured in a 2023 paper that, after 200 years, he had found a better way to count prime numbers than Riemann’s historic equation. And indeed, in a 2025 paper, Xu and Victor Wang, a mathematician now at the Institute of Mathematics in Taiwan, demonstrated that Harper’s conjecture was true. The derivation fell short of a complete proof because it relied on a separate conjecture imported from physicists. “That’s the very funny part,” Xu says. “I’m personally not a big fan of physics, but my work relies on their intuition.”
But what do all these findings really say about the primes? Radziwill is cautious. “If I have a random number generator on a computer, it’s not random to me,” he says. “But if you don’t know how it’s functioning, it’s random to you.” In other words, just as a cloud of gas particles could be described deterministically if a powerful enough computer existed, there may be a highly complex deterministic method that can describe the primes. Until then, mathematicians (and physicists) continue to grapple with the meaning behind the many profound probabilistic patterns.