Most of the article is relevant background information, here’s some of that, and the new stuff:
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.
So… as best I can tell, this is not something that fundamentally breaks cryptography… at least not yet.
Basically, it seems that a new method for analyzing prime numbers and sets of prime numbers, one that is more… informative/precise than doing different kinds of analysis of the 200 year old Riemann Zeta function… that appears to have been achieved.
It is still not a proof, a full fledged solution that describes everything perfectly, all primes or where they would be in any given set of numbers, provably so.
But, it seems to be quite useful given you have a supercomputer to assign to prime number analysis.
I… have no idea at all what the actual computational efficiency of this new method is vs the various ways of anakyzing the Riemann Zeta function are, but it seems that this new function is both more precise/informative, yet also requires substantially more computational power.
There, there’s an attempt at a summary from someone who almost minored in math, but didn’t, because I reoriented toward a double degree in Econ/Econometrics and also Political Science.
Yeah, the… actual descriptions of exactly what they are doing and how they are using it is not present in the article, nor even in what I could see that it linked to, which were largely just abstracts, summaries of speeches to be given.
I don’f think that this would blow up modern cryptography, as that would require a way of basically doing something like an extremely fast and robust factoring of resultant numbers back down into their actual prime factors… where those resultant numbers have 256, 512, etc digits.
It… seems possible that this… could maybe be done… with some kind of reference help from this new method, but would require an unknown, but seemingly extremely large amount of computational power.
I don’t know, I am out my depth, and am not able to find very precise information from the article on that kind of application of the new function.
Going by this, the fastest and totally deterministic prime factorization algo is still from the 80s/90s.
Where this newly discovered function is not deterministic, it is probabalistic, yes more precise in being able to differentiate between randomness and chaos, but still probabalistic, it is not a prime factorization function/method/algo, and it seems to work best … or only really be relevant? … with smaller sets, smaller ranges of numbers, at describing where primes pop up in those constrained sets.
I think we would need an actual math / cryptology / computer science PhD or something like that to appear to be able to explain whether or not anything I just said is remotely accurate or relevant.
Would love an “explain like I minored in Maths”
Doesn’t need to be dumbed down too much. Maybe someone wants to explain it. I understand the basics of Chaos.
Or just on a scale of how fucked is modern encryption on a scale from 1-10.
Well, I would attempt to provide that, but I am not giving scientific american my email.
https://www.removepaywall.com/search?url=https%3A%2F%2Fwww.scientificamerican.com%2Farticle%2Fmathematicians-discover-prime-number-pattern-in-fractal-chaos%2F
Ah ok, ycombo commenter has solved that.
Most of the article is relevant background information, here’s some of that, and the new stuff:
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.
So… as best I can tell, this is not something that fundamentally breaks cryptography… at least not yet.
Basically, it seems that a new method for analyzing prime numbers and sets of prime numbers, one that is more… informative/precise than doing different kinds of analysis of the 200 year old Riemann Zeta function… that appears to have been achieved.
It is still not a proof, a full fledged solution that describes everything perfectly, all primes or where they would be in any given set of numbers, provably so.
But, it seems to be quite useful given you have a supercomputer to assign to prime number analysis.
I… have no idea at all what the actual computational efficiency of this new method is vs the various ways of anakyzing the Riemann Zeta function are, but it seems that this new function is both more precise/informative, yet also requires substantially more computational power.
There, there’s an attempt at a summary from someone who almost minored in math, but didn’t, because I reoriented toward a double degree in Econ/Econometrics and also Political Science.
Yeah, the… actual descriptions of exactly what they are doing and how they are using it is not present in the article, nor even in what I could see that it linked to, which were largely just abstracts, summaries of speeches to be given.
I don’f think that this would blow up modern cryptography, as that would require a way of basically doing something like an extremely fast and robust factoring of resultant numbers back down into their actual prime factors… where those resultant numbers have 256, 512, etc digits.
It… seems possible that this… could maybe be done… with some kind of reference help from this new method, but would require an unknown, but seemingly extremely large amount of computational power.
I don’t know, I am out my depth, and am not able to find very precise information from the article on that kind of application of the new function.
https://mathworld.wolfram.com/PrimeFactorizationAlgorithms.html
Going by this, the fastest and totally deterministic prime factorization algo is still from the 80s/90s.
Where this newly discovered function is not deterministic, it is probabalistic, yes more precise in being able to differentiate between randomness and chaos, but still probabalistic, it is not a prime factorization function/method/algo, and it seems to work best … or only really be relevant? … with smaller sets, smaller ranges of numbers, at describing where primes pop up in those constrained sets.
I think we would need an actual math / cryptology / computer science PhD or something like that to appear to be able to explain whether or not anything I just said is remotely accurate or relevant.