The main numbers are sometimes called “atoms” of mathematics, because they can only be divided into themselves and 1. Over the course of two millennia, mathematicians asked the question of whether the main numbers are really random, or some unknown template underlies their streamlining. Recently, the theorists of numbers proposed several amazing assumptions about the main models – in particular, probabilistic patterns that appear in large groups of mathematical atoms.
Patterns in simple simple 1859 hypothesis With the participation of the legendary function Riemann Zeta. Mathematician Bernhard Rimann received a function that considers the number of simple numbers to the number XIt includes the field of three main ingredients: a smooth assessment, a set of corrective terms coming from the Riemann Zeta function, and a small term of error.
There was a lot Written About the function of Riemann Zeta, but the most important thing you need to know is that it provides a correction for a smooth assessment. To do this, he accepts a wavy scheme, sometimes raising an account, sometimes reducing it. These corrective fluctuations are determined by places zeros Functions Zeta Riemann. In fact, the famous Riman hypothesis claims that all such zeros are on the “critical line”, where the real part is equal 1⁄2Field
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Zeros intrigue mathematicians for two reasons. Firstly, they mean that the Zeta function encodes still updated information about simple numbers. Secondly, they suggest that the distance between the simple numbers, despite the violations, is as ordered as much as possible; Smaller fluctuations will contradict the density of prime numbers.
Taken together, this means that the error in the main formula for Riemann calculation is as minimal as possible.
The hypothesis was checked all the way to trillions, but never proved. It would take only one counter example to update most of the modern theory of numbers, therefore, proving that the hypothesis was a priority in mathematics for decades.
However, within a century after the opening of Riman, mathematics were stuck, it would seem, the random structure of the main numbers. The problem was so difficult and so important that in 2000 the clay mathematics institution created Million reward For those who could prove the hypothesis of Riman.
The main numbers and probability of Oracle
In particular, it was shown that the main numbers are subject to a certain random measuresThe field in mathematics The measure is associated with the statistical behavior of a large number of things. For example, one gas particle can be easy to simulate, but to predict the behavior of a large cloud of billions of particles, it will be outside today's computing power. Instead, the general statistics of the cloud movements can be captured as a specific type Random measureField
North -Wall University Mathematician Maxim Radziwill calls this technique the probability of oracle. “I can quickly rid the truth of probability,” he says. “I can find the right model, and then I can find out what is the correct answer for almost any question.” But Oracle cannot explain the deeper meaning of this answer, leaving mathematicians with a little understanding of how to prove their new discoveries.
To be clear, simple numbers are not random numbers; They are completely determined. But if you choose a large number of simple numbers, their distribution is with the flow, they spread along the number line – in size statistically, like some types of random sequences. But what are the species?
The first measure of simple numbers was discovered in the 1970s during a random discussion between Cambridge University. Student Hugh Montgomery and the famous physicist Freemon Dyson from the Institute of Advanced Education. Montgomery was afraid to disturb the venerable Dyson, but, in the visible one, told him about his work, says John Kiss, a mathematical physicist from Oxford University, familiar with history. Dyson reacted with extreme excitement, realizing that the ideas of Montgomery are related to the projects that he had already worked on.
Dyson was well versed with random measures from cooperation with the physicist of the Nobel Prize Eugene Wigner to understand the mathematics of the nuclei of heavy atoms. The calculation of the permissible energies of such very filled nuclei was directly too complicated, so Wigner Statistically predicted Energy levels. The results showed energy that fell on “regularly” irregular spaces; They were not tightly knocked together or very far from each other.
Montgomery turned out to be amazingly similar to behavior in the main numbers – in particular, the correlation between the positions of the notorious zeros of the Zeta Riemann function. They were not evenly distributed, but they were not completely uncorrected.
In the same shocking discovery as beautifully, it was shown that the spaces between the zeros of the Riemann Zeta function correspond to the same type of random measure that described the quantum systems. For the main numbers, he hinted at thin patterns, woven into other muddy statistical data.
The main numbers and chaos
Since then, close to a dozen random measures have been associated with simple numbers, but many of the results have summarized assumptions. “Many of these results really create your intuition,” says Radziwill. “They tell you what a typical object looks like, but in fact they do not prove the results on their own.”
In September 2025 conferenceAdam Harper, the theorist of the number at the University of Warurik in England, presented evidence of the suitability of another random measure in the desire to find basic laws. Gaussian multiplier chaos reflects a strongly fluctuating, large -scale invariant accident that describes various chaotic systems, from turbulence to quantum gravity and even financial markets. Since fractals are large -scale, it is sometimes also called a “random fractal measure”. Surprisingly, Harper's proof showed that statistics associated with zero zeta function can also be captured by random fractal dimensions.
In addition, Harper, Max Venzian Suy from New York University and Kananan Sundaradajan from Stanford University found a way to predict When This chaotic behavior arose in simple numbers. Random measures describe large collections of the main numbers. But when you consider smaller and smaller collections, statistics change, losing its probabilistic models and returning to a pure, unstructured accident. The group announced during 2025 Summer conference that if random fractal measures described the numbers before Xthen for all intervals during the transition (X To X + y, Where u Small) They can calculate the exact mixture of randomness and chaos. After this interval, statistics returned to random fractal indicators.
When mathematicians tried to look at a short interval (X To X + √X), they were laid in deeper mathematical waters, called “outside the square root barrier.” Harper suggested inside this small section in 2023 paper In 200 years, he found the best way to consider the main numbers than the historical equation of Riman. Indeed, in 2025 paperSuy and Viktor Van, a mathematician, now at the Institute of Mathematics in Taiwan, demonstrated that Harper's hypothesis was true. Drivation did not reach complete evidence, since it relied on a separate hypothesis imported from physicists. “This is a very funny part,” says Xui. “Personally, I am not a big fan of physics, but my work is based on their intuition.”
But what do all these conclusions actually say about simple numbers? Radziwill is careful. “If I have a random number generator on my computer, this is not by chance,” he says. “But if you do not know how it functions, it is by chance for you.” In other words, just as a cloud of gas particles could be described by a deterministic way, if there is a sufficiently powerful computer, there may be a very complex determinated method that can describe the common simple ones. Until then, mathematicians (and physicists) continue to fight the meaning of many deep probabilistic laws.
