Prime numbers and crystal-like materials share a hidden organization

In the end, prime numbers and crystals have a lot in common. A new analysis by Princeton University researchers suggests that the distribution of prime numbers in the digital line is remarkably similar to that of the atomic structure of some crystalline materials. In simple terms, the sequence of prime numbers over long stretches of the number line shows properties remarkably similar to the resulting sequence of shiny X-rays on an object to reveal its internal atomic structure.

The study, published in The Journal of Statistical Mechanics, shows that the organization of prime numbers presents a striking similarity to the atomic organization of quasicrystals, strange crystalline materials endowed with a aperiodic atomic organization. The organization of prime numbers and quasicrystals are both part of a class of models called "hyperuniform". The potential applications of this knowledge are exciting. Mathematicians have long sought an algorithm capable of predicting the location of prime numbers in the numerical line, but no such algorithm has yet been found. According to the researchers, the knowledge gained through the study could help design new methods for predicting prime numbers on the digital right. Salvatore Torquato, principal investigator of the study, said: "There is much more order than ever in prime numbers"

Prime numbers, quasicrystals and hyperuniformity

Prime numbers are numbers whose only factors are one and themselves; numbers like 3, 7, 11 or 2^74,207,281-1. The very large prime numbers are the building blocks of modern cryptography. The prime numbers appear to be more or less randomly distributed on the number line. Currently, there is no algorithm for generating prime numbers, but mathematicians have been able to determine some extremely general characteristics of their organization. The further the number line is, the more prime numbers are spread. moreover, the probability that a randomly selected number is a prime number is inversely proportional to the number of digits in that number.

Torquato, a professional chemist, is familiar with X-ray crystallography, the process of radiating X-rays through an object to map its atomic structure in 3 dimensions. For ordered and periodic crystals, such as diamonds or quartz, crystallography results in a predictable pattern of bright dots called Bragg peaks. Quasicrystals do not have the periodic structure of regular crystals and, compared to ordinary crystals, quasicrystals exhibit a distinct and complex pattern of Bragg peaks.

In a previous study published in February, the team used powerful computer simulations to show what would happen if the prime numbers were treated as atoms in an X-ray array. The results showed that the first crystals order presented a Bragg diagram very similar to that of a quasi-crystal, and another type of system called "periodic order limit". The present article aims to provide a theoretical explanation of the numerical results of the previous experiment crystals.

All these organizations belong to a class of models known as "hyperuniform" materials. A hyperuniform material (sometimes called superhomogenous) seems to show no obvious order but adopts a particular organizational structure on a large scale. Although this may seem contradictory, one could say that a hyperuniform material is a material of "ordered chaos" – the pieces seem randomly arranged at small scales, but there is order in the madness from the point of view of the bird.

Since the genesis of the concept of hyperuniformity in the early 2000s, hyperuniform organization models have been found throughout nature, including in the eyes of chickens, quasicrystals, random number distributions, quantum sets of particles and even the large-scale structure. from the universe. Like the Fibonacci sequence, mathematical models of hyperuniformity are found everywhere in nature and their unexpected discovery almost always takes scientists by surprise.

In this case, the model that Torquato and the cohorts found in prime numbers is similar to that of quasicrystals and other order systems of the limit periods, but sufficiently different to give them the name of "effectively limiting the order ". similar ", which means that between the peaks of equal height, there are smaller groupings of peaks.

All this to say that the study suggests that the same mathematical rules underlie both the distribution of prime numbers in the numerical line and the three-dimensional atomic structure of some materials. It often happens that refined mathematical concepts in one domain are surprisingly applicable in another area. This was certainly a surprise to physicists of the early twentieth century when they discovered that non-Euclidean geometries, at the time an interesting mathematical curiosity, were applicable as models of general relativity. Similarly, the early pioneers of quantum mechanics were surprised when it turned out that matrix algebra did indeed have an application in early quantum theory. Again, we see a particular mathematical concept, that of hyperuniformity, arise in two seemingly unrelated fields: material science and number theory.

The results of the analysis of Torquato and his colleagues are exciting for the scientific community. According to Henry Cohn, a Microsoft researcher who did not participate in the study, "What's fascinating about this article is that it gives us a different perspective of prime numbers: instead of considering them as numbers, we can try them. to map their structure via X-ray diffraction ", it is thought that the new information could make it possible to create algorithms for generating prime numbers, which could change the whole face of mathematics as we know it.

In a way, the new study somewhat reverses the traditional understanding of the relationship between mathematics and science. Generally, scientists use mathematics to make predictions about the physical world, but now it seems like we could use the physical world to make predictions about mathematics! Galileo Galilei is often credited that the book of the universe is written in the language of mathematics and that its characters are triangles, circles and other geometric figures. It seems like this lesson sounds true, even 400 years later.