Researchers Discover a Pattern to the Seemingly Random Distribution of Prime Numbers

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This is fascinating....

Often known as “the building blocks of mathematics,” prime numbers have fascinated mathematicians for centuries due to their highly unpredictable and seemingly random nature. However, a team of researchers at Princeton University have recently discovered a strange pattern in the primes’ chaos. Their novel modelling techniques revealed a surprising similarity between primes and certain naturally occurring crystalline materials, a similarity that may carry significant implications for physics and materials science.
A productive hunch
Chemists and physicists typically study the structure of a material by firing X-rays at a sample and observing how the rays scatter off the atoms within it. This process is known as X-ray diffraction, where different materials produce different patterns depending on how symmetrically their atoms are arranged.

Image: Theoretical Chemistry Group

A liquid, whose atoms are jumbled and moving, will scatter the X-rays all over the place and fail to create a pattern, whereas a crystal’s rigid lattice of atoms—such as in salt or diamond—will diffract them in a more orderly manner due to their repetitive internal structure. Additionally, rare materials known as quasicrystals create similar diffraction patterns without having a repeating atomic structure. In both crystals and quasicrystals, the X-rays typically form a pattern of periodic bright spots known as “Bragg peaks” as the rays constructively interfere with each other over fixed intervals.
Last year, theoretical chemist and Princeton professor Salvatore Torquato had a hunch—what if prime numbers were modelled as atom-like particles? Would they create a pattern too?



Together with his student Ge Zhang and number theorist Matthew de-Courcy-Ireland, Torquato computationally represented the primes as a one-dimensional string of atoms and scattered light off them. The result, published in the Journal of Statistical Mechanics: Theory and Experiment last week, was astounding: not only did they create a quasicrystal-like interference pattern, but it was a type of fractal pattern that has never been seen before. Torquato told Quanta Magazine that this implies prime numbers “are a completely new category of structures” when considered as a physical system.
Curiously, this uniquely fractal pattern only appears over sufficiently long stretches of the number line—over shorter stretches, the pattern fails to emerge. Known as “hyperuniformity”, this is a rare property that only a few materials and systems in nature demonstrate, such as the arrangement of color-detecting cone cells in bird’s eyes, certain emulsions and quasi-crystals, the large-scale structure of the universe, and as it turns out, prime numbers.
While the team’s findings are not groundbreaking for number theory (as most of the relevant mathematics has been seen before in other forms), they may prove useful in a new research area known as “aperiodic order”—the study of non-repeating patterns.

https://motherboard.vice.com/en_us/article/pa8dw8/prime-number-pattern-mimics-crystal-patterns