# Prime number atoms and structures

Are some atoms, particles, structures possibly constructed on prime numbers?

University researchers suggest prime atoms and materials might be a possibility. Physical examples include parts of birds eyes, quasicrystals, meteorites

The seemingly random digits known as prime numbers are not nearly as scattershot as previously thought. A new analysis by Princeton University researchers has uncovered patterns in primes that are similar to those found in the positions of atoms inside certain crystal-like materials.

The researchers found a surprising similarity between the sequence of primes over long stretches of the number line and the pattern that results from shining X-rays on a material to reveal the inner arrangement of its atoms.

"We showed that the primes behave almost like a crystal or, more precisely, similar to a crystal-like material called a 'quasicrystal.'"

Hyperuniform materials have special order at large distances and include crystals, quasicrystals and special disordered systems. Hyperuniformity is found in the arrangement of cone cells in bird eyes, in certain rare meteorites, and in the large-scale structure of the universe.

The discovery may aid research in both mathematics and materials science. "Prime numbers have beautiful structural properties, including unexpected order, hyperuniformity and effective limit-periodic behavior," said Torquato. "The primes teach us about a completely new state of matter."

"What's fascinating about this paper is that it gives us a different perspective on the primes: instead of viewing them as numbers, we can view them as particles and try to map out their structure via X-ray diffraction,"

Surprising hidden order unites prime numbers and crystal-like materials | phys.org

## Prime atomic structures

Remarkably, the structure factor of the primes is characterized by dense Bragg peaks, like a quasicrystal, but positioned at certain rational wavenumbers, like a limit-periodic point pattern. However, the primes show an erratic pattern of occupied and unoccupied sites, very different from the predictable patterns of standard limit-periodic systems.

In summary, by focusing on the scattering characteristics of the primes in certain sufficiently large intervals, we have discovered that prime configurations are hyperuniform of class II and characterized by an unexpected order across length scales. In particular, they provide the first example of an effectively limit-periodic point process, a hallmark of which are dense Bragg peaks in the structure factor. The discovery of this hidden multiscale order in the primes is in contradistinction to their traditional treatment as pseudo-random numbers

Uncovering Multiscale Order in the Prime Numbers via Scattering

Illustration of the superposition of effective multiple periodicities in the primes. We take the primes to be ‘occupied’ sites (black dots) on an integer lattice of spacing 2 that contains all of the positive odd integers. The crosses indicate sites that cannot be occupied because of a certain periodicity 2n (n sites on the odd integer lattice), where n is a square-free odd number.For example, the peak at π/3 with n = 3 and m = 1 corresponds to remainders when dividing by 6. A prime must leave a remainder of 1 or 5 or else it would be divisible by 2 or 3. The lattice in the figure has a spacing of 2, so these allowed sites appear with a period of 3 instead of 6. The forbidden sites appear as red crosses. The other two sites may or may not be prime, and if one averages over many periods in a sufficiently large interval, each of them will have an equal occupation probability (due to Dirichlet’s theorem on arithmetic progressions). The overall effect of this equidistribution of occupied sites is an effective periodicity of 6. Similarly, the primes show an effective periodicity of 10 (blue crosses), 14 (green crosses)

Uncovering Multiscale Order in the Prime Numbers via Scattering

### Another alternative atom model

Edwin Kaal has also proposed a new model for the atom, the Proton-Electron Atom.