Uncovering Multiscale Order in the Prime Numbers via Scattering

TL;DR

This paper reveals that prime numbers exhibit multiscale order and hyperuniformity, resembling quasicrystals and limit-periodic patterns, and introduces an algorithm for predicting primes based on their scattering properties.

Contribution

It uncovers a new class of many-particle systems with pure point diffraction patterns called effectively limit-periodic, demonstrating multiscale order in primes and linking physics concepts to number theory.

Findings

Primes in certain intervals are hyperuniform and display dense Bragg peaks.

The structure factor of primes shows quasicrystal-like features at rational wavenumbers.

An algorithm is developed to predict primes with high accuracy.

Abstract

The prime numbers have been a source of fascination for millenia and continue to surprise us. Motivated by the hyperuniformity concept, which has attracted recent attention in physics and materials science, we show that the prime numbers in certain large intervals possess unanticipated order across length scales and represent the first example of a new class of many-particle systems with pure point diffraction patterns, which we call {\it effectively limit-periodic}. In particular, the primes in this regime are hyperuniform. This is shown analytically using the structure factor $S(k)$, proportional to the scattering intensity from a many-particle system. Remarkably, the structure factor for primes is characterized by dense Bragg peaks, like a quasicrystal, but positioned at certain rational wavenumbers, like a limit-periodic point pattern. We identify a transition between ordered and disordered prime regimes that depends on the intervals studied. Our analysis leads to an algorithm that enables one to predict primes with high accuracy. Effective limit-periodicity deserves future investigation in physics, independent of its link to the primes.