# Simple wave-optical superpositions as prime number sieves

## TL;DR

This paper demonstrates how simple wave superpositions can encode prime numbers and related number theory concepts, revealing a physical wave-based approach to understanding prime distributions and structures.

## Contribution

It introduces a novel wave-optical method to encode primes and their properties, linking diffraction phenomena to number theory in a physically realizable system.

## Key findings

- Primes are identified as zeros with phase singularities in wave fields.
- Diffraction patterns encode primes, twin primes, and Gaussian primes.
- Wave superpositions can produce the irregular distribution of primes from ordered systems.

## Abstract

We encode the sequence of prime numbers into simple superpositions of identical waves, mimicking the archetypal prime number sieve of Eratosthenes. The primes are identified as zeros accompanied by phase singularities in a physically generated wave-field for integer valued momenta. Similarly, primes are encoded in the diffraction pattern from a simple single aperture and in the harmonics of a single vibrating resonator. Further, diffraction physics connections to number theory reveal how to encode all Gaussian primes, twin-primes, and how to construct wave fields with amplitudes equal to the divisor function at integer spatial frequencies. Remarkably, all of these basic diffraction phenomena reveal that the naturally irregular sequence of primes can arise from trivially ordered wave superpositions.