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Simple Wave-Optical Superpositions as Prime Number Sieves
T C Petersen1, M Ceko1, I D Svalbe1
1School of Physics and Astronomy, Monash University, Victoria 3800, Australia.
Physical Review Letters
|April 2, 2019
Summary
Prime numbers can be encoded using wave phenomena, similar to the Eratosthenes sieve. This research demonstrates how wave physics can generate the irregular prime number sequence from simple wave patterns.
Area of Science:
- Wave physics
- Number theory
- Quantum mechanics
Background:
- The distribution of prime numbers is a fundamental problem in mathematics.
- Prime number sieves, like Eratosthenes sieve, are classical methods for identifying primes.
- Wave phenomena exhibit complex patterns that may relate to number sequences.
Purpose of the Study:
- To encode the sequence of prime numbers using wave superpositions.
- To explore the connection between diffraction physics and number theory.
- To demonstrate a physical method for generating prime numbers.
Main Methods:
- Encoding primes into superpositions of identical waves.
- Identifying primes as zeros with phase singularities in wave fields.
- Utilizing diffraction patterns from single apertures.
- Analyzing harmonics of vibrating resonators.
Main Results:
- Primes are encoded as zeros in wave fields with integer momenta.
- Diffraction and resonator harmonics reveal prime number encoding.
- Methods are shown for encoding Gaussian primes and twin primes.
- Wave fields with amplitudes equal to the divisor function are constructed.
Conclusions:
- Basic diffraction phenomena can generate the irregular sequence of prime numbers.
- Simple, trivially ordered wave superpositions can produce prime number sequences.
- Wave physics offers a novel perspective on the generation of prime numbers.
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