The Quantum Mechanical Potential for the Prime Numbers

TL;DR

This paper derives a quantum mechanical potential that corresponds to prime numbers, suggesting primality testing could be achieved through physical laws in principle.

Contribution

It introduces a quantum potential model for prime numbers and demonstrates its explicit form in a semi-classical approximation, linking number theory with quantum physics.

Findings

Prime numbers satisfy a specific quantum spectral criterion.

An explicit quantum potential for primes is computed.

Primality testing could be approached via physical principles.

Abstract

A simple criterion is derived in order that a number sequence ${\cal S}_n$ is a permitted spectrum of a quantized system. The sequence of the prime numbers fulfils the criterion and the corresponding one-dimensional quantum potential is explicitly computed in a semi-classical approximation. The existence of such a potential implies that the primality testing can in principle be resolved by the sole use of physical laws.