A Fowler-Nordheim Integrator can Track the Density of Prime Numbers

TL;DR

This paper introduces a novel physical device based on Fowler-Nordheim quantum tunneling that can emulate prime number distribution, offering a potential new approach to understanding prime number generation and its applications.

Contribution

It demonstrates how a floating-gate device governed by FN tunneling can model prime number statistics, bridging physics and number theory in a scalable, reliable manner.

Findings

Prototype devices match prime number asymptotic statistics

Spectral signatures resemble those of prime number sequences

Device operation remains reliable at extremely low tunneling currents

Abstract

"Does there exist a naturally occurring counting device that might elucidate the hidden structure of prime numbers ?" is a question that has fascinated computer scientists and mathematical physicists for decades. While most recent research in this area have explored the role of the Riemann zeta-function in different formulations of statistical mechanics, condensed matter physics and quantum chaotic systems, the resulting devices (quantum or classical) have only existed in theory or the fabrication of the device has been found to be not scalable to large prime numbers. Here we report for the first time that any hypothetical prime number generator, to our knowledge, has to be a special case of a dynamical system that is governed by the physics of Fowler-Nordheim (FN) quantum-tunneling. In this paper we report how such a dynamical system can be implemented using a counting process that naturally arises from sequential FN tunneling and integration of electrons on a floating-gate (FG) device. The self-compensating physics of the FG device makes the operation reliable and repeatable even when tunneling-currents approach levels below 1 attoamperes. We report measured results from different variants of fabricated prototypes, each of which shows an excellent match with the asymptotic prime number statistics. We also report similarities between the spectral signatures produced by the FN device and the spectral statistics of a hypothetical prime number sequence generator. We believe that the proposed floating-gate device could have future implications in understanding the process that generates prime numbers with applications in security and authentication.