Factorising numbers with a Bose-Einstein condensate

TL;DR

This paper explores a novel approach to integer factorization using a Bose-Einstein condensate analogy, providing asymptotic bounds and statistical characterizations of factorization distributions for large numbers.

Contribution

It introduces a thermodynamic analogy to number factorization, deriving asymptotic bounds and analyzing the distribution of factors for large integers.

Findings

Distribution approaches Gaussian for products of distinct primes

Non-Gaussian statistics arise with repeated prime factors

Asymptotic bounds on skewness and total factorizations are proposed

Abstract

The problem to express a natural number N as a product of natural numbers without regard to order corresponds to a thermally isolated non-interacting Bose gas in a one-dimensional potential with logarithmic energy eigenvalues. This correspondence is used for characterising the probability distribution which governs the number of factors in a randomly selected factorisation of an asymptotically large N. Asymptotic upper bounds on both the skewness and the excess of this distribution, and on the total number of factorisations, are conjectured. The asymptotic formulas are checked against exact numerical data obtained with the help of recursion relations. It is also demonstrated that for large numbers which are the product of different primes the probability distribution approaches a Gaussian, while identical prime factors give rise to non-Gaussian statistics.