Factorising numbers with a Bose-Einstein condensate

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.

17 pages, 7 figures, RevTeX

References in corpus (6)

• Phase Transition in the Number Partitioning Problem
• Condensate Fluctuations in Trapped Bose Gases: Canonical vs. Microcanonical Ensemble
• Fermion-boson transmutation and comparison of statistical ensembles in one dimension
• Asymptotics of the number partitioning distribution
• Partition asymptotics from one-dimensional quantum entropy and energy currents
• Number Fluctuation and the Fundamental Theorem of Arithmetic

Cited by in corpus (8)

• Physics of the Riemann Hypothesis
• Grand-canonical condensate fluctuations in weakly interacting Bose-Einstein condensates of light
• Effective Trap-like Activated Dynamics in a Continuous Landscape
• Prime Factorization of Arbitrary Integers with a Logarithmic Energy Spectrum
• Quantum simulation of the Riemann-Hurwitz zeta function
• On algebraic structure of the set of prime numbers
• Achieving quantum advantage in a search for a violations of the Goldbach conjecture, with driven atoms in tailored potentials
• On a Deterministic Property of the Category of $k$-almost Primes: A Deterministic Structure Based on a Linear Function for Redefining the $k$-almost Primes ($\exists n\in \mathrm{N}$, $1\le k\le n$) in Certain Intervals