Theory of Analogous Force on Number Sets

TL;DR

This paper develops a thermodynamic framework for understanding number sequences, introducing an analogous force concept that explains distributions like Benford's law and offers insights into Fibonacci and prime numbers.

Contribution

It proposes a novel thermodynamic theory treating number sequences as particles, deriving an analogous force, and explaining distribution phenomena such as Benford's law and Fibonacci sequence predictions.

Findings

Derives an analogous force on number sets proportional to probability distribution derivatives.

Explains the emergence of Benford's law in Fibonacci numbers.

Provides recursion relations for predicting Fibonacci sequences.

Abstract

A general statistical thermodynamic theory that considers given sequences of x-integers to play the role of particles of known type in an isolated elastic system is proposed. By also considering some explicit discrete probability distributions p_{x} for natural numbers, we claim that they lead to a better understanding of probabilistic laws associated with number theory. Sequences of numbers are treated as the size measure of finite sets. By considering p_{x} to describe complex phenomena, the theory leads to derive a distinct analogous force f_{x} on number sets proportional to $(\frac{\partial p_{x}}{\partial x} )_{T}$ at an analogous system temperature T. In particular, this yields to an understanding of the uneven distribution of integers of random sets in terms of analogous scale invariance and a screened inverse square force acting on the significant digits. The theory also allows to establish recursion relations to predict sequences of Fibonacci numbers and to give an answer to the interesting theoretical question of the appearance of the Benford's law in Fibonacci numbers. A possible relevance to prime numbers is also analyzed.