Minimal Representations of Natural Numbers Under a Set of Operators

TL;DR

This paper investigates the minimal length representations of natural numbers using specific operator sets, providing bounds, patterns, and computational verification for numbers up to 4.5 million.

Contribution

It introduces a framework for analyzing minimal representations of natural numbers with various operator sets and offers new bounds and computational results.

Findings

Established bounds on complexities of natural numbers under certain operators

Identified patterns in minimal representations and complexities

Verified results through extensive computational analysis of 4.5 million numbers

Abstract

This paper studies the minimal length representation of the natural numbers. Let O be a fixed set of integer-valued functions (primarily hyperoperations). For each n, what is the shortest way of expressing n as a combinations of functions in O to the constant 1? For example, if O contains the two functions Sx (successor of x) and *xy (x times y) then the shortest representation of 12 is *SSS1SS1, with 8 symbols. This is taken to mean that 8 is complexity of 12 under O. We make a study of such minimal representations and complexities in this paper, proving and/or rightly predicting bounds on complexities, discussing some relevant patterns in the complexities and minimal representations of the natural numbers and listing the results gleaned from computational analysis. Computationally, the first 4.5 million natural numbers were probed to verify our mathematically obtained results. Due to the finiteness of the problem, we used the method of exhaustion of possibilities to state some other results as well.