Integer Complexity: Experimental and Analytical Results II

TL;DR

This paper explores the complexity of representing natural numbers with minimal 1's, analyzing the spectrum of logarithmic complexity, its relation to base-3 digit behavior of powers of 2, and introduces P-algorithms for number representation.

Contribution

It provides new insights into the structure of integer complexity spectrum and connects it to digit patterns in base-3 representations, also introducing P-algorithms for number representations.

Findings

Complexity spectrum lies within [3, 4.755]

Connection between complexity and base-3 digit behavior of 2^n

Introduction of P-algorithms for number representation

Abstract

We consider representing of natural numbers by expressions using 1's, addition, multiplication and parentheses. $\left\| n \right\|$ denotes the minimum number of 1's in the expressions representing $n$. The logarithmic complexity $\left\| n \right\|_{\log}$ is defined as $\left\| n \right\|/{\log_3 n}$. The values of $\left\| n \right\|_{\log}$ are located in the segment $[3, 4.755]$, but almost nothing is known with certainty about the structure of this "spectrum" (are the values dense somewhere in the segment etc.). We establish a connection between this problem and another difficult problem: the seemingly "almost random" behaviour of digits in the base 3 representations of the numbers $2^n$. We consider also representing of natural numbers by expressions that include subtraction, and the so-called $P$-algorithms - a family of "deterministic" algorithms for building representations of numbers.