An Application of Markov Chain Analysis to Integer Complexity

TL;DR

This paper applies Markov chain analysis to study integer complexity, providing bounds on the minimal number of ones needed to express integers and improving existing bounds for a large class of algorithms.

Contribution

It introduces Markov chain methods to analyze algorithms for integer complexity, improving upper bounds on the minimal number of ones needed.

Findings

Established bounds for integer complexity using Markov chains.

Improved the upper bound to 3.52 log_3 n for a large class of algorithms.

Analyzed the distribution of integer complexity values.

Abstract

The complexity $f(n)$ of an integer was introduced in 1953 by Mahler & Popken: it is defined as the smallest number of $1$'s needed in conjunction with arbitrarily many +, * and parentheses to write an integer $n$ (for example, $f(6) \leq 5$ since $6 = (1+1)(1+1+1)$). The best known bounds are $$ 3 \log_{3}{n} \leq f(n) \leq 3.635 \log_{3}{n}.$$ The lower bound is due to Selfridge (with equality for powers of 3); the upper bound was recently proven by Arias de Reyna & Van de Lune, and holds on a set of natural density one. We use Markov chain methods to analyze a large class of algorithms, including one found by David Bevan that improves the upper bound to $$ f(n) \leq 3.52 \log_{3}{n}$$ on a set of logarithmic density one.