An Application of Markov Chain Analysis to Integer Complexity

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) ≤ 5 since 6 = (1 + 1)(1 + 1 + 1)). The best known bounds are 3 log 3 n ≤ f (n) ≤ 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) ≤ 3.52 log 3 n on a set of logarithmic density one.