Integer Complexity: Experimental and Analytical Results

TL;DR

This paper explores the properties of integer complexity, defined as the minimum number of ones needed to express a number using addition and multiplication, through experimental and analytical methods, revealing deep conjectures and results.

Contribution

It provides extensive computational data up to 10^12 and offers new analytical insights into the properties and conjectures related to integer complexity.

Findings

Computed ||n|| for n up to 10^12

Suggests a link between integer complexity and the infinitude of Sophie Germain primes

Proves new analytical results about integer complexity

Abstract

We consider representing of natural numbers by arithmetical expressions using ones, addition, multiplication and parentheses. The (integer) complexity of n -- denoted by ||n|| -- is defined as the number of ones in the shortest expressions representing n. We arrive here very soon at the problems that are easy to formulate, but (it seems) extremely hard to solve. In this paper we represent our attempts to explore the field by means of experimental mathematics. Having computed the values of ||n|| up to 10^12 we present our observations. One of them (if true) implies that there is an infinite number of Sophie Germain primes, and even that there is an infinite number of Cunningham chains of length 4 (at least). We prove also some analytical results about integer complexity.