The 2-complexity of even positive integers

TL;DR

The paper introduces l-complexity as a generalization of integer complexity and focuses on the 2-complexity of even integers. It proves a complete classification: for $m=\lceil \log_2 n\rceil-1$, \|n\|_2=m+1$ precisely when $n=2^{m+1}$ or $n=2^m+2^{m'}$, and \|n\|_2=m+2$ only when $n$ is a sum of powers of two in one of four explicit forms. It also derives exact or near-exact values for products with small powers, such as \|2^m6^r\|_2 and \|2^m10^r\|_2, and poses several open questions and conjectures about 2- and general l-complexity. These results illuminate the structure of efficient representations using powers of two and contribute to the broader study of integer complexity and its generalizations.

Abstract

The question of integer complexity asks about the minimal number of $1$'s that are needed to express a positive integer using only addition and multiplication (and parentheses). In this paper, we propose the notion of $l$-complexity of multiples of $l$, which specializes to integer complexity when $l=1$, prove several elementary results on $2$-complexity of even positive integers, and raise some interesting questions on $2$-complexity and in general $l$-complexity.

Theorems & Definitions (28)

• Conjecture 1
• Theorem 1
• Theorem 2
• Conjecture 2
• Proposition 1
• Lemma 1
• proof
• Lemma 2
• proof
• proof : Proof of \ref{['log-growth']}