The 2-complexity of even positive integers

TL;DR

This paper investigates the minimal number of ones needed to express even positive integers using addition and multiplication, introducing the concept of l-complexity for multiples of l and exploring properties specific to 2-complexity.

Contribution

It introduces the notion of l-complexity for multiples of l, proves elementary results on 2-complexity of even integers, and raises open questions about l-complexity.

Findings

Elementary results on 2-complexity of even integers

Introduction of l-complexity concept for multiples of l

Open questions on l-complexity and 2-complexity

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.