Numbers with Integer Complexity Close to the Lower Bound

TL;DR

This paper introduces a classification method for numbers with integer complexity close to the lower bound, proves exact complexity for certain powers of 2 and 3, and analyzes the distribution of complexity defects.

Contribution

It presents a new method to classify numbers with small complexity defect and establishes asymptotic growth of such numbers, advancing understanding of integer complexity.

Findings

Proved $|2^m 3^k|=2m+3k$ for $m extless=21$

Developed a classification method for numbers with $ ext{defect}<r$

Showed the number of such numbers up to $x$ grows as $( ext{log } x)^{ ext{floor}(r)+1}$

Abstract

Define $|n|$ to be the complexity of $n$, the smallest number of 1's needed to write $n$ using an arbitrary combination of addition and multiplication. John Selfridge showed that $|n|\ge 3\log_3 n$ for all $n$. Define the defect of $n$, denoted $\delta(n)$, to be $|n|-3\log_3 n$; in this paper we present a method for classifying all $n$ with $\delta(n)<r$ for a given $r$. From this, we derive several consequences. We prove that $|2^m 3^k|=2m+3k$ for $m\le 21$ with $m$ and $k$ not both zero, and present a method that can, with more computation, potentially prove the same for larger $m$. Furthermore, defining $A_r(x)$ to be the number of $n$ with $\delta(n)<r$ and $n\le x$, we prove that $A_r(x)=\Theta_r((\log x)^{\lfloor r \rfloor+1})$, allowing us to conclude that the values of $|n|-3\log_3 n$ can be arbitrarily large.