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2019 Integer complexity: the integer defect
Harry Altman
Mosc. J. Comb. Number Theory 8(3): 193-217 (2019). DOI: 10.2140/moscow.2019.8.193

## Abstract

Define $\parallel n\parallel$ to be the complexity of $n$, the smallest number of 1s needed to write $n$ using an arbitrary combination of addition and multiplication. John Selfridge showed that $\parallel n\parallel \ge 3{log}_{3}n$ for all $n$, leading this author and Zelinsky to define the defect of $n$, $\delta \left(n\right)$, to be the difference $\parallel n\parallel -3{log}_{3}n$. Meanwhile, in the study of addition chains, it is common to consider $s\left(n\right)$, the number of small steps of $n$, defined as $\ell \left(n\right)-⌊{log}_{2}n⌋$, an integer quantity, where $\ell \left(n\right)$ is the length of the shortest addition chain for $n$. So here we analogously define $D\left(n\right)$, the integer defect of $n$, an integer version of $\delta \left(n\right)$ analogous to $s\left(n\right)$. Note that $D\left(n\right)$ is not the same as $⌈\delta \left(n\right)⌉$.

We show that $D\left(n\right)$ has additional meaning in terms of the defect well-ordering we considered in 2015, in that $D\left(n\right)$ indicates which powers of $\omega$ the quantity $\delta \left(n\right)$ lies between when one restricts to $n$ with $\parallel n\parallel$ lying in a specified congruence class modulo $3$. We also determine all numbers $n$ with $D\left(n\right)\le 1$, and use this to generalize a result of Rawsthorne (1989).

## Citation

Harry Altman. "Integer complexity: the integer defect." Mosc. J. Comb. Number Theory 8 (3) 193 - 217, 2019. https://doi.org/10.2140/moscow.2019.8.193

## Information

Received: 2 August 2018; Revised: 7 April 2019; Accepted: 29 April 2019; Published: 2019
First available in Project Euclid: 13 August 2019

zbMATH: 07095940
MathSciNet: MR3990804
Digital Object Identifier: 10.2140/moscow.2019.8.193

Subjects:
Primary: 11A67

Keywords: arithmetic formulas , integer complexity , well-ordering