Upper Bounds on Integer Complexity

TL;DR

This paper establishes a universal non-trivial upper bound on the integer complexity of all natural numbers, improving understanding of how efficiently numbers can be expressed using only ones, addition, and multiplication.

Contribution

It provides the first universal upper bound for integer complexity applicable to all natural numbers, refining previous bounds that only held for almost all numbers.

Findings

Proves that ||n|| ≤ A log n for all n > 1, with A = 41 / log 55296.

Improves the understanding of the maximum complexity of integers.

Establishes a new universal upper bound on integer complexity.

Abstract

Define $||n||$ to be the \emph{complexity} of $n$, which is the smallest number of $1$s needed to write $n$ using an arbitrary combination of addition and multiplication. John Selfridge showed that $||n|| \geq 3\log_3 n$ for all $n$. Richard Guy noted the trivial upper bound that $||n|| \leq 3\log_2 n$ for all $n>1$ by writing $n$ in base 2. An upper bound for almost all $n$ was provided by Juan Arias de Reyna and Jan Van de Lune. This paper provides the first non-trivial upper bound for all $n$. In particular, for all $n>1$ we have $||n|| \leq A \log n$ where $A = \frac{41}{\log 55296}$.