Zeroless Arithmetic: Representing Integers ONLY using ONE

TL;DR

This paper explores representing integers solely with the number one using recurrence equations, providing enumeration, efficient generation algorithms, and methods to find shortest formulas for positive integers.

Contribution

It introduces a novel approach using recurrence equations to enumerate and generate integer representations with only ones, including algorithms for shortest formula discovery.

Findings

Enumeration formulas for representations using only ones

Efficient algorithms for random generation of formulas

Method to find shortest representations for any positive integer

Abstract

We use recurrence equations (alias difference equations) to enumerate the number of formula-representations of positive integers using only addition and multiplication, and using addition, multiplication, and exponentiation, where all the inputs are ones. We also describe efficient algorithms for the random generation of such representations, and use Dynamical Programming to find a shortest possible formula representing any given positive integer.