Some integer formula-encodings and related algorithms

TL;DR

This paper explores special integer formulas built from addition, multiplication, and exponentiation with input 1, introducing two canonical encoding methods and algorithms for efficiently encoding large sets of integers.

Contribution

It presents two new canonical formula-encodings for integers and algorithms to efficiently determine encodings for large consecutive integer sets.

Findings

Two distinct canonical encoding methods based on decimal and arithmetic theorem

Algorithms for efficient encoding of large consecutive integers

Enhanced understanding of integer formulas and their representations

Abstract

We investigate the special class of formulas made up of arbitrary but finite com- binations of addition, multiplication, and exponentiation gates. The inputs to these formulas are restricted to the integral unit 1. In connection with such formulas, we describe two essen- tially distinct families of canonical formula-encodings for integers, respectively deduced from the decimal encoding and the fundamental theorem of arithmetic. Our main contribution is the de- tailed description of two algorithms which efficiently determine the canonical formula-encodings associated with relatively large sets of consecutive integers.