Characterising Solutions of Anomalous Cancellation

TL;DR

This paper investigates the mathematical properties and structure of solutions involving anomalous digit cancellation in fractions, providing general results and estimates across different bases, especially prime power bases.

Contribution

It offers new general results on the structure of anomalous cancellation solutions and estimates their quantity in various base representations, extending previous specific studies.

Findings

Solutions are connected to the number of digits and base representation.

Saturation of solutions occurs in certain bases, notably prime power bases.

Detailed properties of solutions in prime power bases are established.

Abstract

Anomalous cancellation of fractions is a mathematically inaccurate method where cancelling the common digits of the numerator and denominator correctly reduces it. While it appears to be accidentally successful, the property of anomalous cancellation is intricately connected to the number of digits of the denominator as well as the base in which the fraction is represented. Previous work have been mostly surrounding three digit solutions or specific properties of the same. This paper seeks to get general results regarding the structure of numbers that follow the cancellation property (denoted by $P^*_{\ell; k}$) and an estimate of the total number of solutions possible in a given base representation. In particular, interesting properties regarding the saturation of the number of solutions in general and $p^n$ bases (where $p$ is a prime) have been studied in detail.