Analysis of the High Water Mark Convergents of Champernowne's Constant in Various Bases

TL;DR

This paper investigates patterns in the high water mark convergents of Champernowne's Constant across bases 2 to 124, enabling efficient computation and prediction of properties related to its continued fraction expansion.

Contribution

It extends known patterns of Champernowne's Constant's convergents to various bases and introduces methods for calculating and predicting their properties across these bases.

Findings

Patterns exist in HWMs of Cb in bases 2-124.

Formulas can predict HWM lengths and digit accuracy.

Minor base-specific corrections improve pattern accuracy.

Abstract

In this paper, we show that patterns exist in the properties of the High Water Mark (HWM) convergents of Champernowne's Constant in various bases (Cb), specifically in bases 2 through 124. The convergents are formed by truncating the Continued Fraction Expansion (CFE) of Cb immediately before the CFE HWMs. These patterns have been extended from the known patterns in the CFE HWMs of Champernowne's Constant in base ten. We show that the patterns may be used to efficiently calculate the CFE coefficients of Cb, and to calculate and predict the lengths of the HWM coefficients, the number of correct digits of Cb as calculated by the convergent, and the convergent error. We have discovered that minor corrections to the pattern formulations in base 10 are required for some bases, and these corrections are presented and discussed. The resulting formulations may be used to make the calculations for Cb in any base.