On the High Water Mark Convergents of Champernowne's Constant in Base Ten

TL;DR

This paper investigates patterns in the continued fraction convergents of Champernowne's constant in base 10, formulates conjectures to predict their properties, and explores the structure and lengths of high water mark coefficients and their generations.

Contribution

It introduces new conjectures and formulas for predicting properties of convergents and high water mark coefficients of Champernowne's constant, including their lengths and structural relationships.

Findings

Patterns in convergents near HWMs are identified.

Conjectures for predicting errors and lengths of convergents are formulated.

Existence of multiple generations of HWMs is demonstrated.

Abstract

In this paper we show that numerous patterns exist in the properties of the convergents formed by truncating the Continued Fraction Expansion (CFE) of the Champernowne Constant in base 10 (C10) immediately before the High Water Marks (HWMs). From these patterns, we have formulated conjectures that may be used to predict the first position in C10 that is incorrect as calculated by the convergent, the error of the convergent, the denominator of the convergent, and the number of digits of C10 that are required to give the correct numerator of the convergent. The numerator and denominator then provide a very efficient method for calculating the CFE coefficients. In addition, we have formulated a conjecture that predicts the exact length in decimal digits of each of the HWM coefficients of C10 for HWM numbers greater than 3. Furthermore, it is conjectured that the lengths of certain other large coefficients that appear in the CFE of C10 are related to the lengths of the HWMs and a formula is given for this relationship. We designate these other large coefficients 2nd Generation HWMs, or Child HMWs. It is shown that 3rd, 4th, etc., Generation HWMs also exist. A conjecture regarding the structure and exact length of the denominators of the 2nd Generation HWM Convergents is presented. Finally, a table is presented that shows the Coefficient Numbers and the lengths of the 3rd and 4th Generation coefficients up to but not including HWM number 12.