Champernowne常数在不同基下的高水位转换分析
Analysis of the High Water Mark Convergents of Champernowne's Constant in Various Bases
摘要：
在本文中，我们表明，模式存在于各种基地（Cb），特别是在基地2至124的Champernowne常数的高水位（HWM）收敛的属性。通过截断紧接在CFE HWM之前的Cb的连分数展开（CFE）来形成转换。这些模式是从Champernowne常数以十为底的CFE HWM中已知的模式扩展而来的。我们表明，模式可以用来有效地计算的CFE系数的Cb，并计算和预测的长度的HWM系数，正确的位数的Cb计算的收敛，收敛误差。我们已经发现，轻微的修正，在基地10的图案配方需要一些基地，这些修正提出和讨论。所得公式可用于计算任何基的Cb。
英文摘要：
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.
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