论文标题
关于某些非理性数字的B-ary扩展中的精神分裂症模式
On Schizophrenic Patterns in b-ary Expansions of Some Irrational Numbers
论文作者
论文摘要
在本文中，我们研究了复发$ f_b（n）= b f_b（n-1）+n $定义的$ b $ arive膨胀，其初始值$ f（0）= 0 $在奇数正整数$ n $中取出，其中特殊情况$ b = 10 $通常称为“ schizizophrenic”或“ schizizophrenic”或“ sichizophrenic”或“模拟式”或“模仿”。这些不合理的数字在2004美元的$ 2004 $中定义，并由布朗以2009美元的价格进行了详细研究，其特殊性是在小数扩展中包含长期重复数字的特殊性。本文的主要贡献是，通过正式定义这些数字的$ b $扩展中存在的精神分裂模式，并研究出现在其中出现的非重生和重复的数字序列的长度，将精神分裂数扩展到所有整数$ b \ geq2 $。
In this paper we study the $b$-ary expansions of the square roots of the function defined by the recurrence $f_b(n)=b f_b(n-1)+n$ with initial value $f(0)=0$ taken at odd positive integers $n$, of which the special case $b=10$ is often referred to as the "schizophrenic" or "mock-rational" numbers. Defined by Darling in $2004$ and studied in more detail by Brown in $2009$, these irrational numbers have the peculiarity of containing long strings of repeating digits within their decimal expansion. The main contribution of this paper is the extension of schizophrenic numbers to all integer bases $b\geq2$ by formally defining the schizophrenic pattern present in the $b$-ary expansion of these numbers and the study of the lengths of the non-repeating and repeating digit sequences that appear within.
