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Prouhet-Thue-Morse Research 6: Extended Equivalence¶
2026-05-18
Foundations & Binary Numeric Definitions
Binary Definitions - Textual & Sequential
Binary Definitions - Others
Binary Equivalence
Extending the Alphabet
Extended Equivalence (you are here)
Property Preservation
Complexity and Honorable Mentions
Introduction¶
Proving Equivalence Between Extended Definitions¶
Correlating Definition 1 and Definition 2¶
\[ \begin{align}\begin{aligned}\begin{proof}
\par\noindent\par
\textbf{Observation 1}: For all integers $n$, $\omega_s^n = \omega_s^{n \; \mod{s}}$\\ \textbf{Inference 1}: $\dfrac{\log(\omega_s^n)}{\log(\omega_s)} = n \; \mod{s}$\\\begin{split} \textbf{Conclusion}: \begin{equation}
\begin{aligned}
T_{n,2}(x, s) &= \dfrac{\log(\omega_s^{p_s(x)})}{\log(\omega_s)} \\
&= \dfrac{(p_s(x) \; \mod{s}) \cdot \log(\omega_s)}{\log(\omega_s)} \\
&= \dfrac{(p_s(x) \; \mod{s}) \cdot 2i\pi s^{-1}}{2i\pi s^{-1}} \\
&= p_s(x) \; \mod{s} \\
&= T_{n,1}(x, s)
\end{aligned}
\end{equation}
\end{proof}\end{split}\end{aligned}\end{align} \]
Correlating Definition 4 and Definition 8¶
\[ \begin{align}\begin{aligned}\begin{proof}
\par\noindent\par
\textbf{Observation 1}: For any given row of $L(N)$, it will start with the index of the row\\ \textbf{Observation 2}: For any given row of $L(N)$, it will end with 1 less than the index of the row $\pmod{N}$\\ \textbf{Inference 1}: $L(N)_{x,*} = r(b(N), x)$\\ \textbf{Observation 3}: $T_{n,4}$ is defined as substituting $r(b(N),x)$ for each element $x$ in the previous iteration\\ \textbf{Conclusion}: $T_{n,4} = T_{n,8}$
\end{proof}\end{aligned}\end{align} \]
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@online{appleton_blog_0018, title = { Prouhet-Thue-Morse Research 6: Extended Equivalence }, author = { Olivia Appleton-Crocker }, editor = { Ultralee0 and Ruby }, language = { English }, version = { 1 }, date = { 2026-05-18 }, url = { https://blog.oliviaappleton.com/posts/0019-thue-morse-06 } }
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