Talk:Quintalum

$$ f_2(10)=f_1^{10}(10)=10\times 2^{10}=10\times 1024=10240$$

$$f_2^2(10)= f_2(f_2(10))= 10240\times 2^{10240}=10240\times 10^{10240 \times \log(2)}=3.6095707260 \times 10^{3086}$$

$$f_2^3(10)= f_2(f_2(f_2(10)))=3.6095707260 \times 10^{3086}\times 10^{3.6095707260 \times 10^{3086} \times \log(2)}=$$

$$=10^{1.0865890600 \times 10^{3086}}$$

$$f_2^4(10)= f_2(f_2(f_2(f_2(10))))=10^{1.0865890600 \times 10^{3086}}\times 10^{10^{1.0865890600 \times 10^{3086}}\times \log(2)}=$$

$$=10^{10^{1.0865890600 \times 10^{3086}}}=10^{10^{10^{10^{3.489}}}}=$$

$$=\underbrace{ 10^{10^{...{^{10^{3.489}}}}}}_{4 \quad tens }=E3.489 \# 4$$

and so on.

$$ f_3(10)=f_2^{10}(10)=\underbrace{ 10^{10^{...{^{10^{3.489}}}}}}_{10 \quad tens }=E3.489 \# 10\approx 10\uparrow\uparrow 11$$

$$f_3^2(10)= f_3(f_3(10))= \left. \begin{matrix} &&\underbrace{10^{10^{...{^{10^{3.489}}}}}}\\ & &\underbrace{10^{10^{...{^{10^{3.489}}}}}} \\ & &10\quad tens \end{matrix} \right \} =E3.489 \# 10\# 2 \approx 10\uparrow\uparrow\uparrow 3$$

$$f_3^3(10)= f_3(f_3(f_3(10)))=  \left. \begin{matrix} &&\underbrace{10^{10^{...{^{10^{3.489}}}}}}\\ & &\underbrace{10^{10^{...{^{10^{3.489}}}}}} \\ & &\underbrace{10^{10^{...{^{10^{3.489}}}}}} \\ & &10 \quad tens \end{matrix} \right \} \text {3 lower braces} =$$

$$=E3.489 \# 10\# 3 \approx 10\uparrow\uparrow\uparrow 4$$

and so on.

$$ f_4(10)=f_3^{10}(10)= \left. \begin{matrix} &&\underbrace{10^{10^{...{^{10^{3.489}}}}}}\\ & &\underbrace{10^{10^{...{^{10^{3.489}}}}}} \\ & & \underbrace{\quad \;\; \vdots \quad\;\;}\\ & &\underbrace{10^{10^{...{^{10^{3.489}}}}}} \\ & &10 \quad tens \end{matrix} \right \} \text {10 lower braces}=$$

$$ =E3.489 \# 10\# 10 \approx 10\uparrow\uparrow\uparrow 11 $$

$$ f_4^2(10)=f_4(f_4(10))= \left. \left.\begin{matrix} &&\underbrace{10^{10^{...{^{10^{3.489}}}}}}\\ & &\underbrace{10^{10^{...{^{10^{3.489}}}}}} \\ & & \underbrace{\quad \;\; \vdots \quad\;\;}\\ & &\underbrace{10^{10^{...{^{10^{3.489}}}}}} \\ & & 10\quad tens \end{matrix} \right \} \begin{matrix} &&\underbrace{10^{10^{...{^{10^{3.489}}}}}}\\ & &\underbrace{10^{10^{...{^{10^{3.489}}}}}} \\ & & \underbrace{\quad \;\; \vdots \quad\;\;}\\ & &\underbrace{10^{10^{...{^{10^{3.489}}}}}} \\ & & 10\quad tens \end{matrix} \right \} \text {10 lower braces}=$$

$$=E3.489 \# 10\# 10\#2 \approx 10\uparrow^{4} 3$$

and so on.

$$f_5(10)=f_4^{10}(10)=$$

$$=\underbrace{\left. \left.\left.\begin{matrix} &&\underbrace{10^{10^{...{^{10^{3.489}}}}}}\\ & &\underbrace{10^{10^{...{^{10^{3.489}}}}}} \\ & & \underbrace{\quad \;\; \vdots \quad\;\;}\\ & &\underbrace{10^{10^{...{^{10^{3.489}}}}}} \\ & &10\quad tens \end{matrix} \right \} \begin{matrix} &&\underbrace{10^{10^{...{^{10^{3.489}}}}}}\\ & &\underbrace{10^{10^{...{^{10^{3.489}}}}}} \\ & & \underbrace{\quad \;\; \vdots \quad\;\;}\\ & &\underbrace{10^{10^{...{^{10^{3.489}}}}}} \\ & &10\quad tens \end{matrix} \right \}\cdots \begin{matrix} &&\underbrace{10^{10^{...{^{10^{3.489}}}}}}\\ & &\underbrace{10^{10^{...{^{10^{3.489}}}}}} \\ & & \underbrace{\quad \;\; \vdots \quad\;\;}\\ & &\underbrace{10^{10^{...{^{10^{3.489}}}}}} \\ & &10\quad tens \end{matrix} \right \} \text {10 lower braces}}_{10 \quad right \quad braces}=$$

$$=E3.489 \# 10 \# 10 \# 10=E3.489 \# 10 \# \#3 \approx 10\uparrow^{4} 11$$

In general case

$$f_m(10)=E3.489 \# 10\# \#(m-2) \approx 10\uparrow^{m-1} 11=10 \rightarrow 11 \rightarrow (m-1)=\{10,11,m-1\}$$