User:Wythagoras/Rado's sigma function/BB(9)

\(\Sigma(9)>f_2^{28}(3426462097810920) >f_2^29(45)\). (Green 1964)

Analysis
Define for odd \(n\) the following hierarchy:

\(B_n(m) =\begin{cases}  1 && \text{if  } m=0 \\  m+1 && \text{if  } n=1 \\ B_{n-2}[B_n(m-1) + 1] + 1 && \text{otherwise}\)

Then, Green's numbers \(BB_n\) are defined as:


 * \(BB_n = B_{n-2}[B_{n-2}(1)]\) for odd \(n\)

Definition form S. Ligocki.
 * \(BB_n = B_{n-3}[B_{n-3}(3) + 1] + 1\) for even \(n\)

Ligocki showed \(B_3(m)=3m+1\) and \(B_5(m)=\frac72\cdot 3^m-\frac52\). It can be proven using induction that \(B_5(m)>f_2(m)\). \(B_5(2)=29\). (Note that Ligocki erroneously claims that \(B_5(2)+1=29\)) We have

\[BB_9=B_7(B_7(1))=B_7(B_5(2)+1)=B_7(30)>B_5^{28}(B_7(2)+1)=B_5^{28}(B_5(B_7(1)+1)+2)=B_5^{28}(B_5(B_5(2)+2)+2)=B_5^{28}(B_5(31)+2)=B_5^{28}(2161856886993813+2)=B_5^{28}(2161856886993815)>B_5^{27}(3^{2161856886993816})>B_5^{27}(f_2(3426462097810920))>f_2^{28}(3426462097810920)>f_2^29(46)\]