User:Wythagoras/Rado's sigma function/BB(12)

\(\Sigma(12)>f_4(f_3^2(2161856886993814))\). (Green 1964)

Green's numbers and the \(B\) hierarchy
Define for odd \(n\) the following hierarchy:

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

Then, Green's numbers \(\text{BB}_{\text{Green}}(n)\) are defined as:


 * \(\text{BB}_{\text{Green}}(n) = B_{n-2}[B_{n-2}(1)]\) for odd \(n\)


 * \(\text{BB}_{\text{Green}}(n) = B_{n-3}[B_{n-3}(3) + 1] + 1\) for even \(n\)

Definition form S. Ligocki.

The following things can be observed:


 * \(B_3(m)=3m+1\) and \(B_5(m)=\frac72\cdot 3^m-\frac52\). (As noted by Ligocki)


 * It can be proven using induction that \(B_5(m)>f_2(m)+2\) and that \(B_5(m)>3^{m+1}\) for \(m\geq1\).
 * It can be proven using induction that \(B_7(m)>f_3(m)\) and \(B_7(m)>f_4(m)\)

Showing the bound for \(\text{BB}_{\text{Green}}(12)\)
Using the claims above, we have the following inequalities:

\begin{align*} \text{BB}_{\text{Green}}(12) &> B_9(B_9(3)) \\ &=B_9(B_7(B_7(B_7(2)+1))) \\ &=B_9(B_7(B_7(2161856886993814))) \\  &>f_4(f_3(f_3(2161856886993814))) \\  \end{align*}