User:Wythagoras/Rado's sigma function/BB(11)

\(\Sigma(11)>f_4(2161856886993812)\). (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 \(B_5(m)>f_2(m)\) and that \(B_5(m)>3^{m+1}\) for \(m\geq1\).

Showing the bound for \(\text{BB}_{\text{Green}}(11)\)
Using the claims above, we have the following inequalities:

\begin{align*} \text{BB}_{\text{Green}}(11) &= B_9(B_9(1)) \\ &=B_9(B_7(2)+1) \\ &=B_9(2161856886993814) \\ &>B_7^{2161856886993814}(2) \\ &>B_7^{2161856886993813}(2161856886993813) \\ &>f_3^{2161856886993812}(2161856886993812) \\ &=f_4(2161856886993812) \end{align*}