User:Wythagoras/Rado's sigma function/BB(10)

\(\Sigma(10)>f_3(f_2(3426462097810917))\). (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 \(\text{BB}_{\text{Green}}(10)\)
Using the claims above, we have the following inequalities:

\begin{align*} \text{BB}_{\text{Green}}(10) &> B_7(B_7(3)) \\ &=B_7(B_5(B_5(B_7(1)+1)+1))) \\ &=B_7(B_5(B_5(31)))) \\ &=B_7(B_5(2161856886993813))) \\ &>B_7(3^{2161856886993814}+3) \\ &>B_5^{3^{2161856886993814}+2}(B_7(1)+1)\\ &>B_5^{3^{2161856886993814}+2}(31) \\&>B_5^{3^{2161856886993814}}(3^{2161856886993814}) \\ &> f_3(3^{2161856886993814}) \\ &> f_3(f_2(3426462097810917)) \end{align*}