F(2,n) =
F(1,F(2,n-1)) =
2^F(2,n-1)
==
2^..4 :n-2 =
2^^n
F(3,n) =
F(2,F(3,n-1)) =
2^^F(3,n-1)
==
2^^..4 :n-2 = 2^^^n
F(m,n) = 2^^{..}n ^:m = 2→n→m
a→b→c+1→2 = a→b→(a→b→c→2)
==
a→b→(..a→b→1→2.) :c:
=
a→b→(..a^b.) :c:
N* ≤
2→3→(..12.)
:7:
<
4→2→8→2 ==
4→2→(..16.)
:7:
Graham's number record explained by Gardner
GN' ≤
3→3→(..4.)
:64:
<
2→3→65→2 =
2→3→(..8.)
:64:
Graham & Rothschild's original estimate was less
GN* ≤
2→3→(..12.)
:7:
<
4→2→8→2 <
2→3→9→2
New bounds for Graham's number of hypercube dimensions
13 ≤ GN ≤ 2→6→3
[as of September 2014]
Presented as research material on the topic of Big numbers for which Graham's number(s) is an ongoing inspiration.