I was trying to figure out how big 3^^^3 was, which led to the following interesting math result. How high would a power tower of 3's have to be to surpass a googolplex raised to the googolplexth power? For what value of X is (3^^X)>(googolplex^^2)? I don't have the full answer, but an upper bound for X is 16. A power tower of 3's 16 high is guaranteed to be vastly larger than a googolplex raised to itself. And when you consider that 3^^^3 is a power tower 7.6 trillion 3's tall... it's way larger than I thought.
In what follows, all logs are to base 3.
Definition: [a,b,c,d] := a^(b^(c^d))), etc.
Lemma: log [a,b,...,z] = log a^[b,...,z] = (log a) [b,...,z].
Definition: {n} := [3,...,3] with n 3's.
Lemma: log {n} = {n-1}.
Definition: G := [10,10,100].
OK. So we want to know when {n} > [G,G]. Taking logs, this is the same as {n-1} > G log G = [10,10,100] log [10,10,100]. Taking logs again, it's the same as {n-2} > log log [10,10,100] + log [10,10,100] which (unless it comes out amazingly close) is the same as {n-2} > log [10,10,100] = (log 10) [10,100]. Taking ...
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