Rich politely shielded me from the jeers of the rest of the list in pointing out an "algebraic" identity that applies to 3 as it does to any number, namely: (a^a)^(a^a) = (a^(a^a))^a In general, if K and L are two different exponentiation-towers, then (a^K)^L = a^(KL) = (a^L)^K. This inspires two new questions: (1) Is this "KL law" the only algebraic law that exponentiation-towers obey in general, or are there other such general relations not inferrable from the KL law? (2) In particular for a=3, are there any equivalences that aren't true for general a? (Probably a careful reading of Guy and Selfridge [1973], cited by Neil earlier, would answer both questions.) On Wed, Nov 29, 2017 at 2:30 PM, <rcs@xmission.com> wrote:
(3^3)^(3^3) = (3^(3^3))^3 = 3^(3^4) = 3^81 . The 3 and 3^3 in the exponent are multiplied, which commutes.
Rich
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Quoting Allan Wechsler <acwacw@gmail.com>:
I got my head in a little twist over this whole thread. Reassure me: do all the different parethesizations of 3^3^3^...^3, for n 3's, give different values? The original problem (using principle values of a^b) is interesting because i is special, right?
On Wed, Nov 29, 2017 at 9:46 AM, James Propp <jamespropp@gmail.com> wrote:
Apropos of parenthesizing exponential towers, here's an easier question
(probably not new): Look at all the (Catalan-many) ways to parenthesize a_1 - a_2 - ... - a_n. Not all are distinct as linear functions of a_1, a_2, ..., a_n; e.g.,
(a - (b - c)) - d = a - (b - (c - d)).
How many different functions can be obtained? I don't know the answer. Is it 2^(n-2)?
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