Ah, good. So there are always "special" relations, at least for positive integers, that are true for one number but not any others. That answers my question (2) pretty definitively. On Wed, Nov 29, 2017 at 5:28 PM, Tom Karzes <karzes@sonic.net> wrote:
In the case where a=3, note that:
3^(3^3) = 3^(3*3*3) = ((3^3)^3)^3
Call the first form x and the last form y. They have different numbers of threes, but we can balance them by using equal numbers of x and y, e.g.
x^y = y^x
i.e.:
(3^(3^3))^(((3^3)^3)^3) = (((3^3)^3)^3)^(3^(3^3))
You can do the same thing for any integer. E.g. in the case of a=4:
4^(4^4) = 4^(4*4*4*4) = (((4^4)^4)^4)^4
Tom
Allan Wechsler writes:
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.)
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