From: utzoo!decvax!ucbvax!ucbcad:moore
Newsgroups: net.math
Title: Re: Infinite Exponentiation - (nf)
Article-I.D.: ucbcad.286
Posted: Fri Apr 29 19:38:36 1983
Received: Tue May  3 01:55:59 1983

#R:ucbvax:-48000:ucbcad:11900001:000:717
ucbcad!moore    Apr 29 02:55:00 1983

			...
			x
		       x
		      x = 2. 

	What is the value of x? 


since I don't have a picture mode editor, I will restate this as

x**x**x**x**..... = 2.

if we assume that x**x**x = x**(x**x) (right to left associative)

then x**x**x... = x**(x**x**x...) = x**2 => x**2 = 2. => x = sqrt(2).

    We have implicitly assumed that an answer exists here, in that we
are manipulating x**x**x... as though it was a well-behaved quantity.
As an example of the pitfalls, the problem x**x**x.. = 4 has the same
solution, i.e. x = sqrt(2)! Does anyone know what the largest n is
such that x**x**x.. = n has a solution? I suspect e, but have yet
to prove it.

    Peter Moore
    moore@Berkeley
    ...!ucbvax!ucbcad!moore