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From: tan@ihu1e.UUCP (exit)
Newsgroups: net.puzzle
Subject: Re: x to the x ... :
Message-ID: <448@ihu1e.UUCP>
Date: Thu, 2-May-85 14:00:18 EDT
Article-I.D.: ihu1e.448
Posted: Thu May  2 14:00:18 1985
Date-Received: Fri, 3-May-85 04:41:25 EDT
References: <558@hou2e.UUCP>
Organization: AT&T Bell Laboratories
Lines: 26

> Now I have a question.
> 
> 	x^(x^(x^(x^... = 2  implies  x = sqrt(2)  as shown earlier.
> 
> 	Then in the equation x^(x^(x^(x^... = 4, we can perform a similar
> 	substitution to obtain
> 
> 		x^4 = 4  =>  x = sqrt(2).
> 
> 	How can the L.H.S. using x = sqrt(2) equal both 2 and 4?
> 
> 				Jeff Heatwole
> 				..!hou2e!jlh
The answer is that x^(x^(x^...... for x=sqrt(2) is equal to 2 and
not to 4.  The equation x^(x^(x^..... = 4 has no solution (at least
no real solution- I haven't taken the time to check for complex ones).
The solution given by Heatwole is erroneous, because the substitution
assumes a priori that there is a solution.  If there is indeed no
solution to the equation x^x^x.... = 4, the substitution of 4
for x^x^x..... is invalid.  The same can be said for the "solution"
given for x^x^x.... = 2; the substitution only proves that if there is
a solution, it must be sqrt(2) (ignoring for the moment the possibility
of -sqrt(2)).  Actual computation of the limit shows that sqrt(2) here
IS a valid solution.
-- 
Bill Tanenbaum AT&T Bell Laboratories - Naperville Ill.