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Path: utzoo!linus!philabs!sbcs!debray
From: debray@sbcs.UUCP (Saumya Debray)
Newsgroups: net.math
Subject: Re: A rather deceptive problem.
Message-ID: <542@sbcs.UUCP>
Date: Mon, 5-Dec-83 23:01:09 EST
Article-I.D.: sbcs.542
Posted: Mon Dec  5 23:01:09 1983
Date-Received: Wed, 7-Dec-83 09:01:27 EST
References: <464@dartvax.UUCP>
Organization: SUNY at Stony Brook
Lines: 41


	> If it is given that  x^x = 2, is it possible to obtain an
	> exact expression for x which contains no variables?   ( Or,
	> general case: if x^x = y,  what is y in terms of x?

For values of x > 1, here's a solution:

	for x^x = 2, we have x log x = log 2, whence

		    log 2
		x = -----
		    log x

	which leads to a continued fraction representation for x:

		     		log 2
		x = --------------------------------
		                   log 2
		     log ( ------------------------ )
		                       log 2
			   log ( ------------------ )
			                  log 2
				 log ( ------------ )
				            .
					      .
					        .


The general case (x^x = y) has a similar solution.

Questions: (1) Does anyone have a general solution that would hold for
		all values of x ?

	   (2) It seems to me that since "x^x = 2" is a polynomial of
	       order x, it ought to have x roots, of which the above is
	       just one. Any suggestions for a general solution?
-- 

Saumya Debray
SUNY at Stony Brook
{philabs, ogcvax}!sbcs!debray