[net.math] deceptive problem: cont'd

debray@sbcs.UUCP (Saumya Debray) (12/12/83)

A followup to the "deceptive problem" posted from Dartmouth College (I
forget the submitter's name, I'm sorry):

I posted a solution to the equation "x^x = 2" of the form

	x = log 2 / (log (log 2 / (log (log 2 / ... )) )).

A solution of this form holds, in general, for equations of the form x^x = y
for y > 0.  Does anyone have a solution that will also hold for values of
y < 0 ? 
e.g.:
	x^x = y is satisfied by {x = -1/3, y = - 3^(1/3)}.  

-------------------

Saumya Debray
SUNY at Stony Brook

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------------------
P.S.: a couple of people have pointed out that I boo-boo'd in my previous
posting on this issue by claiming that "x^x = 2" was a polynomial of degree
x. I agree; my head hangs in shame!
-- 

Saumya Debray
SUNY at Stony Brook

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howard@metheus.UUCP (Howard A. Landman) (12/24/83)

But of course!  Instead of taking the logarithm of each side, take the
x'th root instead:

	x^x = y

	x = y^(1/x)

This leads to another infinite expansion, similar to the continued fraction:

	x = y^(1/y^(1/y^(1/y^( ... ))))

But this one doesn't use logs so it works for all values (except possibly at
0).

	Howard A. Landman
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