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        riddles >> medium >> Powerful Equation
        
(Message started by: ThudanBlunder on Jan 20^th, 2009, 5:16pm)

Title: Powerful Equation
Post by ThudanBlunder on Jan 20^th, 2009, 5:16pm

If x http://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/ne.gif y, find all real solutions to x^y = y^x

Which solutions are rational? Which are integral?

Title: Re: Powerful Equation
Post by balakrishnan on Jan 21^st, 2009, 3:24am

[hide]The equation can be written as log(x)/x=log(y)/y.
It is an increasing function from 0 to e and decreases to zero from e to http://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/infty.gif. So if x and y has to be distinct, then x and y must lie in the range (1,http://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/infty.gif). One of the root will lie in the range (1,e) and the other in the range (e,http://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/infty.gif) .
If x and y are integers, then one of them must be 2(since there is only 1 integer in the range (1,e)) and the other is 4.
So the integer solutions are x=2,y=4 and x=4,y=2.

For rational solutions, it is not very difficult to see that x and y would take the form
x=((n+1)/n)^n, y=((n+1)/n)^(n+1) or
x=((n+1)/n)^(n+1), y=((n+1)/n)^(n) or

where n is any integer[/hide]