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        riddles >> medium >> Radical power equality
        
(Message started by: Michael_Dagg on Feb 4th, 2008, 7:20pm)
Title: Radical power equality
Post by Michael_Dagg on Feb 4th, 2008, 7:20pm
Let positive integers  a, n  be given.  Show that
there is a positive integer  b  for which

(sqrt(a) - sqrt(a - 1))^n  =  sqrt(b) - sqrt(b - 1).
Title: Re: Radical power equality
Post by Eigenray on Feb 4th, 2008, 8:04pm
First, if n=2, we take [hide]b=(2a-1)2[/hide].  Now [hide]by replacing a with b, and n by n/2, repeatedly, we can assume n=2k+1 is odd[/hide]  In this case,

[hideb][http://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/surd.gifa - http://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/surd.gif(a-1)]n = chttp://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/surd.gifa - dhttp://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/surd.gif(a-1), where

c = http://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/sum.gifC(2k+1,2i+1) ai(a-1)k-i,
d = http://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/sum.gifC(2k+1,2i) ai(a-1)k-i.

But we have also [http://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/surd.gifa + http://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/surd.gif(a-1)]n = chttp://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/surd.gifa + dhttp://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/surd.gif(a-1), and multiplying these two equations gives

1 = ([http://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/surd.gifa - http://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/surd.gif(a-1)][http://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/surd.gifa + http://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/surd.gif(a-1)])n = [chttp://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/surd.gifa - dhttp://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/surd.gif(a-1)][chttp://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/surd.gifa + dhttp://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/surd.gif(a-1)] = c2a - d2(a-1),

and therefore we take b = c2a:

(http://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/surd.gifa - http://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/surd.gif(a-1))n = http://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/surd.gif(c2a) - http://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/surd.gif(d2(a-1)) = http://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/surd.gif(c2a) - http://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/surd.gif(c2a-1).[/hideb]
Title: Re: Radical power equality
Post by Michael_Dagg on Feb 5th, 2008, 3:42pm
Clever (and economical) !   ;)


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