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 May 14th, 2013, 10:57 PM #1 Math Team     Joined: Mar 2012 From: India, West Bengal Posts: 3,871 Thanks: 86 Math Focus: Number Theory Diophantine Equations Can it be proved that there are no solutions to the equation $x^x + y^y= z^z$ Where (x, y, z) > (1, 1, 1) ? Balarka .
 May 15th, 2013, 05:02 AM #2 Global Moderator     Joined: Nov 2006 From: UTC -5 Posts: 16,046 Thanks: 937 Math Focus: Number theory, computational mathematics, combinatorics, FOM, symbolic logic, TCS, algorithms Re: Diophantine Equations Yes, f(t) = t^t grows too quickly to have sums. Let x <= y <= z-1, then x^x = z^(z-1) * (z - ((z-1)/z)^(z-1)) > z^(z-1) * (z - 1) > y^y * (z-1), a contradiction for y >= 1.

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