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May 14th, 2013, 11:57 PM   #1
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Diophantine Equations

Can it be proved that there are no solutions to the equation



Where (x, y, z) > (1, 1, 1) ?

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May 15th, 2013, 06:02 AM   #2
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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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