Do there exist different positive integers **x, y**, and **z** such that

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Do there exist different positive integers **x, y**, and **z** such that

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There are no integers that satisfy that equation. Lets say xy, lets say that z=y+1 and for x^x + y^y to be as big as possible x=y-1, rearranging we have (y+1)^y+1 -[(y^y + (y-1)^y-1] and when y=1 it is indeterminate and for all y>1 it is greater than 0 therefore no matter what x and y are z^z will always be greater.

Paul.

it doesn’t seem to like the less than sign, in the top line it should read Lets say x is less than y.

I agree