## Is there (x,y,z); x^x + y^y = z^z ?

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

$x^x \; + \; y^y \; = \; z^z$

math grad - Interest: Number theory
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### 3 Responses to Is there (x,y,z); x^x + y^y = z^z ?

1. paul says:

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.

• paul says:

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

• benvitalis says:

I agree