visual diagram of proof

This is the section of number line where x,y, z, x+y located relative to each other (order is important, not distance)

_______ X ________ Y ___ Z ___ X + Y ____

Here is section of number line where X^2 to (X+Y)^2, with Y^2, Z^2, (X^2 +Y^2), etc


___ X^2 ___Y^2__ (X^n + Y^n)^2/n __ (X^2 + Y^2)^2/2


Now Z^2 =(X^n + Y^n)^2/n and is less than (X^2 + Y^2)^2/2 =(X^2 + Y^2)^1

So Z^2 < (X^2 + Y^2)^1

There must exist a value, call it a, where 0 < a < 1 where

Z^2 = (X^2 + Y^2)^a

But if X,Y, and Z are whole numbers, then the left hand side of equation is a whole number, while the right hand side is an non-whole number.
This is true because a whole number raised to a non-whole number is an non-number number.

Thus this contradiction proves there is no X,Y,Z all being whole numbers that can meet this requirement.