What happens when Z is presumed to be whole

Let's see when Z is presumed to be a whole number

Let's look at the section of the number line from Y^2 to (X + Y)^2

It should look like this:

------- Y^2 ------ Z^2 ----- Z^2 + 1 ----------(X^2 + Y^2)----(X + Y)^2 ----

We know there is at least one number that corresponds to Z^2 + 1 since (X^2 + Y^2) is a member of that set of whole numbers between Z^2 and (X + Y)^2. There may be many numbers,there may be only one number, but there is at least one.

Let us now look at the number line from Y^n to (X + Y)^n

Z^n
--------Y^n -------+ ------Z^n + 1 -------- (X^2 + Y^2)n/2 -----(X + Y)^n----
(X^n + Y^n)

If you inspect closely, in the first line segment Z^2 + 1 < (X^2 + Y^2) and in the

second (X^n + Y^n) < Z^n +1.

What this implies is that there is some Z^h where

where 2 < h < n, where Z^h = X^h + Y^h.

We also know h is 'close' to n,

as the nth root of Z^h is close to Z, less than Z+1 , in fact.

If we now use some information about square roots, the importance of there being a Z^h becomes significant.

Because of the equation between Z^h and X^h, Y^h, if we increase the left hand side by a power to make Z^h*m = Z^n, according to Pythagorus theorem, we would would have to increase both X, Y by more than than a factor of m in order for the relationship to still hold. Z is a larger number than X,Y , so increasing Z by a factor, means that X,Y will also increase but by the same factor that will make X,Y larger but not enough to be equal to a whole number.

If you increase the hypotenuse by a factor m, the 2 legs of the triangle have to increase by more than a factor m, in order that the relationship remain the same, ie. that whole numbers remain whole numbers.

So if Z^h = X^h + Y^h then Z^h*p = (X^n*m + Y^n*m) where p > 1, m > p, if Z,XY remain whole numbers. Because we chose to look at Z^n +1, the nth root is very close to Z. This also implies that nth roots of X^n*m and Y^n*m are very close to X,Y.

So for all cases of Z^n, n > 2, X^n + Y^n < Z^n < X^n + Y^n +1
It is impossible for Z^n to be a whole number, being between two consecutive whole numbers.