okay, so in one of those shower revelation moments, it occurred to me that the theory I was using earlier did not say that a square is a product of two squares...it said that a number to the nth power was a product of two numbers to the nth power. Using this glaringly obvious information I relooked at the problem and found:

allows for either odd or even y. when z and x are both odd, y is even. When z and x are not both odd or even, then y is odd. When z and x are both even, y is even and (x,y,z)=2 and is a contradiction of (x,y,z)=1.

So for this first case, where y is even, then z-x and z+x are both even. And if 2|y, then 2^3|y^3 and the math gives:

, divide this through by 8 and you get

setting and

the math gives me

The second case is when y is odd and z+x and z-x are both odd.

So for this case the math gives:

setting and

the math gives me

The answer in the back of the book takes it a step further, which had me confused, by stating both answers in one combined format of:

where either d=2, u>0, v>0, (u,v)=1, u odd or d=1, u>v>0, (u,v)=1, u odd, v odd.

So my answers match up, I just need to take a look at the u odd vs u and v both odd when I am a little more awake! One more problem and I will only have one more lesson to go.