Theory of numbers: Should the proof of the last theorem of an elemental Fermat be related to squares and triangles?

I ask this in my development of a framework based on a method of discovery. As I have memory and learning problems, I had to teach myself in many ways.

The statement of the problem.

it has no solutions when x, y and z are natural numbers (positive whole numbers (integers) except 0 or "count numbers" such as 1, 2, 3 …). This means that there are no natural numbers x, y and z for which this equation is true (that is, the values ​​on both sides can never be equal if x, y, z are natural numbers and n is an integer greater than 2).

IF these are the rules without hidden variables. I have a potential elementary solution with a positive integer that is the sum of the squares and the hypotenuse of the Pythagorean triples. I came back for fun after a wild adventure in math, numbers, equations and the construction of a dynamic system.

So I just want to make sure that the logic is correct. Thanks in advance.