And, your proof using Diophantine equations has a mistake. You use 2) incorrectly at "If we substitute value for X into 2) we get". In 2), as you give it, Z is squared. When you use it, Z is not squared. And, you have given no explanation of the origin of your two equations relating x, y, and...
And, your proof using Diophantine equations has a mistake. You use 2) incorrectly at "If we substitute value for X into 2) we get". In 2), as you give it, Z is squared. When you use it, Z is not squared. And, you have given no explanation of the origin of your two equations relating x, y, and...