Hello. As an amateur, I am working on the history of the Last Fermat Theorem $(z^n≠x^n+y^n,\forall n \gt 2 )$, mainly trying to trace how Fermat came to it. He wrote the theorem in an observation of a Diophantus article about dividing squares $(z^2=x^2+y^2)$. Retrospectively we can wonder why instead he didn't assert $(z^2\neq x^n+y^n, \forall n>3)$ or more generally $(z^2\neq x^p+y^q, p>3, q>3)$, which are still today unresolved conjectures ? What was the very particular context that led him to his "admirable" discovery? This story needs to be clarified. If you want to help, feel free to contact me!