Ethernos Posts 1 & 2: You do not seem to understand that Fermat was relating to integer solutions. His conjecture was proven recently: Perhaps circa 5-15 years ago. I do not remember when. The proof used methods not known by Fermat or any of his contemporaries. He was a genius in the pertinent mathematical discipline (Number Theory, I think) & some believe that he had a proof not using the modern methods. There might be serious mathematicians trying to develop a proof consistent with methods known to Fermat. There is at least one proof which was accepted by the mathematical community until a very subtle flaw was discovered. Some believe that this was Fermatâ€™s proof.

BTW: The conjecture was called Fermat's Last Theorem because it was the last to be proven, not because it was chronologically last. In Fermat's era, mathematicians were secretive about some of their methods. They played games with each other, trying to be able to answer questions that other mathematicians could not answer.