I had this rare privilege of being able to pursue in my adult life, what had been my childhood dream.
I really believed that I was on the right track, but that did not mean that I would necessarily reach my goal.
I tried to fit it in with some previous broad conceptual understanding of some part of mathematics that would clarify the particular problem I was thinking about.
I don't believe Fermat had a proof. I think he fooled himself into thinking he had a proof.
Pure mathematicians just love to try unsolved problems - they love a challenge.