I was often thinking about "whether Fermat really had a proof of his famous theorem":
Theorem. There are no positive integers x, y, z, and n > 2 such that x^n + y^n = z^n.
Today I found an answer in the mathematical FAQ:
"Did Fermat prove this theorem? No he did not. Fermat claimed to have found a proof of the theorem at an early stage in his career. Much later he spent time and effort proving the cases n = 4 and n = 5. Had he had a proof to his theorem earlier, there would have been no need for him to study specific cases."
This is followed by an interesting discussion about different mistakes that Fermat may have done (see section 3.1.4 of FAQ).
But, anyway, Pierre de Fermat never had his theorem proven.