My favorite theorem is Fermat's Last Theorem.
In many ways it is related and similar to the Pythagorean Theorem stating that:
$ a^n + b^n = c^n $ has no non-zero integer solutions for any integer n > 2
This "theorem" was actually a challenge problem that Fermat left for the rest of the world. He claimed: "I have a truly marvelous proof of this proposition which this margin is too narrow to contain." The proof to this conjecture was not found for over 350 years after his death. Finally, in 1995, after using and developing mathematical techniques far beyond those that Fermat had, Andrew Wiles published a correct proof.
