Fermat's theorem was a mathematical statement. The Eleventh Doctor used a proof of it to convince a body of geniuses they should take his advice about how to save the world from annihilation by the Atraxi. He uploaded the proof — which the Doctor called "the real one, never been seen before" — to a secure video conference from a laptop computer in Leadworth. Simultaneously, he muttered an acknowledgment to the originator of the theorem, Fermat, and mentioned that Evariste got killed in a duel before he could write down his own proof. He claimed responsibility for the death of the latter, saying that it had only happened because he had slept in that morning. (TV: The Eleventh Hour)
Behind the scenes
This reference appears to be to Fermat's Last Theorem, but details in the Doctor's speech indicate differences from the recorded history in the real world. The theorem was actually proved using modern mathematics in the 1990s, but the Doctor's clarification that he was sending "the real one", which leads us to believe that he meant the proof that Fermat supposedly had but couldn't write it down and not the proof of Dr. Andrew Wiles who used modern mathematics. The other reference is to Évariste Galois, a 19th century mathematician, who was killed in a duel, whose work on the theorem was fundamental to the 1990s proof.