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Cubum autem in duos cubos, aut quadratoquadratum in duos quadratoquadratos, et generaliter nullam in infinitum ultra quadratum potestatem in duos eiusdem nominis fas est dividere cuius rei demonstrationem mirabilem sane detexi. Hanc marginis exiguitas non caperet. 
("It is impossible for a cube to be the sum of two cubes, a fourth power to be the sum of two fourth powers, or in general for any number that is a power greater than the second to be the sum of two like powers. I have discovered a truly marvelous demonstration of this proposition that this margin is too narrow to contain.")^{[1]}
—Pierre de Fermat

Fermat, a prominent 17thcentury amateur mathematician, wrote the above note in his copy of a number theory textbook. By the time he died, the textbook was full of such teasing notes; his son published a new annotated edition of the book containing all of these notes in their proper places.
For nearly all the notes, it didn't take long for other mathematicians to figure out what Fermat was talking about. The quoted one was the exception. As such, it became known as Fermat's last theorem"last" not in the sense that it was the last mathematics he ever did (he almost certainly wrote the note fairly early in his life) but in the sense that it was the last claim he made to remain unproven. It took until nearly 350 years after Fermat's death until mathematicians Andrew Wiles and Richard Taylor released a proof in 1994.
In fact, it's almost certain that Fermat himself didn't really have a proof. Wiles' proof certainly would have been inaccessible to Fermat; it relies on mathematical concepts which weren't developed until after the Second World War. Romantically, one might imagine that Fermat had come up with some simple proof that has since eluded everyone else. In reality it's far more likely that he was mistaken, especially since later in life he went to the effort of working out a proof for a certain special case (that no fourth power can be written as the sum of two fourth powers). In fact, 19thcentury mathematician Gabriel Lamé had a flawed proof attempt that could have been much like Fermat's — the idea is just about practicable for a brilliant 17th century mathematician, whereas the flaw in it is a rather subtle technical matter that escaped just about everyone even in the 19th century.
There's often this idea in fiction that Wiles' proof is somehow incomplete or not good enough (mostly for being utterly inelegantnote, however, that mathematics is full of theorems whose bestknown proof is massively more difficult and complex than the statement of the theorem itself; Fermat's last theorem is by no means unique in this regard). No currently unsolved problem in mathematics has a story behind it that's nearly as good as Fermat's mysterious margin note, so it can be useful to pretend that Fermat's last theorem remains unsolved.
Not to be confused with Fermat's Little Theorem, which can be proved convincingly on the back of a postcard.
Instances of Fermat's last theorem in fiction:
 In Arthur Porges' short story "The Devil and Simon Flagg", a mathematician bets his soul that the Devil cannot prove Fermat's last theorem in twentyfour hours. He wins.
 A problem that might be substituted for Fermat's Last Theorem if reusing this plot would be to ask for Ramsey numbers. Extra bonus for them being associated with an Alien Invasion anecdote.
 Star Trek:
 In Star Trek the Next Generation, Picard spends some time trying to prove Fermat's last theorem. He says he finds it humbling that an 800yearold problem, first posed by a French mathematician without a computer, still eludes solution. (The episode in question was broadcast five years before Wiles' proof was released.)
 In Star Trek Deep Space Nine, Jadzia says that one of Dax's earlier hosts had the most original approach to Fermat's last theorem since Wiles. This may be an attempted Hand Wave for the TNG example, by showing that people are still working on the problem in the Star Trek universe even though it's been solved.
 In the Doctor Who episode The Eleventh Hour, the Doctor uses Fermat's original proof of Fermat's last theorem^{[2]} to get a team of scientists to take him seriously after hacking into their videoconference.
 Arthur C. Clarke's The Last Theorem is about a Sri Lanka mathematician trying to find a simpler proof to the problem.
 The Irish band BATS have a song about Andrew Wiles and the theorem.
 Irregular Webcomic posits that Fermat was a time traveler.
 "Prove Fermat's last theorem" occurs as a problem in an Only Smart People May Pass setup in Gash Bell. It's posed to the dumbest member of the party, and the rest force the guardian to give a simpler question by making him admit that he doesn't know the answer.
 Shows up in Tom Stoppard's Arcadia; as a joke Septimus assigns the Teen Genius Thomasina to solve it. She eventually comes to the conclusion that Fermat was Trolling. Interestingly, Arcadia was published mere months before Wiles' proof.
 In The Millennium Trilogy, Lisbeth spends most of the second book puzzling over the Theorem. At the end of the book, she understands what he meant, but after the ending of the book, forgets it.
 Appears briefly on a blackboard in the 2000 remake of Bedazzled. Satan (Elizabeth Hurley as a Hot Teacher) erases it from the list of homework assignments while commenting, "You'll never use this stuff."
 In GetBackers, Lucky, the genius dog, is given a problem like this to solve. The dog answers that it's unsolveable (x = "nothing"), which is what really clues Ban in to the fact that the whole "genius dog" thing isn't a parlor trick... the dog's actually been infected with the same virus that caused apes to mutate into humans, the socalled "Missing Link Virus." It... doesn't make sense in context, but there is an explanation.
 ↑ In Layman's Terms, take this equation: x^n plus y^n equals z^n. The Last Theorm dictactes that if n is a number above 2, you can't use whole numbers (2, 3, 4, etc.) for the x, y and z.
 ↑ (along with an explanation of why electrons have mass and a description of an FTL drive)