Line #14 UsefulNotes/FermatsLastTheorem


1	[[quoteright:300:https://static.tvtropes.org/pmwiki/pub/images/fermatproof_1095.jpg]]
2	[-[[caption-width-right:300:[[TheNineties The USSR, Apartheid, and now this]]. IsNothingSacred]]-]
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4	->''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.''
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6	->("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.")
7	-->-- '''Pierre de Fermat'''
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9	In LaymansTerms, take this equation: a[[superscript:n]] + b[[superscript:n]] = c[[superscript:n]], where a, b, c and n are all positive whole numbers. While there are infinitely many cases where this equation is true when n = 1 (which is simple addition) or n = 2 (called the [[https://en.wikipedia.org/wiki/Pythagorean_triple Pythagorean triples]]), the Last Theorem says that there is no solution if n is greater than 2. The problem was to solve the theorem, either by proving it or by producing a counterexample. Despite monumental interest and attention from the mathematical community, nobody managed it for three and a half centuries.
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11	Fermat, a prominent 17th-century 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. The proof's effect in fiction that referenced it was a mess-up -- most works set in the future just assumed the Theorem would remain unproven for centuries, millennia, or even forever.
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13	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 UsefulNotes/WorldWarII. 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)[[note]]Though he could still have had a general proof for all odd numbers or odd primes, since proving the case for all odd numbers/primes and 4 would prove Fermat's last theorem for all cases[[/note]]. In fact, 19th-century 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.
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15	Interestingly, Wiles didn't actually prove Fermat's last theorem directly. His proof is a proof by contradiction revolving around a completely separate concept, the [[https://en.wikipedia.org/wiki/Modularity_theorem Taniyama-Shimura Conjecture]], which states that all [[https://en.wikipedia.org/wiki/Elliptic_curve elliptic curves]] have an associated [[https://en.wikipedia.org/wiki/Modular_form modular form]]. By the mid-20th century, the remaining unproven case for the equation was for all prime values of n, any of which will be referred to as ''p''. Additionally, it was shown that if ''p'' is odd (and thus a counterexample to the theorem since 2 is the only even prime number), then a, b, c, and ''p'' can be used to produce a specific type of elliptic curve called "semistable". [[https://en.wikipedia.org/wiki/Ribet%27s_theorem It was proven in 1986]] by Ken Ribet, building on prior work by Gerhard Frey and Jean-Pierre Serre, that the ''p'' elliptic curve could never have a modular form. [[https://en.wikipedia.org/wiki/Wiles%27s_proof_of_Fermat%27s_Last_Theorem Wiles was eventually able to prove]] that the Taniyama-Shimura Conjecture was true for semistable elliptic curves, which created a contradiction with the parameters established by Ribet's theorem. This meant that the ''p'' elliptic curve couldn't actually exist, and so an odd prime counterexample to Fermat's last theorem couldn't exist, thus proving it. The "eventually" is because, when Wiles' first proof was being peer-reviewed, a flaw was discovered. Even with help from Richard Taylor, the problem stymied him so badly that he almost gave up on it before having a EurekaMoment regarding two of the mathematical techniques he'd used and how they could shore each other up, which led to the fix he needed. The updated proof was published a year later and was found to be correct.
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17	There's often this idea in fiction that Wiles' proof is somehow incomplete or not good enough. 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. Admittedly, the complexity of the proof compared to the simplicity of the statement makes it appear inelegant. Note, however, that [[http://en.wikipedia.org/wiki/Catalan_conjecture mathematics]] [[http://en.wikipedia.org/wiki/Four_color_theorem is]] [[http://en.wikipedia.org/wiki/Jordan_curve_theorem full]] [[http://en.wikipedia.org/wiki/Graph_minor_theorem of]] [[http://en.wikipedia.org/wiki/Classification_of_finite_simple_groups theorems]] whose best-known 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.
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19	Among remaining unsolved problems in math, the [[https://en.wikipedia.org/wiki/Riemann_hypothesis Riemann Hypothesis]] probably comes closest to having a story behind it nearly as good as Fermat's last theorem, though understanding its statement requires rather more background.[[note]]For those with a better understanding of math, the big factor is the zeta function, which is represented as ζ(x) = [[subscript:n = 1]][[superscript:∞]]Σ 1/(n^x). It only works when x > 1, but analytic continuation allows it to also work if x < 0. x isn't just limited to real numbers, but also imaginary and complex numbers. The Riemann Hypothesis focus on when ζ(x) = 0, of where there are two types, the trivial and non-trivial zeros. The trivial zeros are when x is a negative even number, like -2 or -4; the non-trivial zeros are located between 0 and 1 on the real number side. The conjecture of the hypothesis is that all of these non-trivial zeros have their real part as 0.5, regardless of how big or small the imaginary part is.[[/note]] Also worth mentioning is the [[https://en.wikipedia.org/wiki/Collatz_conjecture Collatz Conjecture]],[[note]]In LaymansTerms, take any positive whole number. If it's even, divide it by 2. If it's odd, multiply it by 3 and add 1. The conjecture goes that if you rinse and repeat with whatever you come up with, you will eventually end up at the 4->2->1->4 loop no matter how big your starting number is.[[/note]] which has been so fiendishly difficult to prove that even mathematicians like Paul Erdős said that we don't have the proper tools to do so yet.
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21	Not to be confused with Fermat's [[http://en.wikipedia.org/wiki/Fermat%27s_little_theorem Little Theorem]], which can be proved convincingly on the back of a postcard.
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23	----
24	!! Instances of Fermat's last theorem in fiction:
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26	[[AC:{{Anime}} & {{Manga}}]]
27	* In ''Manga/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 [[InsufferableGenius Ban]] in to the fact that the whole "genius dog" thing isn't a parlor trick... the dog's actually been [[spoiler:infected with the same virus that caused apes to mutate into humans, the so-called "Missing Link Virus."]] It... doesn't make ''sense'' in context, but there is an explanation.
28	* ''Manga/ScienceFellInLoveSoITriedToProveIt'': The second half of episode 7 involves several fairy tales being given a scientific spin courtesy of the cast. Their version of ''Literature/TheTaleOfTheBambooCutter'' involves Kaguya challenging her suitors to solve Fermat's Last Theorem. The first suitor, who isn't good at math, is stumped. The second, with average ability, feels confident that he can solve it with assistance. The third, who is proficient, [[ImpossibleTask realizes that she doesn't intend to marry]]. After the tale's end, Himuro continues to a spiel on the history of the proof of the theorem before Kanade cuts her off.
29	* The theorem is mentioned in an episode of ''Anime/YuGiOhArcV'', in which Yuya is challenged to prove it during a Quiz Duel. Given that he's terrible at math, he declines to answer... But really, given that the quiz only gives you ''five seconds'' to respond, it's doubtful anyone would have been able to prove the theorem in time.
30	* "Prove Fermat's last theorem" occurs as a problem in an OnlySmartPeopleMayPass setup in ''Manga/ZatchBell''. 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.
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32	[[AC:{{Comics}}]]
33	* In the Dutch comic ''[[ComicBook/StormDonLawrence Storm: De Kronieken van Pandarve]]'' [[note]]Storm: The Chronicles of Pandarve[[/note]], the [[GeniusLoci planetary intelligence Pandarve]] tries to solve Fermat's theorem to pass the time. When Storm needs her full attention to deal with an incoming meteor, he reveals that the theorem was solved, and that he knew that all the time but never told her. Pandarve gets quite enraged at this, partially because a mere human proved smarter than her, but mostly because she is now ''bored''. She calms down when Storm tells her about another unsolved problem, [[http://en.wikipedia.org/wiki/Goldbach%27s_conjecture Goldbach's conjecture]].
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35	[[AC:{{Fanfic}}]]
36	* In the ''WesternAnimation/MyLittlePonyFriendshipIsMagic'' fic ''FanFic/AnAcademicVisit'', a pony math professor named Silver Compass occasionally works on "Starswirl's Unsolved Theorem". It happens to be identical to Fermat's Last Theorem. Some human characters show him the proof, making him extremely grateful. Silver Compass notes that the mathematical concepts needed for the proof have not yet been developed in Equestria.
37	* ''Fanfic/RocketshipVoyager'' is ostensibly a 1950s sci-fi short story set in 2020, and has Dr. Zimmerman complaining that his chief mathematician was reassigned just as he was on the verge of solving the theorem, as a MythologyGag on the ''Series/StarTrekTheNextGeneration'' example below.
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39	[[AC:{{Film}}]]
40	* Appears briefly on a blackboard in the 2000 remake of ''Film/{{Bedazzled 2000}}''. Satan (Elizabeth Hurley as a HotTeacher) erases it from the list of homework assignments while commenting, "You'll never use this stuff."
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42	[[AC:{{Literature}}]]
43	* In Arthur Porges' short story "The Devil and Simon Flagg", a mathematician [[ChessWithDeath bets his soul]] that the Devil cannot prove Fermat's last theorem in twenty-four hours. [[spoiler:He wins.]]
44	* When the general public gains access to {{Chronoscope}} technology in ''Literature/TheLightOfOtherDays'', a school student uses it to view Fermat's original notes and thus reconstruct the original proof.
45	* In the Literature/DoctorWhoMissingAdventures novel ''Millennial Rites'', it's mentioned in passing that the CorruptCorporateExecutive villain has an algebraic proof of the Theorem that he's keeping secret.
46	* Creator/ArthurCClarke's ''The Last Theorem'' is about a Sri Lankan mathematician who discovers a new proof of the Theorem that is not only considerably more concise than Wiles' version but doesn't rely on any mathematics that post-date Fermat, and thus might be Fermat's own proof.
47	* In ''Literature/TheMillenniumTrilogy'', Lisbeth spends most of the second book puzzling over the Theorem. At the end of the book, she [[EurekaMoment understands what he meant]], but after the ending of the book, forgets it.
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49	[[AC:LiveActionTV]]
50	* ''Franchise/StarTrek'':
51	** In ''[[Series/StarTrekTheNextGeneration The Next Generation]]'' (TNG), Picard spends some time trying to prove Fermat's last theorem. He says he finds it humbling that an 800-year-old problem, first posed by a lone French mathematician without a computer, still eludes solution. The episode would become [[ScienceMarchesOn out of date five years later when Wiles' proof was released]]. This was simply unfortunate for the writers as at the time the problem had remained unsolved for over 350 years, and there had been no indication that a solution would ever be found, let alone so soon.
52	** In ''[[Series/StarTrekDeepSpaceNine Deep Space Nine]]'', Jadzia says that one of Dax's earlier hosts had the most original approach to Fermat's last theorem "since Wiles over 300 years ago". This is likely an attempt to {{retcon}} the TNG example by indicating that people in the ''Star Trek'' universe are still working on the theorem even though it's been proved -- perhaps always looking for new and better proofs (just as there are hundreds of known proofs of Pythagoras' theorem), even though this would seemingly make Picard's use of it as a lesson in humility completely moot.
53	* In the ''Series/DoctorWho'' episode [[Recap/DoctorWhoS31E1TheEleventhHour "The Eleventh Hour"]], the Doctor uses Fermat's original proof of the theorem (along with an explanation of why electrons have mass and a description of an {{FTL}} drive) to get a team of scientists to take him seriously after hacking into their videoconference. He also admits that the unfinished stuff was his fault as he "slept in".
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55	[[AC:{{Music}}]]
56	* The Irish band BATS have a song about Andrew Wiles and the theorem.
57	* Music/TomLehrer's "That's Mathematics" mentions Wiles' proof of "what Fermat jotted down in that margin, which could've used some enlargin'", though he makes it sound a ''lot'' simpler than it is. That mention is edited out of the version featured on ''The Remains of Tom Lehrer'' box set to avoid [[UnintentionalPeriodPiece dating the song]].
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59	[[AC:{{Theatre}}]]
60	* Shows up in ''Theatre/{{Arcadia}}''; as a joke Septimus assigns the TeenGenius Thomasina to solve it. She eventually comes to the conclusion that Fermat was {{Troll}}ing. Interestingly, ''Arcadia'' was published mere months before Wiles' proof.
61	* The Musical ''Fermat's Last Tango'' is a NoCelebritiesWereHarmed version of a modern mathematician using computers to find the proof, while taunted by Fermat's ghost, returned from the afterlife (a specific one for mathematicians, called the [[IncrediblyLamePun After Math]]). (Was originally to be called ''Proof'', but premiered at the same time as ''{{Theatre/Proof}}''.)
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63	[[AC:VideoGames]]
64	* In ''{{WesternAnimation/Futurama}}'' World's of Tomorrow, during the first Halloween event, Farnsworth asks Bender's ghost to find Fermat's spirit and find out if he was just trolling us.
65	--> '''Farnsworth:''' "Meanwhile, as long as you're dead, be a sport and poke around the afterlife for me. See if you can find Fermat and get him to say whether he was just jerking us around."
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67	[[AC:WebComics]]
68	* ''WebComic/IrregularWebcomic'' posits that Fermat was a [[http://www.irregularwebcomic.net/1807.html time traveler]].
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70	[[AC:WesternAnimation]]
71	* ''WesternAnimation/TheSimpsons'': The creators have on two occasions inserted apparent counterexamples to Fermat's theorem as a background InJoke.[[note]]The book ''The Simpsons And Their Mathematical Secrets'' details more examples of EasterEggs like this. The author has also written a book about Fermat's Last Theorem, subtitled "The Story Of A Riddle That Confounded The World's Greatest Minds For 358 Years".[[/note]] Anyone attempting to verify these equations on a pocket calculator would find that they are apparently true, but that is because pocket calculators are not precise enough for such astronomically large numbers:
72	** In [[Recap/TheSimpsonsS7E6TreehouseOfHorrorVI the Halloween episode which aired in 1995]], a few months after Wiles published his proof, the equation 1782[[superscript:12]] + 1841[[superscript:12]] = 1922[[superscript:12]] can be seen in the Third Dimension. With a little reflection it is clear why this cannot hold: raising any integer ''n'' to any power produces a number of the same parity as ''n'', and summing an even number and an odd number gives an odd number, but the number to the right of the equals sign is clearly even.
73	** [[Recap/TheSimpsonsS10E2TheWizardOfEvergreenTerrace An episode from a few years later]] more prominently features another apparent counterexample: a blackboard Homer is writing on displays the equation 3987[[superscript:12]] + 4365[[superscript:12]] = 4472[[superscript:12]]. Here the fallacy is less transparent, but it is still simple to disprove, without need for a super powerful calculator: 3987 and 4365 are multiples of 3, and so is the result of raising each to the 12th (or any) power, as is their sum. But 4472 is not divisible by 3, so neither is the stated result.

Context UsefulNotes / FermatsLastTheorem