History UsefulNotes / FermatsLastTheorem

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 in a 4-2-1 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.

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. 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 in a 4-2-1 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.

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.

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 elliptic curves have an associated 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 an elliptic curve. [[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 the specific type of elliptic curve which the equation was tied to. This created a contradiction with the parameters established by Ribet's theorem, meaning that the ''p'' elliptic curve couldn't actually exist. This meant 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.

** 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 the whole point of Picard's use of it as a lesson in humility completely moot.

** 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. Rather embarrassingly, the episode, which was broadcast at a time when the problem had remained unsolved for over 350 years, would become [[ScienceMarchesOn out of date only five years later when Wiles' proof was released]] -- though that's more a testament to Wiles' genius than a lack of foresight on the part of the writers.
** 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) -- though it does require some imagination to reinterpret Picard's statement that "for 800 years people have been trying to solve it".

* ''Fanfic/RocketshipVoyager'' is ostensibly a 1950's 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.

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 = 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.

* ''Fanfic/RocketshipVoyager'' is ostensibly a 1950's 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 Fermat's Last Theorem, as a MythologyGag on the ''Series/StarTrekTheNextGeneration'' example below.

* ''Fanfic/RocketshipVoyager'' is ostensibly a 1950's serial set in 2020, and has Dr. Zimmerman complaining that his chief mathematician was reassigned just as he was on the verge of solving Fermat's Last Theorem, as a MythologyGag on the ''Series/StarTrekTheNextGeneration'' example below.

* ''Fanfic/RocketshipVoyager'' is ostensibly a 1950's serial set in 2020, and has Dr. Zimmerman complaining that one of his mathematicians was reassigned just as they were on the verge of solving Fermat's Last Theorem (as a MythologyGag on the ''Series/StarTrekTheNextGeneration'' example).

* ''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 UsefulNotes/FermatsLastTheorem. 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.

* 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.

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 elliptic curves have an associated modular form. By the mid-20th century, the remaining unproven case for the equation was for all prime values of n, 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 an elliptic curve. [[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 the specific type of elliptic curve which the equation was tied to. This created a contradiction with the parameters established by Ribet's theorem, meaning that the ''p'' elliptic curve couldn't actually exist. This meant 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.

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 elliptic curves have an associated modular form. By the mid-20th century, the remaining unproven case for the equation was for all prime values of n, 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 an elliptic curve. [[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 the specific type of elliptic curve which the equation was tied to. This created a contradiction with the parameters established by Ribet's theorem, meaning that the ''p'' elliptic curve couldn't actually exist. This meant an odd prime counterexample to Fermat's last theorem can'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.