Fermat's last theorem

[[image:Pierre de Fermat.jpg|thumb|right|165px|Pierre de Fermat]]
{{cquote|I have a truly marvelous demonstration of this proposition which this margin is too narrow to contain.|||Pierre de Fermat}}
{{cquote|I have an elegant refutation of your attempted proof, but unfortunately this page is not large enough to contain it.|||Anon}}
{{Mathematics}}
'''Fermat's last theorem'''<ref>Singh, Simon (1997). Fermat's Last Theorem. Fourth Estate. ISBN 1-85702-669-1. </ref> states that in the equation, <math>A^n+B^n=C^n</math>, if <math>A,B,C</math> are positive integers, <math>n</math> cannot be an integer greater than 2. Fermat wrote in the margin of a book that he had a proof of the theorem, but it would not fit in the margin. He died in 1665 before ever revealing it, and his note was discovered posthumously by his son.<ref>http://www-groups.dcs.st-and.ac.uk/~history/HistTopics/Fermat's_last_theorem.html</ref><ref>http://www.simonsingh.net/Fermat_Corner.html</ref> Fermat must have realized that his imagined proof was flawed, because he later{{when}} wrote a proof of the special case n = 4 (in which he introduced the method of proof by infinite descent).

In 1753, Euler claimed to have a proof where n=3, but this was flawed; in the early 19th century Sophie Germain, Dirichlet, and Adrien-Marie Legendre proved special cases. Gabriel Lamé claimed to have a proof in 1847 but it was shown to be incorrect. Despite a lot of interest it proved challenging even to the best minds: one study showed that in the years from 1908 and 1912, 1000 false proofs were published.<ref>[https://mathshistory.st-andrews.ac.uk/HistTopics/Fermat's_last_theorem/ Fermat's Last Theorem], J J O'Connor and E F Robertson, Maths History, St Andrews Uni, 1996</ref>

In 1994, Andrew Wiles presented a proof of the theorem, using 20th century [[mathematics|mathematical]] principles (modular forms, Galois theory, algebraic geometry). A mistake was discovered, which Wiles repaired after about a year of work with his former student Richard Taylor, with the final proof published in 1995. The theorem built on an earlier theorem of Ken Ribet following a program of Serre. The full modularity theorem was proved in 1998 based on the techniques of Wiles.

==Crank solutions==
Because it seems simple and yet was unproven for so long, and because Fermat claimed to have a simple solution, it has long been a favorite of amateur mathematicians and [[crank]]s.<ref>[https://www.newscientist.com/article/mg15420825-200-review-riddle-of-the-curves/ Review: Riddle of the curves], Richard Pinch, 1999</ref> There's a possibly apocryphal story about a visitor at the University of Chicago:<blockquote>In the Math department office there were a lot of filing cabinets. One was labelled “Form Letters”. He opened it and saw a lot of folders. One folder was labelled “Fermat’s Last Theorem.” He opened it and saw a lot of identical form letters. He looked at the letter. It began: “Dear Sir, We have received your proof of Fermat’s Last Theorem…” There were other folders, like “Goldbach’s Conjecture”, “Angle Trisecting”, etc. Each of the form letters offered to have a graduate student examine the claimed proof for $50, up to finding the first error or non sequitur. Even for $50, graduate students were reluctant to do this work because the “proofs” were generally indecipherable.<ref>[https://www.quora.com/What-kind-of-math-crank-proofs-have-you-gotten-from-amateur-mathematicians What kind of math "crank" proofs have you gotten from amateur mathematicians?], David Grossman, Quora, 2019</ref></blockquote>Even after Wiles's success, the ill-conceived attempts at a simpler proof continued.<ref>[http://www.crank.net/fermat.html Fermat], Crank.net</ref> For instance, in 2005 a Russian professor claimed to have a 3-line proof.<ref name=freeman/> There has even been hoax reports that Wiles's solution had been disproved, such as one in Manila in 2005, possibly intended to give succor to wannabe solvers.<ref name=freeman>[http://fermatslasttheorem.blogspot.com/2005/08/another-false-proof-russian-professor.html Another False Proof: Russian Professor claims to have solved Fermat's Last Theorem], Larry Freeman, Fermat's Last Theorem blog, Aug 24, 2005</ref>

==See also==
*[[Fun:Fermat Primes]]
==References==
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