# Fermat's Last Theorem

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Pierre de Fermat

Fermat's Last Theorem is a very famous idea in mathematics. It says that:

If ${\displaystyle n}$ is a whole number larger than 2, then the equation ${\displaystyle x^{n}+y^{n}=z^{n}}$ has no solutions when x, y and z are natural numbers.

This means that there are no examples where ${\displaystyle x}$, ${\displaystyle y}$ and ${\displaystyle z}$ are natural numbers, i.e. whole numbers larger than zero, and where ${\displaystyle n}$ is a whole number bigger than 2. Pierre de Fermat wrote about it in 1637 inside his copy of a book called Arithmetica. He said "I have a proof of this theorem, but there is not enough space in this margin". However, no correct proof was found for 357 years. It was finally proven in 1995. Mathematicians everywhere think that Fermat, in fact, did not have a good proof of this theorem.

## Overview

Fermat's Last Theorem is a more general form of the Pythagorean theorem,[1] which is an equation that says:

${\displaystyle a^{2}+b^{2}=c^{2}}$

When ${\displaystyle a}$, ${\displaystyle b}$ and ${\displaystyle c}$ are whole numbers this is called a "Pythagorean triple". For example, ${\displaystyle 3^{2}+4^{2}=9+16=25}$, and since ${\displaystyle {\sqrt[{2}]{25}}=5}$ we can say ${\displaystyle 3^{2}+4^{2}=5^{2}}$ is a Pythagorean triple. Fermat's Last Theorem rewrites this as

${\displaystyle x^{n}+y^{n}=z^{n}}$

and claims that, if you make the ${\displaystyle n}$ a larger whole number than 2, then ${\displaystyle a}$, ${\displaystyle b}$ and ${\displaystyle c}$ cannot all be natural numbers. For example, ${\displaystyle 3^{3}+4^{3}=27+64=91}$ and ${\displaystyle {\sqrt[{3}]{91}}=4.49794144528}$, and so ${\displaystyle 3^{3}+4^{3}=4.49794144528^{3}}$ is an example that confirms this.

## Proof

British mathematician Andrew Wiles

The proof was made for some values of ${\displaystyle n}$, such as ${\displaystyle n=3}$, ${\displaystyle n=4}$, ${\displaystyle n=5}$ and ${\displaystyle n=7}$, which was managed by many mathematicians including Fermat, Euler, Sophie Germain. However, since there are an infinite number of Pythagorean triples,[2] as numbers count upwards forever, this made Fermat's Last Theorem hard to prove or disprove; the full proof must show that the equation has no solution for all values of ${\displaystyle n}$ (when ${\displaystyle n}$ is a whole number bigger than 2) but it is not possible to simply check every combination of numbers if they continue forever.

An English mathematician named Andrew Wiles found a solution in 1995, 358 years after Fermat wrote about it.[3][4][5] Richard Taylor helped him find the solution[source?]. The proof took eight years of research. He proved the theorem by first proving the modularity theorem, which was then called the Taniyama–Shimura conjecture. Using Ribet's Theorem, he was able to give a proof for Fermat's Last Theorem. He received the Wolfskehl Prize from Göttingen Academy in June 1997: it amounted to about \$50,000 U.S. dollars.[6]

After a few years of debate, people agreed that Andrew Wiles had solved the problem. Andrew Wiles used a lot of modern mathematics and even created new maths when he made his solution. This mathematics was unknown when Fermat wrote his famous note, so Fermat could not have used it. This leads one to believe that Fermat did not in fact have a complete solution of the problem.

## References

1. Stark, pp. 151–155.
2. Stillwell J (2003). Elements of Number Theory. New York: Springer-Verlag. pp. 110–112. ISBN 0-387-95587-9.
3. Wiles, Andrew (1995). "Modular elliptic curves and Fermat's Last Theorem" (PDF). Annals of Mathematics. Annals of Mathematics. 141 (3): 443–551. doi:10.2307/2118559. JSTOR 2118559. OCLC 37032255.
4. Taylor R, Wiles A (1995). "Ring theoretic properties of certain Hecke algebras". Annals of Mathematics. Annals of Mathematics. 141 (3): 553–572. doi:10.2307/2118560. JSTOR 2118560. OCLC 37032255. Archived from the original on 2001-11-27. Retrieved 2011-10-18.
5. Neil Pieprzak. "Fermat's last theorem and Andrew Wiles". Plus Magazine. Retrieved 2012-04-30.
6. Singh S 1998. Fermat's Enigma, p284. New York: Anchor Books. ISBN 978-0-385-49362-8