Fermat's Last Theorem

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Theorem

$\forall a, b, c, n \in \Z_{>0}, \; n > 2$, the equation $a^n + b^n = c^n$ has no solutions.


Proof

The proof of this theorem is beyond the current scope of $\mathsf{Pr} \infty \mathsf{fWiki}$, and indeed, is beyond the understanding of many high level mathematicians.

For the curious reader, the proof can be found here, in a paper published by Andrew Wiles, entitled Modular elliptic curves and Fermat's Last Theorem, in volume 141, issue 3, pages 443 through 551 of the Annals of Mathematics.

It is worth noting that Wiles' proof was indirect in that he proved a special case of the Taniyama-Shimura Conjecture, which then along with the already proved Epsilon Conjecture implied that integral solutions of the theorem were impossible.


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Cubic

The Diophantine equation $a^3 + b^3 = c^3$ has no solutions in strictly positive integers.


Fourth Powers

$\forall a, b, c \in \Z_{>0}$, the equation $a^4 + b^4 = c^4$ has no solutions.


Source of Name

This entry was named for Pierre de Fermat.


Historical Note

Many of Fermat's theorems were stated, mostly without proof, in the margin of his copy of Bachet's translation of Diophantus's Arithmetica.

In $1670$, his son Samuel published an edition of this, complete with Fermat's marginal notes.


Fermat's Note

As Fermat himself put it, sometime around $1637$:

Cubum autem in duos cubos, aut quadratoquadratum in duos quadratoquadratos et generaliter nullam in infinitum ultra quadratum potestatem in duos ejusdem nominis fas est dividere: cujus rei demonstrationem mirabilem sane detexi. Hanc marginis exiguitas non caperet.

Loosely translated from the Latin, that means:

The equation $x^n + y^n = z^n$ has no integral solutions when $n > 2$. I have discovered a perfectly marvellous proof, but this margin is not big enough to hold it.


Nobody managed to find such a proof, until it was finally proved by Andrew Wiles in $1994$.

It is seriously doubted that Fermat actually had found a general proof of it.

It is almost impossible that he found Wiles' proof, since it uses areas of mathematics that were not yet invented in Fermat's time.


Sources

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