Fermat's Last Theorem
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Theorem
$\forall a, b, c, n \in \Z^*, \; n > 2$, the equation $a^n + b^n = c^n$ has no solutions.
Proof
The proof of this theorem is beyond the current scope of ProofWiki, 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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Historical Note
As Fermat himself put it:
- "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."
Fermat wrote this in the margin of his copy of Diophantus' Arithmetica and claimed to have found a proof of it. Nobody managed to find such a proof, till it was finally proved by Andrew Wiles in 1994. It is seriously doubted that Fermat actually had found a general proof of it, and it is almost impossible that he found Wiles' proof since it uses areas of mathematics that were not yet invented in Fermat's time.
Theorem
- Donald E. Knuth: The Art of Computer Programming: Volume 1: Fundamental Algorithms (1968): Notes on the Exercises: Exercise $4$
- Donald E. Knuth: The Art of Computer Programming: Volume 3: Sorting and Searching (1973): Notes on the Exercises: Exercise $3$
