Fermat's last theorem

Fermat's last theorem (sometimes abbreviated as FLT and also called Fermat's great theorem) is one of the most famous theorems in the history of mathematics. It states that:

Did Fermat really have a proof?

This is the note that Fermat wrote in the margin of Arithmetica:

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Cubum autem in duos cubos, aut quadrato-quadratum in duos quadrato-quadratos,et generaliter nullam in infinitum ultra quadratum potestatem in duos eiusdem nominis fas est dividere cuius rei demonstrationem mirabilem sane detexi. Hanc marginis exigitas non caperet.

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(It is impossible to separate a

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cube into two cubes, or a fourth power into twofourth powers, or in general, any power higher than the second into two likepowers. I have discovered a truly marvelous proof of this, which this marginis too narrow to contain.)

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There is considerable doubt over whether Fermat's claim to have "a truly marvellous proof" was correct. The length of Wiles's proof is about 200 pages and is beyond the understanding of most mathematicians today. It is quite possible that there is a proof that is both essentially shorter, and more elementary in its methods; initial proofs of major results are typically not the most direct. Math institutions still receive many papers, some say in the thousands, claiming to have found such a proof and these are often subject to media attention.

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The methods used by Wiles were unknown when Fermat was writing, and most believe it is unlikely that Fermat managed to derive all the necessary mathematics to demonstrate a solution. In the words of Andrew Wiles, "it's impossible; this is a 20th century proof". Alternatives are that there is a simpler proof that all other mathematicians up until this point have missed, or that Fermat was mistaken.

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A plausible faulty proof that might have been accessible to Fermat has been suggested. It is based on the mistaken assumption that unique factorization works in all rings of integral elements of algebraic number fields. This is an acceptable explanation to many experts in number theory, on the grounds that subsequent mathematicians of stature working in the field followed the same path.

Related Topics:
Unique factorization - Rings - Integral - Algebraic number field - Number theory

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The fact that Fermat never published an attempted proof, or even publicly announced that he had one, does suggest that he may have had later thoughts, and simply neglected to cross out his private marginal note. In addition, later in his life, Fermat published a proof for the case

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: a^4 + b^4 = c^4.

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If he really had come up with a proof for the general theorem, it is perhaps less likely that he would have published a proof for a special case, unless this special case could be used to prove the general theorem.

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On the other hand, the academic conventions of his time were not those that applied from the middle of the eighteenth century, and this argument cannot be taken as definitive. Academic publishing was only then just starting to develop, and mathematicians commonly withheld mathematical techniques to maintain their superiority to other mathematicians. Fermat did not publish proofs for the vast majority of his theorems, including those theorems for which mathematical historians believe he actually had a proof.

Related Topics:
Eighteenth century - Academic publishing

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