Fermat's last theorem Guide, Meaning , Facts, Information and Description
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:
There are no positive natural numbers a, b, and c such that where n is a natural number greater than 2.
The 17th-century mathematician Pierre de Fermat wrote about this in 1637 in his copy of Claude-Gaspar Bachet's translation of the famous Arithmetica of Diophantus: "I have discovered a truly remarkable proof but this margin is too small to contain it". (Original Latin: "Cuius rei demonstrationem mirabilem sane detexi hanc marginis exiguitas non caperet.") However, no correct proof was found for 357 years.
This statement is significant because all the other theorems proposed by Fermat were settled, either by proofs he supplied, or by rigorous proofs found afterwards. Mathematicians were long baffled, for they were unable either to prove or to disprove it. The theorem was therefore not the last that Fermat conjectured, but the last to be proved. The theorem is generally thought to be the mathematical result that has provoked the largest number of incorrect proofs.
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2 Early history 3 The proof 4 Did Fermat really have a proof? 5 See also 6 External links and references 7 Bibliography |
Fermat's last theorem is a generalization of the Diophantine equation a2 + b2 = c2, which is linked to the Pythagorean theorem. Ancient Greeks and Babylonians knew that this equation has integer solutions, such as (3,4,5) (32 + 42 = 52) or (5,12,13). These solutions are known as Pythagorean triples, and there exist an infinity of them (excluding trivial solutions for which a, b and c have a common divisor). According to Fermat's last theorem, no such solution exists when the exponent 2 is replaced by a larger integer number.
While the theorem itself has no known direct use (e.g., it has not been used to prove any other theorem), it has been shown to be connected to many other topics in mathematics, and is not only an unimportant mathematical curiosity. Moreover, the search for a proof has initiated research about many important mathematical topics.
The theorem needs only to be proven for n=4 and in the case where n is an odd prime number. For various special exponents n, the theorem had been proved over the years, but the general case remained elusive.
Fermat himself proved the case n=4, while Euler proved the theorem for n=3. The case n=5 was proved by Dirichlet and Legendre in 1825, and the case n=7 by Gabriel Lamé in 1839.
In 1983 Gerd Faltings proved the Mordell conjecture, which implies that for any n > 2, there are at most finitely many coprime integers a, b and c with an + bn = cn.
Using sophisticated tools from algebraic geometry (in particular elliptic curves and modular forms), Galois theory and Hecke algebras, the English mathematician Andrew Wiles, from Princeton University, with help from his former student Richard Taylor, devised a proof of Fermat's last theorem that was published in 1995 in the journal Annals of Mathematics.
In 1986, Ken Ribet had proved Gerhard Frey's epsilon conjecture that every counterexample an + bn = cn to Fermat's last theorem would yield an elliptic curve defined as:
which would provide a counterexample to the Taniyama-Shimura conjecture.
This latter conjecture proposes a deep connection between elliptic curves and modular forms.
Wiles and Taylor were able to establish a special case of the Taniyama-Shimura conjecture sufficient to exclude such counterexamples arising from Fermat's last theorem.
The story of the proof is almost as remarkable as the mystery of the theorem itself. Wiles spent seven years working out nearly all the details by himself and with utter secrecy. When he announced his proof in June 1993, he amazed his audience with the number of ideas and constructions used in his proof. Unfortunately, upon closer inspection a serious error was discovered: it seemed to lead to the breakdown of this original proof. Wiles and Taylor then spent about a year trying to revive the proof. In September 1994, they were able to resurrect the proof with some different, discarded techniques that Wiles had used in his earlier attempts.
There is considerable doubt over whether Fermat's "truly remarkable proof" was correct. The length of Wiles's proof is about 200 pages and is beyond the understanding of most mathematicians today. Moreover, the methods used by Wiles were unknown when Fermat was writing, and it seems inconceivable that Fermat managed to derive all the necessary mathematics to demonstrate the same solution (in the words of Andrew Wiles, "it's impossible; this is a 20th century proof"). The alternatives are that there is a simpler proof that all other mathematicians up until this point have missed, or that Fermat was mistaken. In fact, 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. The fact that Fermat never published an attempted proof, or even publicly announced that he had one, suggests that he may have found his own error and simply neglected to cross out his marginal note. In addition, later in his life, Fermat published a proof for the case a4 + b4 = c4. If he really had come up with a proof for the general theorem, it is unlikely that he would have published a proof for a special case.
This is an Article on Fermat's last theorem. Page Contains Information, Facts Details or Explanation Guide About Fermat's last theorem Mathematical context
Early history
The proof
Did Fermat really have a proof?
See also
External links and references
Bibliography