Fermat's Enigma: The Epic Quest to Solve the World's Greatest Mathematical Problem
2006; Volume: 40; Issue: 2 Linguagem: Inglês
ISSN
0730-8639
Autores Tópico(s)Advanced Mathematical Theories
ResumoFERMAT'S ENIGMA: THE EPIC QUEST TO SOLVE THE WORLD'S GREATEST MATHEMATICAL PROBLEM by Simon Singh, Anchor Books, New York, 307 pp., ISBN: 0385493622 (1997). Simon Singh has skillfully integrated the story of the quest for the solution to the world's greatest mathematical problem with a fascinating history of mathematics, including information on some of the great mathematicians and their most beautiful work. The story is compelling and the story-telling equally thrilling. Pierre Fermât, a lawyer and councilor to the Parliament of Toulouse as well as a brilliant mathematician, initiated the quest to solve the world's greatest mathematical problem in 1630 when he scribbled the following note in his personal copy of Diophantus' Arithmetica beside the Pythagorean proposition about expressing a perfect square as the sum of two other perfect squares: But it is impossible to divide a cube into two cubes, or a fourth power into two fourth powers, or generally any power beyond the square. I have found a remarkable demonstration. This margin is too small to contain it. Fermat's Last Theorem x^sup n^ + y^sup n^ = z^sup n^ is a more general form of the Pythagorean triple x^sup 2^ + y^sup 2^ = z^sup 2^ where x, y, and z are whole numbers. Infinitely many Pythagorean triples can be found, but Fermat asserted that x^sup n^ + y^sup n^ = z^sup n^ had no solutions for n > 2. The approaches to proving Fermat's Theorem have included proofs for selected values of n, then for different classes of n, and general solutions for all n. About 367 years after Fermat wrote his note in the margin, Andrew Wiles, an Englishman who first encountered Fermat's Last Theorem at age 10, was officially recognized for proving Fermat's Last Theorem on June 27, 1997 with the award of the Wolfskehl Prize. More that a century had passed between the time Fermat made his note and the presentation of solutions of Fermat's theorem for the cases for n = 4 and n = 3 by Leonhard Euler. Euler used the method of infinite descent to demonstrate that the equation had no solution for n = 4, and the concept of imaginary numbers to demonstrate that the equation had no solution for n = 3. Proof for the case for n = 4 also proved the case for 8, 12, 16, 20, ... and the case for n = 3 proved the case for 6, 9, 12, 15,... . This would suggest that to prove Fermat's Last Theorem for all values of n, one only had to prove it for each of the prime values of n. Unfortunately, this does not allow for proof for the general case because of the infinitude of the primes. Sophie Germain focused on a particular type of prime such as 2p + 1, when p is prime, and demonstrated the case of n = 5 and 11, a result confirmed by Dirichlet and Legendre. Gabriel Lame, taking a cue from Germain, proved the case for n = 7. Following Germain's breakthrough, the French Academy of Sciences offered prizes, medals, and money for the solution of Fermat's Last Theorem. …
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