A theorem first proposed by Fermat in the form of a note scribbled in the margin of his copy of the ancient
Greek text *Arithmetica* by Diophantus. The scribbled note was discovered posthumously, and the original is
now lost. However, a copy was preserved in a book published by Fermat's son. In the note, Fermat claimed to have discovered a
proof that the Diophantine Equation has no Integer solutions for .

The full text of Fermat's statement, written in Latin, reads ``Cubum autem in duos cubos, aut quadratoquadratum in duos quadratoquadratos & generaliter nullam in infinitum ultra quadratum potestatem in duos eiusdem nominis fas est diuidere cuius rei demonstrationem mirabilem sane detexi. Hanc marginis exiguitas non caperet.'' In translation, ``It is impossible for a cube to be the sum of two cubes, a fourth power to be the sum of two fourth powers, or in general for any number that is a power greater than the second to be the sum of two like powers. I have discovered a truly marvelous demonstration of this proposition that this margin is too narrow to contain.''

As a result of Fermat's marginal note, the proposition that the Diophantine Equation

(1) |

Note that the restriction is obviously necessary since there are a number of elementary formulas for generating an
infinite number of Pythagorean Triples satisfying the equation for ,

(2) |

A first attempt to solve the equation can be made by attempting to factor the equation, giving

(3) |

(4) |

(5) |

(6) |

It is sufficient to prove Fermat's Last Theorem by considering Prime Powers only, since the arguments can
otherwise be written

(7) |

(8) |

The so-called ``first case'' of the theorem is for exponents which are Relatively Prime to , , and ( ) and was considered by Wieferich. Sophie Germain proved the first case of Fermat's Last Theorem for any Odd Prime when is also a Prime. Legendre subsequently proved that if is a Prime such that , , , , or is also a Prime, then the first case of Fermat's Last Theorem holds for . This established Fermat's Last Theorem for . In 1849, Kummer proved it for all Regular Primes and Composite Numbers of which they are factors (Vandiver 1929, Ball and Coxeter 1987).

Kummer's attack led to the theory of Ideals, and Vandiver developed Vandiver's
Criteria for deciding if a given Irregular Prime satisfies the theorem. Genocchi (1852) proved that the first
case is true for if is not an Irregular Pair. In 1858, Kummer showed that the first case
is true if either or is an Irregular Pair, which was subsequently extended to include and by Mirimanoff (1905). Wieferich (1909) proved that if the equation is solved in integers
Relatively Prime to an Odd Prime , then

(9) |

(10) |

(11) |

(12) |

(13) |

(14) |

The ``second case'' of Fermat's Last Theorem (for ) proved harder than the first case.

Euler proved the general case of the theorem for , Fermat , Dirichlet and Lagrange . In 1832, Dirichlet established the case . The case was proved by Lamé (1839), using the identity

(15) |

A recent false alarm for a general proof was raised by Y. Miyaoka (Cipra 1988) whose proof, however,
turned out to be flawed. Other attempted proofs among both professional and amateur mathematicians are discussed by vos
Savant (1993), although vos Savant erroneously claims that work on the problem by Wiles (discussed below) is
invalid. By the time 1993 rolled around, the general case of Fermat's Last Theorem had been shown to be true for all
exponents up to (Cipra 1993). However, given that a proof of Fermat's Last Theorem requires truth for *all* exponents, proof for any finite number of exponents does not constitute any significant progress towards a proof of
the general theorem (although the fact that no counterexamples were found for this many cases is highly suggestive).

In 1993, a bombshell was dropped. In that year, the general theorem was partially proven by Andrew Wiles (Cipra 1993, Stewart 1993) by proving the Semistable case of the Taniyama-Shimura Conjecture. Unfortunately, several holes were discovered in the proof shortly thereafter when Wiles' approach via the Taniyama-Shimura Conjecture became hung up on properties of the Selmer Group using a tool called an ``Euler system.'' However, the difficulty was circumvented by Wiles and R. Taylor in late 1994 (Cipra 1994, 1995ab) and published in Taylor and Wiles (1995) and Wiles (1995). Wiles' proof succeeds by (1) replacing Elliptic Curves with Galois representations, (2) reducing the problem to a Class Number Formula, (3) proving that Formula, and (4) tying up loose ends that arise because the formalisms fail in the simplest degenerate cases (Cipra 1995a).

The proof of Fermat's Last Theorem marks the end of a mathematical era. Since virtually all of the tools which were eventually brought to bear on the problem had yet to be invented in the time of Fermat, it is interesting to speculate about whether he actually was in possession of an elementary proof of the theorem. Judging by the temerity with which the problem resisted attack for so long, Fermat's alleged proof seems likely to have been illusionary.

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© 1996-9

1999-05-26