Fermat's last theorem is 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
*n* > 2 and

The full text of Fermat's statement, written in Latin, reads "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 exiguitas non caperet" (Nagell 1951, p. 252). 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) |

where *x*, *y*, *z*, and *n* are integers, has no nonzero solutions for
*n* > 2 has come to be known as Fermat's Last Theorem. It was
called a "theorem"
on the strength of Fermat's statement, despite the fact that no other
mathematician was able to prove it for hundreds of years.

Note that the restriction *n* > 2 is obviously necessary since
there are a number of elementary formulas for generating an infinite
number of Pythagorean
triples satisfying the equation for *n* = 2,

(2) |

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

(3) |

Since the product is an exact power,

(4) |

Solving for *y* and *z* gives

(5) |

which give

(6) |

However, since solutions to these equations in rational numbers are no easier to find than solutions to the original equation, this approach unfortunately does not provide any additional insight.

If an odd prime *p* divides *n*, then the reduction

(7) |

can be made, so redefining the arguments gives

(8) |

If no odd prime divides *n*, then *n* is a power of 2, so
and, in this case, equations (7) and (8) work with
4 in place of *p*. Since the case *n* = 4 was proved by Fermat
to have no solutions, it is sufficient to prove Fermat's last theorem by
considering odd prime powers only.

Similarly, is sufficient to prove Fermat's last theorem by considering
only relatively
prime *x*, *y*, and *z*, since each term in equation
(1) can then be divided by by

The so-called "first case" of the theorem is for exponents which are relatively
prime to *x*, *y*, and *z* ( ) and was considered by Wieferich. Sophie Germain
proved the first case of Fermat's Last Theorem for any odd prime *p*
when is also a prime. Legendre
subsequently proved that if *p* is a prime such that
*p*. This established
Fermat's Last Theorem for *p* < 100. In 1849, Kummer
proved it for all regular primes
and composite
numbers of which they are factors (Vandiver 1929, Ball and Coxeter
1987).

The "second case" of Fermat's last theorem is "*p* divides exactly
one of *x*, *y*, *z*. Note that is ruled out by *x*, *y*, *z* being
relatively prime, and that if *p* divides two of *x*, *y*,
*z*, then it also divides the third, by equation (8).

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 *p* 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). Vandiver (1920ab) pointed out
gaps and errors in Kummer's memoir which, in his view, invalidate Kummer's
proof of Fermat's Last Theorem for the irregular primes 37, 59, and 67,
although he claims Mirimanoff's proof of FLT for exponent 37 is still
valid.

Wieferich (1909) proved that if the equation is solved in integers relatively
prime to an odd
prime *p*, then

(9) |

(Ball and Coxeter 1987). Such numbers are called Wieferich primes. Mirimanoff (1909) subsequently showed that

(10) |

must also hold for solutions relatively
prime to an odd
prime *p*, which excludes the first two Wieferich
primes 1093 and 3511. Vandiver (1914) showed

(11) |

and Frobenius extended this to

(12) |

It has also been shown that if *p* were a prime of the form

(13) |

which raised the smallest possible *p* in the "first case" to by 1941 (Rosser 1941). Granville and
Monagan (1988) showed if there exists a prime *p*
satisfying Fermat's Last Theorem, then

(14) |

for *q* = 5, 7, 11, ..., 71. This establishes that the first case
is true for all prime exponents
up to (Vardi 1991).

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

Euler
proved the general case of the theorem for *n* = 3, Fermat *n* =
4, Dirichlet
and Lagrange
*n* = 5. In 1832, Dirichlet
established the case *n* = 14. The *n* = 7 case was proved by
Lamé (1839; Wells 1986, p. 70), using the identity

(15) |

Although some errors were present in this proof, these were subsequently fixed by Lebesgue (1840). Much additional progress was made over the next 150 years, but no completely general result had been obtained. Buoyed by false confidence after his proof that pi is transcendental, the mathematician Lindemann proceeded to publish several proofs of Fermat's Last Theorem, all of them invalid (Bell 1937, pp. 464-465). A prize of German marks, known as the Wolfskehl Prize, was also offered for the first valid proof (Ball and Coxeter 1987, p. 72; Barner 1997; Hoffman 1998, pp. 193-194 and 199).

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 tenacity with which
the problem resisted attack for so long, Fermat's alleged proof seems
likely to have been illusionary. This conclusion is further supported by
the fact that Fermat searched for proofs for the cases *n* = 4 and
*n* = 5, which would have been superfluous had he actually been in
possession of a general proof.

abc Conjecture, Beal's Conjecture, Bogomolov-Miyaoka-Yau Inequality, Euler System, Fermat-Catalan Conjecture, Generalized Fermat Equation, Mordell Conjecture, Pythagorean Triple, Ribet's Theorem, Selmer Group, Sophie Germain Prime, Szpiro's Conjecture, Taniyama-Shimura Conjecture, Vojta's Conjecture, Waring Formula

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