In number theory Fermat's Last Theorem (sometimes
called Fermat's conjecture, especially in
older texts) states that no three positive
integers a, b, and c satisfy the equation
an + bn = cn for any integer value of n greater
than 2. The cases n = 1 and n = 2 have been
known to have infinitely many solutions since
antiquity.This theorem was first conjectured
by Pierre de Fermat in 1637 in the margin
of a copy of Arithmetica where he claimed
he had a proof that was too large to fit in
the margin. The first successful proof was
released in 1994 by Andrew Wiles, and formally
published in 1995, after 358 years of effort
by mathematicians. The proof was described
as a 'stunning advance' in the citation for
his Abel Prize award in 2016. The proof of
Fermat's Last Theorem also proved much of
the modularity theorem and opened up entire
new approaches to numerous other problems
and mathematically powerful modularity lifting
techniques.
The unsolved problem stimulated the development
of algebraic number theory in the 19th century
and the proof of the modularity theorem in
the 20th century. It is among the most notable
theorems in the history of mathematics and
prior to its proof, it was in the Guinness
Book of World Records as the "most difficult
mathematical problem", one of the reasons
being that it has the largest number of unsuccessful
proofs.
== Overview ==
=== 
Pythagorean origins ===
The Pythagorean equation, x2 + y2 = z2, has
an infinite number of positive integer solutions
for x, y, and z; these solutions are known
as Pythagorean triples. Around 1637, Fermat
wrote in the margin of a book that the more
general equation an + bn = cn had no solutions
in positive integers, if n is an integer greater
than 2. Although he claimed to have a general
proof of his conjecture, Fermat left no details
of his proof, and no proof by him has ever
been found. His claim was discovered some
30 years later, after his death. This claim,
which came to be known as Fermat's Last Theorem,
stood unsolved in mathematics for the following
three and a half centuries.
The claim eventually became one of the most
notable unsolved problems of mathematics.
Attempts to prove it prompted substantial
development in number theory, and over time
Fermat's Last Theorem gained prominence as
an unsolved problem in mathematics.
=== Subsequent developments and solution ===
With the special case n = 4 proved by Fermat
himself, it suffices to prove the theorem
for exponents n that are prime numbers. Over
the next two centuries (1637–1839), the
conjecture was proved for only the primes
3, 5, and 7, although Sophie Germain innovated
and proved an approach that was relevant to
an entire class of primes. In the mid-19th
century, Ernst Kummer extended this and proved
the theorem for all regular primes, leaving
irregular primes to be analyzed individually.
Building on Kummer's work and using sophisticated
computer studies, other mathematicians were
able to extend the proof to cover all prime
exponents up to four million, but a proof
for all exponents was inaccessible (meaning
that mathematicians generally considered a
proof impossible, exceedingly difficult, or
unachievable with current knowledge).Entirely
separately, around 1955, Japanese mathematicians
Goro Shimura and Yutaka Taniyama suspected
a link might exist between elliptic curves
and modular forms, two completely different
areas of mathematics. Known at the time as
the Taniyama–Shimura–Weil conjecture,
and (eventually) as the modularity theorem,
it stood on its own, with no apparent connection
to Fermat's Last Theorem. It was widely seen
as significant and important in its own right,
but was (like Fermat's theorem) widely considered
completely inaccessible to proof.In 1984,
Gerhard Frey noticed an apparent link between
these two previously unrelated and unsolved
problems. An outline suggesting this could
be proved was given by Frey. The full proof
that the two problems were closely linked
was accomplished in 1986 by Ken Ribet, building
on a partial proof by Jean-Pierre Serre, who
proved all but one part known as the "epsilon
conjecture" (see: Ribet's Theorem and Frey
curve). In plain English, these papers by
Frey, Serre and Ribet showed that if the Modularity
Theorem could be proven for at least the semi-stable
class of elliptic curves, a proof of Fermat's
Last Theorem would also follow automatically.
The connection is described below: any solution
that could contradict Fermat's Last Theorem
could also be used to contradict the Modularity
Theorem. So if the modularity theorem were
found to be true, then by definition no solution
contradicting Fermat's Last Theorem could
exist, which would therefore have to be true
as well.
Although both problems were daunting problems
widely considered to be "completely inaccessible"
to proof at the time, this was the first suggestion
of a route by which Fermat's Last Theorem
could be extended and proved for all numbers,
not just some numbers. Also important for
researchers choosing a research topic was
the fact that unlike Fermat's Last Theorem
the Modularity Theorem was a major active
research area for which a proof was widely
desired and not just a historical oddity,
so time spent working on it could be justified
professionally. However, general opinion was
that this simply showed the impracticality
of proving the Taniyama–Shimura conjecture.
Mathematician John Coates' quoted reaction
was a common one:
"I myself was very sceptical that the beautiful
link between Fermat’s Last Theorem and the
Taniyama–Shimura conjecture would actually
lead to anything, because I must confess I
did not think that the Taniyama–Shimura
conjecture was accessible to proof. Beautiful
though this problem was, it seemed impossible
to actually prove. I must confess I thought
I probably wouldn’t see it proved in my
lifetime." On hearing that Ribet had proven
Frey's link to be correct, English mathematician
Andrew Wiles, who had a childhood fascination
with Fermat's Last Theorem and had a background
of working with elliptic curves and related
fields, decided to try to prove the Taniyama–Shimura
conjecture as a way to prove Fermat's Last
Theorem. In 1993, after six years working
secretly on the problem, Wiles succeeded in
proving enough of the conjecture to prove
Fermat's Last Theorem. Wiles's paper was massive
in size and scope. A flaw was discovered in
one part of his original paper during peer
review and required a further year and collaboration
with a past student, Richard Taylor, to resolve.
As a result, the final proof in 1995 was accompanied
by a smaller joint paper showing that the
fixed steps were valid. Wiles's achievement
was reported widely in the popular press,
and was popularized in books and television
programs. The remaining parts of the Taniyama–Shimura–Weil
conjecture, now proven and known as the Modularity
theorem, were subsequently proved by other
mathematicians, who built on Wiles's work
between 1996 and 2001. For his proof, Wiles
was honoured and received numerous awards,
including the 2016 Abel Prize.
=== Equivalent statements of the theorem ===
There are several alternative ways to state
Fermat's Last Theorem that are mathematically
equivalent to the original statement of the
problem.
In order to state them, we use mathematical
notation: let N be the set of natural numbers
1, 2, 3, ..., let Z be the set of integers
0, ±1, ±2, ..., and let Q be the set of
rational numbers a/b where a and b are in
Z with b≠0. In what follows we will call
a solution to xn + yn = zn where one or more
of x, y, or z is zero a trivial solution.
A solution where all three are non-zero will
be called a non-trivial solution.
For comparison's sake we start with the original
formulation.
Original statement. With n, x, y, z ∈ N
(meaning n, x, y, z are all positive whole
numbers) and n > 2 the equation xn + yn = zn
has no solutions.Most popular treatments of
the subject state it this way. In contrast,
almost all math textbooks state it over Z:
Equivalent statement 1: xn + yn = zn, where
integer n ≥ 3, has no non-trivial solutions
x, y, z ∈ Z.The equivalence is clear if
n is even. If n is odd and all three of x,
y, z are negative then we can replace x, y,
z with −x, −y, −z to obtain a solution
in N. If two of them are negative, it must
be x and z or y and z. If x, z are negative
and y is positive, then we can rearrange to
get (−z)n + yn = (−x)n resulting in a
solution in N; the other case is dealt with
analogously. Now if just one is negative,
it must be x or y. If x is negative, and y
and z are positive, then it can be rearranged
to get (−x)n + zn = yn again resulting in
a solution in N; if y is negative, the result
follows symmetrically. Thus in all cases a
nontrivial solution in Z would also mean a
solution exists in N, the original formulation
of the problem.
Equivalent statement 2: xn + yn = zn, where
integer n ≥ 3, has no non-trivial solutions
x, y, z ∈ Q.This is because the exponent
of x, y and z are equal (to n), so if there
is a solution in Q then it can be multiplied
through by an appropriate common denominator
to get a solution in Z, and hence in N.
Equivalent statement 3: xn + yn = 1, where
integer n ≥ 3, has no non-trivial solutions
x, y ∈ Q.A non-trivial solution a, b, c
∈ Z to xn + yn = zn yields the non-trivial
solution a/c, b/c ∈ Q for vn + wn = 1. Conversely,
a solution a/b, c/d ∈ Q to vn + wn = 1 yields
the non-trivial solution ad, cb, bd for xn
+ yn = zn.
This last formulation is particularly fruitful,
because it reduces the problem from a problem
about surfaces in three dimensions to a problem
about curves in two dimensions. Furthermore,
it allows working over the field Q, rather
than over the ring Z; fields exhibit more
structure than rings, which allows for deeper
analysis of their elements.
Equivalent statement 4 – connection to elliptic
curves: If a, b, c is a non-trivial solution
to xp + yp = zp , p odd prime, then y2 = x(x
− ap)(x + bp) (Frey curve) will be an elliptic
curve.Examining this elliptic curve with Ribet's
theorem shows that it does not have a modular
form. However, the proof by Andrew Wiles proves
that any equation of the form y2 = x(x − an)(x
+ bn) does have a modular form. Any non-trivial
solution to xp + yp = zp (with p an odd prime)
would therefore create a contradiction, which
in turn proves that no non-trivial solutions
exist.In other words, any solution that could
contradict Fermat's Last Theorem could also
be used to contradict the Modularity Theorem.
So if the modularity theorem were found to
be true, then it would follow that no contradiction
to Fermat's Last Theorem could exist either.
As described above, the discovery of this
equivalent statement was crucial to the eventual
solution of Fermat's Last Theorem, as it provided
a means by which it could be 'attacked' for
all numbers at once.
== Mathematical history ==
=== 
Pythagoras and Diophantus ===
==== Pythagorean triples ====
In ancient times it was known that a triangle
whose sides were in the ratio 3:4:5 would
have a right angle as one of its angles. This
was used in construction and later in early
geometry. It was also known to be just one
example of a general rule that any triangle
where the length of two sides, each squared
and then added together (32 + 42 = 9 + 16
= 25), equals the square of the length of
the third side (52 = 25), would also be a
right angle triangle.
This is now known as the Pythagorean theorem,
and a triple of numbers that meets this condition
is called a Pythagorean triple – both are
named after the ancient Greek Pythagoras.
Examples include (3, 4, 5) and (5, 12, 13).
There are infinitely many such triples, and
methods for generating such triples have been
studied in many cultures, beginning with the
Babylonians and later ancient Greek, Chinese,
and Indian mathematicians. Mathematically,
the definition of a Pythagorean triple is
a set of three integers (a, b, c) that satisfy
the equation
a
2
+
b
2
=
c
2
.
{\displaystyle a^{2}+b^{2}=c^{2}.}
==== Diophantine equations ====
Fermat's equation, xn + yn = zn with positive
integer solutions, is an example of a Diophantine
equation, named for the 3rd-century Alexandrian
mathematician, Diophantus, who studied them
and developed methods for the solution of
some kinds of Diophantine equations. A typical
Diophantine problem is to find two integers
x and y such that their sum, and the sum of
their squares, equal two given numbers A and
B, respectively:
A
=
x
+
y
{\displaystyle A=x+y}
B
=
x
2
+
y
2
.
{\displaystyle B=x^{2}+y^{2}.}
Diophantus's major work is the Arithmetica,
of which only a portion has survived. Fermat's
conjecture of his Last Theorem was inspired
while reading a new edition of the Arithmetica,
that was translated into Latin and published
in 1621 by Claude Bachet.Diophantine equations
have been studied for thousands of years.
For example, the solutions to the quadratic
Diophantine equation x2 + y2 = z2 are given
by the Pythagorean triples, originally solved
by the Babylonians (c. 1800 BC). Solutions
to linear Diophantine equations, such as 26x
+ 65y = 13, may be found using the Euclidean
algorithm (c. 5th century BC).
Many Diophantine equations have a form similar
to the equation of Fermat's Last Theorem from
the point of view of algebra, in that they
have no cross terms mixing two letters, without
sharing its particular properties. For example,
it is known that there are infinitely many
positive integers x, y, and z such that xn
+ yn = zm where n and m are relatively prime
natural numbers.
=== Fermat's conjecture ===
Problem II.8 of the Arithmetica asks how a
given square number is split into two other
squares; in other words, for a given rational
number k, find rational numbers u and v such
that k2 = u2 + v2. Diophantus shows how to
solve this sum-of-squares problem for k = 4
(the solutions being u = 16/5 and v = 12/5).Around
1637, Fermat wrote his Last Theorem in the
margin of his copy of the Arithmetica next
to Diophantus's sum-of-squares problem:
After Fermat’s death in 1665, his son Clément-Samuel
Fermat produced a new edition of the book
(1670) augmented with his father’s comments.
Although not actually a theorem at the time
(meaning a mathematical statement for which
proof exists), the margin note became known
over time as Fermat’s Last Theorem, as it
was the last of Fermat’s asserted theorems
to remain unproved.It is not known whether
Fermat had actually found a valid proof for
all exponents n, but it appears unlikely.
Only one related proof by him has survived,
namely for the case n = 4, as described in
the section Proofs for specific exponents.
While Fermat posed the cases of n = 4 and
of n = 3 as challenges to his mathematical
correspondents, such as Marin Mersenne, Blaise
Pascal, and John Wallis, he never posed the
general case. Moreover, in the last thirty
years of his life, Fermat never again wrote
of his "truly marvelous proof" of the general
case, and never published it. Van der Poorten
suggests that while the absence of a proof
is insignificant, the lack of challenges means
Fermat realised he did not have a proof; he
quotes Weil as saying Fermat must have briefly
deluded himself with an irretrievable idea.
The techniques Fermat might have used in such
a "marvelous proof" are unknown.
Taylor and Wiles's proof relies on 20th-century
techniques. Fermat's proof would have had
to be elementary by comparison, given the
mathematical knowledge of his time.
While Harvey Friedman's grand conjecture implies
that any provable theorem (including Fermat's
last theorem) can be proved using only 'elementary
function arithmetic', such a proof need be
‘elementary’ only in a technical sense
and could involve millions of steps, and thus
be far too long to have been Fermat’s proof.
=== Proofs for specific exponents ===
==== 
Exponent = 4 ====
Only one relevant proof by Fermat has survived,
in which he uses the technique of infinite
descent to show that the area of a right triangle
with integer sides can never equal the square
of an integer. His proof is equivalent to
demonstrating that the equation
x
4
+
y
4
=
z
2
{\displaystyle x^{4}+y^{4}=z^{2}}
has no primitive solutions in integers (no
pairwise coprime solutions). In turn, this
proves Fermat's Last Theorem for the case
n = 4, since the equation a4 + b4 = c4 can
be written as a4 + b4 = (c2)2.
Alternative proofs of the case n = 4 were
developed later by Frénicle de Bessy (1676),
Leonhard Euler (1738), Kausler (1802), Peter
Barlow (1811), Adrien-Marie Legendre (1830),
Schopis (1825), Terquem (1846), Joseph Bertrand
(1851), Victor Lebesgue (1853, 1859, 1862),
Theophile Pepin (1883), Tafelmacher (1893),
David Hilbert (1897), Bendz (1901), Gambioli
(1901), Leopold Kronecker (1901), Bang (1905),
Sommer (1907), Bottari (1908), Karel Rychlík
(1910), Nutzhorn (1912), Robert Carmichael
(1913), Hancock (1931), and Vrǎnceanu (1966).For
other proofs for n=4 by infinite descent,
see Infinite descent: Non-solvability of r2
+ s4 = t4, Grant and Perella (1999), Barbara
(2007), and Dolan (2011).
==== Other exponents ====
After Fermat proved the special case n = 4,
the general proof for all n required only
that the theorem be established for all odd
prime exponents. In other words, it was necessary
to prove only that the equation an + bn = cn
has no integer solutions (a, b, c) when n
is an odd prime number. This follows because
a solution (a, b, c) for a given n is equivalent
to a solution for all the factors of n. For
illustration, let n be factored into d and
e, n = de. The general equation
an + bn = cnimplies that (ad, bd, cd) is a
solution for the exponent e
(ad)e + (bd)e = (cd)e.Thus, to prove that
Fermat's equation has no solutions for n > 2,
it would suffice to prove that it has no solutions
for at least one prime factor of every n.
Each integer n > 2 is divisible by 4 or by
an odd prime number (or both). Therefore,
Fermat's Last Theorem could be proved for
all n if it could be proved for n = 4 and
for all odd primes p.
In the two centuries following its conjecture
(1637–1839), Fermat's Last Theorem was proved
for three odd prime exponents p = 3, 5 and
7. The case p = 3 was first stated by Abu-Mahmud
Khojandi (10th century), but his attempted
proof of the theorem was incorrect. In 1770,
Leonhard Euler gave a proof of p = 3, but
his proof by infinite descent contained a
major gap. However, since Euler himself had
proved the lemma necessary to complete the
proof in other work, he is generally credited
with the first proof. Independent proofs were
published by Kausler (1802), Legendre (1823,
1830), Calzolari (1855), Gabriel Lamé (1865),
Peter Guthrie Tait (1872), Günther (1878),
Gambioli (1901), Krey (1909), Rychlík (1910),
Stockhaus (1910), Carmichael (1915), Johannes
van der Corput (1915), Axel Thue (1917), and
Duarte (1944). The case p = 5 was proved independently
by Legendre and Peter Gustav Lejeune Dirichlet
around 1825. Alternative proofs were developed
by Carl Friedrich Gauss (1875, posthumous),
Lebesgue (1843), Lamé (1847), Gambioli (1901),
Werebrusow (1905), Rychlík (1910), van der
Corput (1915), and Guy Terjanian (1987). The
case p = 7 was proved by Lamé in 1839. His
rather complicated proof was simplified in
1840 by Lebesgue, and still simpler proofs
were published by Angelo Genocchi in 1864,
1874 and 1876. Alternative proofs were developed
by Théophile Pépin (1876) and Edmond Maillet
(1897).Fermat's Last Theorem was also proved
for the exponents n = 6, 10, and 14. Proofs
for n = 6 were published by Kausler, Thue,
Tafelmacher, Lind, Kapferer, Swift, and Breusch.
Similarly, Dirichlet and Terjanian each proved
the case n = 14, while Kapferer and Breusch
each proved the case n = 10. Strictly speaking,
these proofs are unnecessary, since these
cases follow from the proofs for n = 3, 5,
and 7, respectively. Nevertheless, the reasoning
of these even-exponent proofs differs from
their odd-exponent counterparts. Dirichlet's
proof for n = 14 was published in 1832, before
Lamé's 1839 proof for n = 7.All proofs for
specific exponents used Fermat's technique
of infinite descent, either in its original
form, or in the form of descent on elliptic
curves or abelian varieties. The details and
auxiliary arguments, however, were often ad
hoc and tied to the individual exponent under
consideration. Since they became ever more
complicated as p increased, it seemed unlikely
that the general case of Fermat's Last Theorem
could be proved by building upon the proofs
for individual exponents. Although some general
results on Fermat's Last Theorem were published
in the early 19th century by Niels Henrik
Abel and Peter Barlow, the first significant
work on the general theorem was done by Sophie
Germain.
==== Sophie Germain ====
In the early 19th century, Sophie Germain
developed several novel approaches to prove
Fermat's Last Theorem for all exponents. First,
she defined a set of auxiliary primes θ constructed
from the prime exponent p by the equation
θ = 2hp + 1, where h is any integer not divisible
by three. She showed that, if no integers
raised to the pth power were adjacent modulo
θ (the non-consecutivity condition), then
θ must divide the product xyz. Her goal was
to use mathematical induction to prove that,
for any given p, infinitely many auxiliary
primes θ satisfied the non-consecutivity
condition and thus divided xyz; since the
product xyz can have at most a finite number
of prime factors, such a proof would have
established Fermat's Last Theorem. Although
she developed many techniques for establishing
the non-consecutivity condition, she did not
succeed in her strategic goal. She also worked
to set lower limits on the size of solutions
to Fermat's equation for a given exponent
p, a modified version of which was published
by Adrien-Marie Legendre. As a byproduct of
this latter work, she proved Sophie Germain's
theorem, which verified the first case of
Fermat's Last Theorem (namely, the case in
which p does not divide xyz) for every odd
prime exponent less than 270, and for all
primes p such that at least one of 2p+1, 4p+1,
8p+1, 10p+1, 14p+1 and 16p+1 is prime (specially,
the primes p such that 2p+1 is prime are called
Sophie Germain primes). Germain tried unsuccessfully
to prove the first case of Fermat's Last Theorem
for all even exponents, specifically for n
= 2p, which was proved by Guy Terjanian in
1977. In 1985, Leonard Adleman, Roger Heath-Brown
and Étienne Fouvry proved that the first
case of Fermat's Last Theorem holds for infinitely
many odd primes p.
==== 
Ernst Kummer and the theory of ideals ====
In 1847, Gabriel Lamé outlined a proof of
Fermat's Last Theorem based on factoring the
equation xp + yp = zp in complex numbers,
specifically the cyclotomic field based on
the roots of the number 1. His proof failed,
however, because it assumed incorrectly that
such complex numbers can be factored uniquely
into primes, similar to integers. This gap
was pointed out immediately by Joseph Liouville,
who later read a paper that demonstrated this
failure of unique factorisation, written by
Ernst Kummer.
Kummer set himself the task of determining
whether the cyclotomic field could be generalized
to include new prime numbers such that unique
factorisation was restored. He succeeded in
that task by developing the ideal numbers.
(Note: It is often stated that Kummer was
led to his "ideal complex numbers" by his
interest in Fermat's Last Theorem; there is
even a story often told that Kummer, like
Lamé, believed he had proven Fermat's Last
Theorem until Lejeune Dirichlet told him his
argument relied on unique factorization; but
the story was first told by Kurt Hensel in
1910 and the evidence indicates it likely
derives from a confusion by one of Hensel's
sources. Harold Edwards says the belief that
Kummer was mainly interested in Fermat's Last
Theorem "is surely mistaken". See the history
of ideal numbers.)
Using the general approach outlined by Lamé,
Kummer proved both cases of Fermat's Last
Theorem for all regular prime numbers. However,
he could not prove the theorem for the exceptional
primes (irregular primes) that conjecturally
occur approximately 39% of the time; the only
irregular primes below 270 are 37, 59, 67,
101, 103, 131, 149, 157, 233, 257 and 263.
==== Mordell conjecture ====
In the 1920s, Louis Mordell posed a conjecture
that implied that Fermat's equation has at
most a finite number of nontrivial primitive
integer solutions, if the exponent n is greater
than two. This conjecture was proved in 1983
by Gerd Faltings, and is now known as Faltings's
theorem.
==== Computational studies ====
In the latter half of the 20th century, computational
methods were used to extend Kummer's approach
to the irregular primes. In 1954, Harry Vandiver
used a SWAC computer to prove Fermat's Last
Theorem for all primes up to 2521. By 1978,
Samuel Wagstaff had extended this to all primes
less than 125,000. By 1993, Fermat's Last
Theorem had been proved for all primes less
than four million.However despite these efforts
and their results, no proof existed of Fermat's
Last Theorem. Proofs of individual exponents
by their nature could never prove the general
case: even if all exponents were verified
up to an extremely large number X, a higher
exponent beyond X might still exist for which
the claim was not true. (This had been the
case with some other past conjectures, and
it could not be ruled out in this conjecture.)
=== Connection with elliptic curves ===
The strategy that ultimately led to a successful
proof of Fermat's Last Theorem arose from
the "astounding" Taniyama–Shimura–Weil
conjecture, proposed around 1955—which many
mathematicians believed would be near to impossible
to prove, and was linked in the 1980s by Gerhard
Frey, Jean-Pierre Serre and Ken Ribet to Fermat's
equation. By accomplishing a partial proof
of this conjecture in 1994, Andrew Wiles ultimately
succeeded in proving Fermat's Last Theorem,
as well as leading the way to a full proof
by others of what is now the modularity theorem.
==== Taniyama–Shimura–Weil conjecture
====
Around 1955, Japanese mathematicians Goro
Shimura and Yutaka Taniyama observed a possible
link between two apparently completely distinct
branches of mathematics, elliptic curves and
modular forms. The resulting modularity theorem
(at the time known as the Taniyama–Shimura
conjecture) states that every elliptic curve
is modular, meaning that it can be associated
with a unique modular form.
The link was initially dismissed as unlikely
or highly speculative, but was taken more
seriously when number theorist André Weil
found evidence supporting it, though not proving
it; as a result the conjecture was often known
as the Taniyama–Shimura–Weil conjecture.
It became a part of the Langlands programme,
a list of important conjectures needing proof
or disproof.Even after gaining serious attention,
the conjecture was seen by contemporary mathematicians
as extraordinarily difficult or perhaps inaccessible
to proof. For example, Wiles's doctoral supervisor
John Coates states that it seemed "impossible
to actually prove", and Ken Ribet considered
himself "one of the vast majority of people
who believed [it] was completely inaccessible",
adding that "Andrew Wiles was probably one
of the few people on earth who had the audacity
to dream that you can actually go and prove
[it]."
==== Ribet's theorem for Frey curves ====
In 1984, Gerhard Frey noted a link between
Fermat's equation and the modularity theorem,
then still a conjecture. If Fermat's equation
had any solution (a, b, c) for exponent p
> 2, then it could be shown that the semi-stable
elliptic curve (now known as a Frey-Hellegouarch)
y2 = x (x − ap)(x + bp)would have such unusual
properties that it was unlikely to be modular.
This would conflict with the modularity theorem,
which asserted that all elliptic curves are
modular. As such, Frey observed that a proof
of the Taniyama–Shimura–Weil conjecture
might also simultaneously prove Fermat's Last
Theorem. By contraposition, a disproof or
refutation of Fermat's Last Theorem would
disprove the Taniyama–Shimura–Weil conjecture.
In plain English, Frey had shown that, if
this intuition about his equation was correct,
then any set of 4 numbers (a, b, c, n) capable
of disproving Fermat's Last Theorem, could
also be used to disprove the Taniyama–Shimura–Weil
conjecture. Therefore if the latter were true,
the former could not be disproven, and would
also have to be true.
Following this strategy, a proof of Fermat's
Last Theorem required two steps. First, it
was necessary to prove the modularity theorem
– or at least to prove it for the types
of elliptical curves that included Frey's
equation (known as semistable elliptic curves).
This was widely believed inaccessible to proof
by contemporary mathematicians. Second, it
was necessary to show that Frey's intuition
was correct: that if an elliptic curve were
constructed in this way, using a set of numbers
that were a solution of Fermat's equation,
the resulting elliptic curve could not be
modular. Frey showed that this was plausible
but did not go as far as giving a full proof.
The missing piece (the so-called "epsilon
conjecture", now known as Ribet's theorem)
was identified by Jean-Pierre Serre who also
gave an almost-complete proof and the link
suggested by Frey was finally proved in 1986
by Ken Ribet.Following Frey, Serre and Ribet's
work, this was where matters stood:
Fermat's Last Theorem needed to be proven
for all exponents n that were prime numbers.
The modularity theorem – if proved for semi-stable
elliptic curves – would mean that all semistable
elliptic curves must be modular.
Ribet's theorem showed that any solution to
Fermat's equation for a prime number could
be used to create a semistable elliptic curve
that could not be modular;
The only way that both of these statements
could be true, was if no solutions existed
to Fermat's equation (because then no such
curve could be created), which was what Fermat's
Last Theorem said. As Ribet's Theorem was
already proved, this meant that a proof of
the Modularity Theorem would automatically
prove Fermat's Last theorem was true as well.
==== Wiles's general proof ====
Ribet's proof of the epsilon conjecture in
1986 accomplished the first of the two goals
proposed by Frey. Upon hearing of Ribet's
success, Andrew Wiles, an English mathematician
with a childhood fascination with Fermat's
Last Theorem, and a prior study area of elliptical
equations, decided to commit himself to accomplishing
the second half: proving a special case of
the modularity theorem (then known as the
Taniyama–Shimura conjecture) for semistable
elliptic curves.Wiles worked on that task
for six years in near-total secrecy, covering
up his efforts by releasing prior work in
small segments as separate papers and confiding
only in his wife. His initial study suggested
proof by induction, and he based his initial
work and first significant breakthrough on
Galois theory before switching to an attempt
to extend horizontal Iwasawa theory for the
inductive argument around 1990–91 when it
seemed that there was no existing approach
adequate to the problem. However, by the summer
of 1991, Iwasawa theory also seemed to not
be reaching the central issues in the problem.
In response, he approached colleagues to seek
out any hints of cutting edge research and
new techniques, and discovered an Euler system
recently developed by Victor Kolyvagin and
Matthias Flach that seemed "tailor made" for
the inductive part of his proof. Wiles studied
and extended this approach, which worked.
Since his work relied extensively on this
approach, which was new to mathematics and
to Wiles, in January 1993 he asked his Princeton
colleague, Nick Katz, to help him check his
reasoning for subtle errors. Their conclusion
at the time was that the techniques Wiles
used seemed to work correctly.By mid-May 1993,
Wiles felt able to tell his wife he thought
he had solved the proof of Fermat's Last Theorem,
and by June he felt sufficiently confident
to present his results in three lectures delivered
on 21–23 June 1993 at the Isaac Newton Institute
for Mathematical Sciences. Specifically, Wiles
presented his proof of the Taniyama–Shimura
conjecture for semistable elliptic curves;
together with Ribet's proof of the epsilon
conjecture, this implied Fermat's Last Theorem.
However, it became apparent during peer review
that a critical point in the proof was incorrect.
It contained an error in a bound on the order
of a particular group. The error was caught
by several mathematicians refereeing Wiles's
manuscript including Katz (in his role as
reviewer), who alerted Wiles on 23 August
1993.The error would not have rendered his
work worthless – each part of Wiles's work
was highly significant and innovative by itself,
as were the many developments and techniques
he had created in the course of his work,
and only one part was affected. However without
this part proved, there was no actual proof
of Fermat's Last Theorem. Wiles spent almost
a year trying to repair his proof, initially
by himself and then in collaboration with
his former student Richard Taylor, without
success. By the end of 1993, rumours had spread
that under scrutiny, Wiles's proof had failed,
but how seriously was not known. Mathematicians
were beginning to pressure Wiles to disclose
his work whether or not complete, so that
the wider community could explore and use
whatever he had managed to accomplish. But
instead of being fixed, the problem, which
had originally seemed minor, now seemed very
significant, far more serious, and less easy
to resolve.Wiles states that on the morning
of 19 September 1994, he was on the verge
of giving up and was almost resigned to accepting
that he had failed, and to publishing his
work so that others could build on it and
find the error. He adds that he was having
a final look to try and understand the fundamental
reasons why his approach could not be made
to work, when he had a sudden insight that
the specific reason why the Kolyvagin–Flach
approach would not work directly also meant
that his original attempts using Iwasawa theory
could be made to work, if he strengthened
it using his experience gained from the Kolyvagin–Flach
approach. Fixing one approach with tools from
the other approach would resolve the issue
for all the cases that were not already proven
by his refereed paper. He described later
that Iwasawa theory and the Kolyvagin–Flach
approach were each inadequate on their own,
but together they could be made powerful enough
to overcome this final hurdle.
"I was sitting at my desk examining the Kolyvagin–Flach
method. It wasn't that I believed I could
make it work, but I thought that at least
I could explain why it didn’t work. Suddenly
I had this incredible revelation. I realised
that, the Kolyvagin–Flach method wasn't
working, but it was all I needed to make my
original Iwasawa theory work from three years
earlier. So out of the ashes of Kolyvagin–Flach
seemed to rise the true answer to the problem.
It was so indescribably beautiful; it was
so simple and so elegant. I couldn't understand
how I'd missed it and I just stared at it
in disbelief for twenty minutes. Then during
the day I walked around the department, and
I'd keep coming back to my desk looking to
see if it was still there. It was still there.
I couldn't contain myself, I was so excited.
It was the most important moment of my working
life. Nothing I ever do again will mean as
much."
— Andrew Wiles, as quoted by Simon SinghOn
24 October 1994, Wiles submitted two manuscripts,
"Modular elliptic curves and Fermat's Last
Theorem" and "Ring theoretic properties of
certain Hecke algebras", the second of which
was co-authored with Taylor and proved that
certain conditions were met that were needed
to justify the corrected step in the main
paper. The two papers were vetted and published
as the entirety of the May 1995 issue of the
Annals of Mathematics. These papers established
the modularity theorem for semistable elliptic
curves, the last step in proving Fermat's
Last Theorem, 358 years after it was conjectured.
=== Subsequent developments ===
The full Taniyama–Shimura–Weil conjecture
was finally proved by Diamond (1996), Conrad,
Diamond & Taylor (1999), and Breuil et al.
(2001) who, building on Wiles's work, incrementally
chipped away at the remaining cases until
the full result was proved. The now fully
proved conjecture became known as the modularity
theorem.
Several other theorems in number theory similar
to Fermat's Last Theorem also follow from
the same reasoning, using the modularity theorem.
For example: no cube can be written as a sum
of two coprime n-th powers, n ≥ 3. (The
case n = 3 was already known by Euler.)
== Exponents other than positive integers
==
=== 
Reciprocal integers (inverse Fermat equation)
===
The equation
a
1
/
m
+
b
1
/
m
=
c
1
/
m
{\displaystyle a^{1/m}+b^{1/m}=c^{1/m}}
can be considered the "inverse" Fermat equation.
All solutions of this equation were computed
by Lenstra in 1992. In the case in which the
mth roots are required to be real and positive,
all solutions are given by
a
=
r
s
m
{\displaystyle a=rs^{m}}
b
=
r
t
m
{\displaystyle b=rt^{m}}
c
=
r
(
s
+
t
)
m
{\displaystyle c=r(s+t)^{m}}
for positive integers r, s, t with s and t
coprime.
=== Rational exponents ===
For the Diophantine equation
a
n
/
m
+
b
n
/
m
=
c
n
/
m
{\displaystyle a^{n/m}+b^{n/m}=c^{n/m}}
with n not equal to 1, Bennett, Glass, and
Székely proved in 2004 for n > 2, that if
n and m are coprime, then there are integer
solutions if and only if 6 divides m, and
a
1
/
m
{\displaystyle a^{1/m}}
,
b
1
/
m
,
{\displaystyle b^{1/m},}
and
c
1
/
m
{\displaystyle c^{1/m}}
are different complex 6th roots of the same
real number.
=== Negative exponents ===
==== n = –1 ====
All primitive integer solutions (i.e., those
with no prime factor common to all of a, b,
and c) to the optic equation
a
−
1
+
b
−
1
=
c
−
1
{\displaystyle a^{-1}+b^{-1}=c^{-1}}
can be written as
a
=
m
k
+
m
2
,
{\displaystyle a=mk+m^{2},}
b
=
m
k
+
k
2
,
{\displaystyle b=mk+k^{2},}
c
=
m
k
{\displaystyle c=mk}
for positive, coprime integers m, k.
==== n = –2 ====
The case n = –2 also has an infinitude of
solutions, and these have a geometric interpretation
in terms of right triangles with integer sides
and an integer altitude to the hypotenuse.
All primitive solutions 
to
a
−
2
+
b
−
2
=
d
−
2
{\displaystyle a^{-2}+b^{-2}=d^{-2}}
are given by
a
=
(
v
2
−
u
2
)
(
v
2
+
u
2
)
,
{\displaystyle a=(v^{2}-u^{2})(v^{2}+u^{2}),}
b
=
2
u
v
(
v
2
+
u
2
)
,
{\displaystyle b=2uv(v^{2}+u^{2}),}
d
=
2
u
v
(
v
2
−
u
2
)
,
{\displaystyle d=2uv(v^{2}-u^{2}),}
for coprime integers u, v with v > u. The
geometric interpretation is that a and b are
the integer legs of a right triangle and d
is the integer altitude to the hypotenuse.
Then the hypotenuse itself is the integer
c
=
(
v
2
+
u
2
)
2
,
{\displaystyle c=(v^{2}+u^{2})^{2},}
so (a, b, c) is a Pythagorean triple.
==== Integer n < –2 ====
There are no solutions in integers for
a
n
+
b
n
=
c
n
{\displaystyle a^{n}+b^{n}=c^{n}}
for integers n < –2. If there were, the
equation could be multiplied through by
a
|
n
|
b
|
n
|
c
|
n
|
{\displaystyle a^{|n|}b^{|n|}c^{|n|}}
to obtain
(
b
c
)
|
n
|
+
(
a
c
)
|
n
|
=
(
a
b
)
|
n
|
{\displaystyle (bc)^{|n|}+(ac)^{|n|}=(ab)^{|n|}}
, which is impossible by Fermat's Last Theorem.
== Base values other than positive integers
==
Fermat's last theorem can easily be extended
to positive rationals:
(
a
x
)
n
+
(
b
y
)
n
=
(
c
z
)
n
{\displaystyle \left({\frac {a}{x}}\right)^{n}+\left({\frac
{b}{y}}\right)^{n}=\left({\frac {c}{z}}\right)^{n}}
can have no solutions with n > 2, because
any solution could be rearranged as:
(
a
y
z
)
n
+
(
b
x
z
)
n
=
(
c
x
y
)
n
{\displaystyle (ayz)^{n}+(bxz)^{n}=(cxy)^{n}}
,to which Fermat's Last Theorem applies.
== Monetary prizes ==
In 1816, and again in 1850, the French Academy
of Sciences offered a prize for a general
proof of Fermat's Last Theorem. In 1857, the
Academy awarded 3000 francs and a gold medal
to Kummer for his research on ideal numbers,
although he had not submitted an entry for
the prize. Another prize was offered in 1883
by the Academy of Brussels.In 1908, the German
industrialist and amateur mathematician Paul
Wolfskehl bequeathed 100,000 gold marks—a
large sum at the time—to the Göttingen
Academy of Sciences to offer as a prize for
a complete proof of Fermat's Last Theorem.
On 27 June 1908, the Academy published nine
rules for awarding the prize. Among other
things, these rules required that the proof
be published in a peer-reviewed journal; the
prize would not be awarded until two years
after the publication; and that no prize would
be given after 13 September 2007, roughly
a century after the competition was begun.
Wiles collected the Wolfskehl prize money,
then worth $50,000, on 27 June 1997. In March
2016, Wiles was awarded the Norwegian government's
Abel prize worth €600,000 for "his stunning
proof of Fermat’s Last Theorem by way of
the modularity conjecture for semistable elliptic
curves, opening a new era in number theory."Prior
to Wiles's proof, thousands of incorrect proofs
were submitted to the Wolfskehl committee,
amounting to roughly 10 feet (3 meters) of
correspondence. In the first year alone (1907–1908),
621 attempted proofs were submitted, although
by the 1970s, the rate of submission had decreased
to roughly 3–4 attempted proofs per month.
According to F. Schlichting, a Wolfskehl reviewer,
most of the proofs were based on elementary
methods taught in schools, and often submitted
by "people with a technical education but
a failed career". In the words of mathematical
historian Howard Eves, "Fermat's Last Theorem
has the peculiar distinction of being the
mathematical problem for which the greatest
number of incorrect proofs have been published."
== 
In popular culture ==
In The Simpsons episode "The Wizard of Evergreen
Terrace" Homer writes the equation
3987
12
+
4365
12
=
4472
12
{\displaystyle 3987^{12}+4365^{12}=4472^{12}}
on a blackboard, which appears to be a counterexample
to Fermat's Last Theorem. The equation is
incorrect but appears to be correct if it
is tested on a hand held calculator that only
displays 10 significant figures.In "The Royale",
a 1989 episode of the 24th-century-set TV
series Star Trek: The Next Generation, Picard
tells Commander Riker about his attempts to
solve the theorem, "still unsolved" after
800 years. He concludes, "In our arrogance,
we feel we are so advanced. And yet we cannot
unravel a simple knot tied by a part-time
French mathematician working alone without
a computer." (Andrew Wiles's insight leading
to his breakthrough proof happened four months
after the series ended.)
== 
See also ==
Beal's conjecture
Diophantus II.VIII
Euler's sum of powers conjecture
Modularity theorem
Proof of impossibility
Pythagorean triple
Sophie Germain prime
Sums of powers, a list of related conjectures
and theorems
Wall–Sun–Sun prime
== Footnotes
