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Fermat's Last Theorem

For 300 years mathematicians have tried without success to prove

a theorem Pierre de Fermat asserted he could prove: No nth power
can be the sum oftwo other nth powers, where n is greater than

by Harold M. Edwards

he 1 7th-century French mathema

tician Pierre de Fermat was the
father of modern number theory.
the branch of mathematics that deals
with the properties of whole numbers.
Like many other scholars of his time he
studied the classics of antiquity. In num
ber theory he drew inspiration from
Diophantus. the Greek mathematician
whose Arithmetica had been rediscov
ered by Europeans in the mid-16th cen
tury. Fermat made extensive marginal
notes in his copy of the book. and af
ter his death in 1665 his son published
a new edition of Arithmetica including
Fermat's notes. One of the notes has
become one of the most famous state
ments in the history of mathematics.
Next to a problem on finding squares
that are sums of two other squares (for
example. 25 equals 9 plus 1 6). Fermat
wrote (to translate from the Latin): "On
the other hand. 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."
The proposition has come to be called
Fermat's last theorem. and although the
keenest mathematicians of the past
three centuries have tried to prove it. no
such demonstration has been found.
Fermat's last theorem remains one of
the great unsolved problems of modern
Did Fermat really have a "marvelous
proof" of the theorem? He was certainly
a prodigious mathematician. helping to
establish the disciplines of analytic ge
ometry (with Descartes). infinitesimal
calculus (with Leipniz and Newton) and
probability theory (with Pascal). He was
not a mathematician by profession but
a jurist at Toulouse in the south of
France. His extensive participation in
the intellectual life of his time was con
ducted entirely through his private cor
respondence with other scholars in the
major European centers of learning.
It is charming to think of a provincial

17th-century jurist outsmarting the best

mathematical minds of three centuries
with his theorem. but the facts suggest
that he did not have a proof. Except for
the famous note in the margin of Dio
phantus' book there is no mention of a
proof of the theorem in any of Fermat's
writings that have survived. He did state
elsewhere that he could show it was im
possible to find solutions to the equa
tions x3 + y3 = z3 and X4 + y4 = Z4. If
he discovered a valid proof of the full
theorem (that xn + yn = zn is impossi
ble for positive whole numbers x. y. z
and n. where n is greater than 2). it is
surprising that he did not mention it too.
It seems most likely that at the time he
wrote the note Fermat had an idea for a
proof he later realized was faulty. His
notes were almost certainly not intend
ed for publication. and there may have
been no occasion for him to go back and
delete or amend the note.
It is of course more exciting to believe
Fermat did have a rigorous proof of the
theorem. and it is just possible that he
did. In either case the theorem has
played an important role in the devel
opment of number theory. Some of
the greatest creations of mathematical
thought have been prompted by the
study of the theorem. and the techniques
that have been developed in the effort to
prove it have contributed to the solution
of many other problems.
Finally. the history of Fermat's last
theorem provides an excellent illustra
tion of the real nature of mathemati
cal inquiry. Mathematicians are often

asked: "How is it possible to do research

in mathematics?" I dou!;>t that physicists
or astronomers or biofogists are ever
asked such a question. but to many peo
ple mathematics seems to be such a cut
and-dried subject that work in it should
amount to nothing more than an orderly
recording of the facts. Of course. noth
ing could be further from the truth. In
mathematics. as in any other field. unan
swered questions are everywhere. and
the difficulty mathematicians face is
finding questions they can answer. not
ones they cannot. It is difficult. however.
to illustrate the point for a nonmathe
matician. because the formulation of in
teresting mathematical questions usual
ly calls for specialized terminology and
background. Fermat's last theorem is a
rare exception to this rule.

Proof by Infin ite Descent

Part of the fascination of Fermat's
last theorem is due to the fact that the
theorem is unusually simple to state and
understand: It is impossible to find posi
tive whole numbers x: y, z and n for
which n is greater than 2 and the equa
tion xn + yn = zn holds. Nonmathema
ticians usually approach the theorem in
what seems to be a very reasonable way:
by "trying it out." Consider the case
where n is equal to 3. that is. the case
that consists in proving that x3 + y3 = z3
has no solution. The cubes of the first 1 0
positive integers are 1. 8. 27, 6 4 , 1 25,
2 1 6 , 343, 5 1 2, 729 and 1 .000. It is not
difficult to see that none of these num-

PIERRE DE FERMAT is called the father of number theory, the branch of mathematics that
deals with the properties of the whole numbers. Born in


near Toulouse, Fermat spent all

his life in the south of France, far from the great European centers of learning. He worked as a
jurist, not as a professional mathematician, and none of his mathematical works was published
until after his death. His extensive participation in the mathematics of his time was conducted
entirely through his private correspondence with other scholars. Fermat formulated many
challenging and perceptive theorems that were not proved until long after his death. By


only one of these theorems remained to be proved. It has come to be called Fermat's last theo
rem: There are no solutions in whole numbers to the equation
er than


xn + yn

Z n, where 1/ is great

Fermat's last theorem is one of the most famous unsolved problems of modern math

ematics. This painting of Fermat is in the collection of the Academie des Sciences, Inscriptions
et Belles Lettres de Toulouse; it is reproduced here with the kind permission of Robert Gillis.




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about 1500 D.C. is one of the oldest-known documents dealing with
number theory. The tablet lists (in slightly disguised form) several sets
of Pythagorean triples, or distinct, positive whole numbers x, y and
z, for which x2 + y2 = Z2, for example 4,961, 6,480 and 8,161. Fer
mat formulated his last theorem while he was considering a prob
lem concerning Pythagorean triples (see illustration on next page).
Any triple provides an immediate proof that the theorem is not true
when the exponent n is equal to 2. Pythagorean triples are named

bers can be expressed as the sum of two

other cubes. For example. if 512 were
equal to the sum of two other cubes. the
cubes would be smaller than 512 and
would precede it on the list of cubes.
The cubes 216 and 343 are good candi
dates. but their sum is 559. which is too
large. The next smaller sum. 216 plus
216. equals 432 and not 512. Hence 512
is not equal to the sum of any two cubes.
It is easy to verify by this procedure
that any particular cube is not the sum
of two other cubes. The required cal
culations could be carried out at high
speed on a computer. and it could be
shown very quickly that no cube with
fewer than. say. 10 digits is the sum of
two other cubes. There are. however. an
infinite number of cubes to be tested. so
that with this procedure even the largest
computer could never resolve the ques
tion of whether any cube can be the sum
of two other cubes.

for their relation to the Pythagorean theorem, which states that in a

squares of the other two sides. It is not easy to find triples by trial
and error, particularly those that involve large numbers, and so tbe
Babylonians probably had a method for finding them. It seems most
likely that their interest stemmed from the geometric applications of
the sets of numbers and that therefore they knew of the Pythagore
an theorem 1,000 years before Pythagoras. The tablet is part of the
Plimpton Collection in the Butler Library at Columbia University.

Once nonmathematicians have seen

that the imposs.ibility of x3 + y3 = z3
cannot be proved simply by "trying it."
they may swing to the opposite extreme
and ask how such a statement can ever
be proved. The answer is that it can be
proved by the method of reasoning to a
contradiction: One assumes that there is
a solution to the equation. and one de
duces from this assumption a statement
known to be false. Arriving at a false
statement. or contradiction. shows that
the original assumption was false and
therefore that there can be no solution.
More specifically. in the case of state
ments. such as Fermat's last theorem.
that are concerned with positive whole
numbers a proof by contradiction often
takes the form of a proof by infinite de
scent. Fermat maintained he had invent
ed this method. which he said was the
basis of all his proofs in the field of num
ber theory. In a proof by infinite descent

one shows that if a solution in positive

whole numbers to the equation in ques
tion is given. it can be used to produce a
smaller 'solution. also in positive whole
numbers. The same argument then
shows that the smaller solution can be
used to produce a still smaller one. and
the process can be repeated indefinitely.
Since the solutions are all positive whole
numbers. however. it is obviously im
possible to find increasingly small solu
tions ad infinitum. Hence there can be
no solution at all.
Early Findings
In all Fermat's writings on number
theory that have been preserved there is
only one proof. found in another of the
marginal notes in Diophantus' Arithme
tica. It concerns Pythagorean triangles.
or right triangles with sides of integer
length. (They derive their name from


their relation to the Pythagorean theo

rem, which states that the sides x and y
and the hypotenuse z of a right triangle
are related by the equation x2 + y2 =
z2.) Fermat proved that the area of
such a triangle cannot be a square: If x,
y and z are positive integers and x2 +
)'2 = z2, then ( l!2)xy is not the square

of an integer. It is easy to show that ei

ther x or y must be even and therefore
that ( l!2)xy must be a whole number.
Fermat used the method of infinite de
scent in the proof. Specifically, he gave
an explicit method by which, given posi
tive whole numbers x, y, z and u that
satisfy the equations X2 + y2 = z2 and




COMA1E:A(r.A1{jJS c. g. 73,dCHETJ. 17. C

&obforuationihu.rV. FER MAT Senatorh<Tolofani.


Acceaic DoCl:rinre Analyticre inuentum nouum,colleaum

ex varijs eiufdem D. de F E RMAT Epiftolis.

Excudebat B ERN A R D V S B 0 S C ) e Regione Coltegij Societatis leut.
- --------- -


D.,.J5 ..

FERMAT'S LAST THEOREM was first set down in the margin of Arithmetica, a work on
number theory by the ancient Greek mathematician Diophantus. Fermat had studied the book
closely, making many marginal notes in his copy of it, a translation into Latin by C. G. Bachet.
After Fermat's death in 1665 his son published a new edition of Bachet's translation of Arith

metica that included Fermat's marginal notes in an appendix. The title page of the book is
shown in this illustration. Translated from the Latin, it announces Six Books of Arithmetica
and One Book on Polygonal Numbers by Diophantus of Alexandria, "with notes by the distin
guished gentleman C. G. Bachet and observations by Doctor P. de Fermat, Senator of Tou
louse," and "a new discovery of analytical doctrine collected from the various letters of the
aforementioned Doctor de Fermat." Fermat's historic note referred to a problem concerning
Pythagorean triples. He wrote: "On the other hand, it is impossible . . for any number that is a
power greater than the second to be the sum of two like powers. I have discovered a truly mar
velous demonstration of this proposition that this margin is too narrow to contain." To this
day it is not known whether Fermat really had a rigorous proof of the theorem. In any case
no other mathematician has been able to prove the theorem or to find a counterexample to it.


(1!2)xy = u2, it is possible to derive an

other set of positive whole numbers X.
Y. Z and V such that X2 + Y2 = Z2 and
( l!2)XY= V2. and such that the trian
gle with sides X. Yand Z is smaller than
the triangle with sides x, y and z in the
sense that the hypotenuse Z is less than
the hypotenuse z. Fermat's method for
deriving X. Y. Z and V is subtle and
would call for a long explanation. The
fact that such a method exists, however,
shows that it is impossible to find any
positive whole-number solutions to the
equations x2 + y2 = z2 and (1!2)xy =
u2, because one solution x, y, z and u
would lead to a smaller solution X. Y. Z
and V. which in turn would lead to a
still smaller solution X', Y', Z' and V',
and so on. That would imply the ex
istence of infinitely decreasing sequen
ces of positive whole numbers z> Z>
Z'> .. . , which is impossible. (The
sign"> stands for "greater than.")
It is interesting and probably not coin
cidental that the fact that the area of a
Pythagorean triangle cannot be a square
immediately implies that X4 + y4 = Z4
has no solution, or that Fermat's last
theorem is true for the case where n
equals 4. A simple but ingenious trick
suffices to connect the two propositions:
assume that X4 + y4 = Z4 for some posi
tive integers x. yand z, and set a equal to
y4, b equal to 2X2Z2, c equal to Z4 + X4
and d equal to y2XZ. Then the simple
algebraic identity (r + S)2= r2 + 2rs + S2
implies that a2 + b2 equals (Z4 - X4)2 +
4X4Z4, or Z8 - 2X4Z4 + x8 + 4X4Z4, or
(Z4 + X4)2, which is equal to c2. Further
more, ( l!2)ab is equal to (1!2)y42x2z2
or (y2xz)2, which is equal to d2. Thus
a2 + b2 = c2 and (1!2)ab = d2, which
Fermat proved was impossible. Hence
the original assumption, that X4 + y4=
Z4 has a solution, must be incorrect,
and Fermat's last theorem is established
for the case n equals 4. In essence. then,
Fermat himself proved his last theorem
for the case of fourth powers.
This proof also establishes that Fer
mat's last theorem is true whenever n is
a multiple of 4, because if n is equal to
4k for some positive integer k. then
xn + yn = zn implies that (Xk)4 + (yk)4=
(zk)4, which is impossible because a
fourth power cannot be the sum of two
other fourth powers. In exactly the same
way, if the theorem can be proved for
any given exponent m, then it is true
for all multiples of m. Therefore. since
every whole number n greater than 2
is divisible either by an odd prime (a
prime number other than 2) or by 4. it
suffices to prove Fermat's last the
orem in all cases where the exponent is
an odd prime number.
Fermat maintained he could prove
the theorem in the case n equals 3, but
no published proof of the impossibility
of finding a solution for x3 + y3 = z3
appeared until about 100 years later.
The proof. which was the work of the


Swiss mathematician Leonhard Euler,

did, however, have a serious flaw.
Euler's Proof
Euler proved the case n equals 3 by
infinite descent, setting forth a method
by which it is possible to derive from
any given solution x, y and z for the
equation x3 + y3
z3 a new solution X.
Yand Z such that Z is less than z. The
method is too long to explain in detail
here, but in outline, by making calcula
tions concerning various characteristics
of x, y and z, Euler reduced the problem
of deriving a smaller solution to one of
proving the following proposition: If p
and q are relatively prime integers (if
they have no common factor, or divisor,
greater than 1) and p2 + 3q2 is a cube,
then there must be integers a and b such
that p equals a3- 9ab2 and q equals
3a2b - 3b2. The proposition is quite

correct and can be proved by apply

ing simple variations of methods found
elsewhere in Euler's published work. In
this instance, however, Euler chose to
employ a novel type of argument that
involved introducing numbers of the
form a + b v=1, where a and b are in
To understand why Euler found the
numbers a + b v=1 useful, expand the
expression (a + b v=1 )3. It is equal
to a3 + 3a2bv=1- 9ab2- 3b3v=1,
or (a3- 9ab2) + (3a2b- 3b3) v=1, or
p + q v=1, where p and q are defined
as they are in the conclusion of the
l?roposition. In other words, the conclu
sion of the proposition states that
(a + b v=1)3 p + q v=1 . Now. the
proposition assumes that p2 + 3q2 is
a cube. If p2 + 3q2 is rewritten as
(P + qY-3)(P- qv=1), then the prop
osition can be restated: If (P + qY-3)
(P- qv=1) is a cube, then p + qv=1

must be a cube, that is, must have the

form (a + b Y-3 )3 for some integers
a and b. Thus the introduction of the
numbers a + b v=1 makes it possi
ble to write the proposition much more
simply and naturally.
The numbers a + b v=1 form a
number system much like that of the
integers. In either case adding, subtract
ing or multiplying any two numbers in
the system gives another number in the
system, whereas dividing two numbers
usually does not. For example, 4 does
not divide 5 in the system of integers
(that is, no integer can be multiplied by 4
to give 5), and given two numbers of the
form a + b v=1 it is not usually possi
ble to find a third number of that form
that is their quotient. The similarities of
the two systems led Euler to take an in
novative and incorrect step in his proof.
He applied an established property of
the integers to the numbers a + b v=1.
The property of the integers that Eu
ler needed for his proof is derived from
the unique factorization of integers into
primes. A positive integer can be writ
ten as a product of prime factors in just
one way. For example, 124 is equal to
2 X 2 X 3 1, and no prime number other
than 2 or 3 1 divides 124 evenly. Unique
factorization implies the following prop
erty of the integers: A product of rel
atively prime positive integers can be
a cube only if each of the integers is a
cube. For example, assume that c and d
are relatively prime and that cdis equal
to 1,000, or 103. Expressed as a product
of prime factors, 1,000 equals 23 X 53.
The factorizations of 1,000 are obtained
by splitting the factors into two subsets,
for example (2 X 2 X 5)(2 X 5 X 5),
or 20 X 50. If the splitting must be done
in such a way that the two subsets are rel
atively prime, then all the 2's must go in
one subset and all the 5's must go in the
other. Hence the only possible values of
the factors c and dare 23 X 53, 23, 53
and 1, all of which are cubes. Similarly.
the product of two relatively prime pos
itive integers can be an nth power only
if each of the integers is an nth power.
Euler assumed that this property of
the integers was also a property of the
numbers a + b v=1 , and he proved
the proposition by the following line of
argument. The proposition states that
if p and q are relatively prime and
(P + q Y-3 )(P - qY-3) is a cube,
then p + q vi 3 is a cube. Euler first
showed that if p and q are relatively
prime, then p + q Y-3 and p
q v=1
are also relatively prime. By extension
of the property of the integers, the prod
uct of relatively prime numbers of the
form a + b v=1 can be a cube only
if the numbers themselves are cubes.
Hence the assumption that (P + qv=1)
(P - q Y-3 ) is a cube implies that p +
qv=1 is a cube. and the proposition
is proved.
The argument certainly gets Euler to
his desired conclusion. The problem is

LEONHARD EULER proved Fermat's last theorem for the case of the first odd prime expo
nent. n equals 3. This painting, made in about 1745, shows the Swiss mathematician during his
lifetime. Before Euler's proof Fermat's last theorem had been established (by a direct applica
tion of Fermat's own work) in only one case: n equals 4. It is easy to show that if the theorem is
true when n equals some integer r, then it is true when n equals any multiple of r. Since every
integer is divisible either by 4 or by an odd prime, it remained only to prove the theorem in the
cases where n is an odd prime. The first successful effort in this direction was Euler's, but al
though his proof was ingenious, it had a serious flaw. The flaw was easily corrected in the proof
of the case n equals 3, but it turned out to be a central issue in many later attacks on theorem.


that reasoning by analogy to the arith

metic of the integers does not constitute
a valid proof. Arguments by analogy are
extremely suggestive and. as the history
of mathematics demonstrates. can yield
useful ideas. but they cannot be taken as
proof of anything. It is particularly sur
prising that Euler was not more circum
spect in his use of analogy. because
although the numbers a + b v=3 do
have many properties in common with
the integers. there are also many proper
ties in which the two number systems
differ. For example. the integers have
a natural ordering. .. . -2. -1. O. 1.
2. . . . . but the numbers a + b v=1 do
not. There is only one way to be sure the
numbers a + b v=1 share with the in
tegers the property that the product of
two relatively prime numbers is a cube
only if each of the numbers is a cube.
and that is to prove they do.
The Flaw in Euler's Work
Even the best mathematicians. how
ever. succumb to the temptation of
proof by analogy from time to time.
often failing to be sufficiently critical
of an argument because they know the
conclusion it reaches is correct. The
temptation is particularly strong when
the argument has the appealing sim
plicity of Eulers. It is quite possible
that Euler neglected to be more rigor
ous because he knew from other con
siderations that the final conclusion
p + q -v'=3
(a + b Y-3)3 was cor
rect. Long before he published his proof
of the case n equals 3 of Fermat's last
theorem Euler worked on other un
proved assertions of Fermat's concern
ing the representation of numbers in the
form X2 + 3y2. In particular he proved
Fermat's assertion that every prime
number p that is one more than a multi
ple of 3 (p equals 3n + 1) has a unique
representation as a square plus three
times a square (p equals x2 + 3y2); for
example. 7 equals 2 X 3 + 1 and 22 +
3 X 12. The techniques Euler devel
oped in the proof of this theorem are
easily applied to prove the proposition
in the case n equals 3 of Fermat's last
theorem. It is possible Euler realized he
could prove the proposition with the es
tablished techniques and therefore did
not subject his more unusual proof to
sufficiently careful scrutiny.
In the earlier proof Euler was ex
tremely cautious about employing ques
tionable arguments. For example. an in
termediate step in the proof requires a
proof that if a and b are relatively prime
integers. then every odd prime factor of
a2 + 3b2 can be written in the form
c2 + 3d2 In this instance Euler did not
have the dangerous assurance that the
statement to be proved was correct. and
his demonstration is a model of clarity
and rigor. It is curious. however. that his
experience with this proof did not alert
him to the unreliability of his later argu=


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ment. The argument by analogy can be

used to prove that every odd prime fac
tor of a2 + 3b2 is of the form c2 + 3d2 ,
but it can also be used to prove that
every odd prime factor of a2 + 5b2 is of
the form c2 + 5d2, which is false. (Note
that 42 + 5 X 12 equals 21 and that nei
ther of the factors of 21. 3 and 7. can be
written as the sum of a square and five
times a square,) Such considerations
may have prompted the rigor that char
acterizes Euler's earlier proof. He had
evidently forgotten them when he re
turned to the subject many years later.
Euler's lapse in rigor is closely related
to his great strengths. namely his ex
traordinary imagination and inventive
ness, His ability to see new connections
and formulate problems in original and
perceptive ways made his work a great
source of inspiration for generations of
mathematicians. His factoring of the
numbers a + b Y-J into primes by
analogy to the factorization of the inte
gers certainly showed great ingenuity. In
fact. although his application- of the idea
in his proof of the case n equals 3 of
Fermat's last theorem was premature.
later events would prove the idea to be
an inspired one. Indeed. the flaw in Eu
ler's argument-the fact that a property
derived from the unique factorization of
the integers does not necessarily hold
for number systems resembling the inte
gers in other ways-would turn out to be
a central issue in more sophisticated in
vestigations of the theorem.

Two More Cases

Euler's proof needed some patching.
but it essentially established Fermat's
last theorem for the case where n equals
the first odd prime 3. In the 1820's Gus
tav Lejeune Dirichlet and Adrien-Marie
Legendre proved the theorem for the
case where n equals the next prime 5.
Their method of proof was basically an
extension of the one Euler had used in
the proof of the case n equals 3. and
the analogue of the crucial equation
p + qv=3 = (a + bv=3)3 was the
equation p + q V5 = (a + b VS)5. (As
n increases, the equations encountered
in this type of proof become much more
complex and the method fails to work.)
For the case n equals 5 in order to prove
that p + q V5 is a fifth power it must be
assumed not only that p2 - 5q2 is a fifth
power and p and q are relatively prime.
as in the case n equals 3. but also that
they have opposite parity (that one is
odd and the other is even ) and that q is
divisible by 5. Dirichlet's proof of this
fact was based on a completely rigor
ous study of the numbers of the form
x2 - 5y2. The proof, modeled on other
works of Eulers. including his rigorous
study of numbers of the form x2 + 3 y2 .
and on works by Joseph-Louis Lagrange
and Carl Friedrich Gauss, did not rely in
any way on analogy to the factorization
of the integers.


Some 15 years passed before G;;tbriel

Lame proved Fermat's last theorem for
the case of the next prime: 7. The proof
was a great accomplishment. but it did
not bode well for future proofs because
it was long. difficult and closely bound
to the number 7. There was little hope
that the proof could be applied to the
next case. n equals 11. or to any other
cases of the theorem. It seemed. then.
that substantial progress would not be
made in the study of Fermat's last theo
rem until a new approach to the prob
lem was discovered.
Lame himself proposed such an ap
proach in 1847. He tried to prove the
general theorem. introducing a complex
nth root of unity: a complex number a
for which an equals 1. but ak does not
equal 1 for any positive integer k less
than n. The idea was not new. In the
preceding century Lagrange had point
ed out that introducing a into the study
of Fermat's last theorem makes it possi
ble to split xn + yn = zn into n factors.
each one containing x and y to the first
power. (It is usually easier to deal with'
lower powers of variables.)
To obtain the factorization note that
l.a,a2, are the roots. or solu
tions. of the equation Xn - 1. and so
by the fundamental theorem of alge
bra Xn - 1 = (X - 1 )(X -a)(X - a2) . . .
(X-an-I). Now set X equal to - xly
and multiply both sides of the equation
by yn. Since only the cases where n is
odd are under consideration. the result
ingequation is xn + yn = (x + y)(x + ay)
(x + a2y) . . . (x + a n-Iy).

The Cyclotomic Integers

Each of the factors of xn + yn is a
number of the form ao + ala + a2a2 +
... + an_Ian-I,
ao, .... an-I

are integers. Today the numbers of this

type-the numbers in the system made
up of the integers and powers of a
are called the cyclotomic integers. be
cause the nth root of unity a is close
ly related to the problem of dividing
a circle into n equal parts. (The com
plex number a can be interpreted as a
point on a circle of unit radius whose
center is at the origin of the complex
plane; the arc of the circle that lies be
tween 1 and a is lInth of the entire
circle.) Like the numbers a + b v=3 .
the cyclotomic integers form a num
ber system resembling the integers in
that adding. subtracting or multiplying
two cyclotomic integers results in a cy
clotomic integer but dividing usually
does not.
Lame's treatment of the arithmetic
of the cyclotomic integers resembles
Euler's treatment of the numbers
a + b v=3. although it may have been
an independent invention. Given the
factorization of xn + yn into cyclotomic
integers. Lame proposed applying the
"fact" that the product of relatively
prime numbers (here by numbers he

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meant cyclotomic integers ) can be an

nth power only if each of the numbers is
an nth power. From this point Lame saw
a fairly direct route to completing the
proof that xn + yn = zn is impossible.
The problem is that a property derived
from the unique factorization of the in-

tegers is no more automatically applica

ble to the cyclotomic integers than it is
to the numbers a + b -v'"=3 . In fact, al
though the application is valid for the
numbers a + b -v'"=3 (for a reason that
is not obvious ), it is usually not valid
for the cyclotomic integers, that is, it is








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ERNST EDUARD KUMMER made the greatest advance that has ever been made on Fer
mat's last theorem. In 1847 the German number theorist. working on a system of numbers
called the cyclotomic integers, discovered a sufficient condition for a prime number p to be an

exponent for which Fermat's last theorem is true, that is, if a prime p satisfies Kummer's con
dition, then xP + yP = zP is impossible. Today prime numbers that satisfy the condition are said

to be regular. When regular primes were discovered, Fermat's last theorem had been estab
lished only for the cases where n equals 3, 4, Sand 7, but Kummer was able to prove the
theorem true for aU but three of the primes (37, S9 and 67) smaller than 100. More recently suf
ficient conditions that are far more inclusive than Kummer's have been discovered. All primes
within the range of computation, even on the largest computers, have been proved to be ex
ponents for which Fermat's last theorem holds. The theorem has still not been proved, how
ever. In fact, it has not even been shown to be true for an infinite number of prime exponents.


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valid for only a few values of n, where

n is an odd prime and a is an nth root
of unity.
The unfortunate Lame was so carried
away by his own optimism that he an
nounced to a meeting of the French
Academy of Sciences that he had
proved Fermat's last theorem by this
method. As soon as he had presented a
sketch of his proof, however. Joseph
Liouville rose to object to Lame's appli
cation of the properties of the ordinary
integers to the arithmetic of the cyclo
tomic integers. It is not clear whether
Liouville knew of Euler's similar error.
In any case it is remarkable that he was
able to instantly spot the weakest point
in Lame 's argument. There were other
weak points, however. Lame 's enthusi
asm was so extreme that he overlooked
several other serious difficulties. In fact,
his method turned out to be unwork
able, even for the exceptional values of
n for which his main assumption holds:
n equals 3 , 5 , 7 , 1 1 , 13 , 1 7 and 19.
Naturally Lame was embarrassed by
the foolishness of his errors and by hav
ing them published in the proceedings
of the French Academy for the entire
mathematical world to see. "If only you
had been in Paris or I had been in Ber
lin," he wrote to his friend Dirichlet in
Berlin, "all of this would not have hap
pened. " Actually it would have been
enough for Lame to have read the pro
ceedings of the Berlin Academy of Sci
ences, where a few months earlier there
had appeared the first announcement of
a new and important theory of the arith
metic of cyclotomic integers.
K ummer ' s Contribution

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The author of the new theory was

Ernst Eduard Kummer. Several years
earlier Kummer had realized that for
problems in number theory such as Fer
mat's last theorem the important prop
erty of ordinary integers is their unique
factorization into primes, and so he had
attempted to prove that the property ap
plied to the cyclotomic integers as well.
What he proved, however, was that
unique factorization usually does not
hold for those numbers. (He had pub
lished the finding in 1844, but in a rath
er obscure place. ) As Kummer contin
ued to work on the cyclotomic integers,
however, it became clear to him that
the full power of unique factorization,
which could not be obtained in the cy
clotomic integers, was not really nec
essary. His theory of 1847 showed that
there was a way to modify the concept
of unique factorization so that the mod
ified version could be applied to prove
some subtle and useful properties of the
cyclotomic integers.
The basis of Kummer's theory was
the introduction of what he called ideal
prime factors into the arithmetic of cy
clotomic integers, in a manner some-


what analogous to the introduction of

the imaginary number ;, or Y-1 , into
the arithmetic of ordinary integers. I
shall not discuss the character of Kum
mer's ideal numbers except to say they
restored some of the most important
properties derived from unique factori
zation to the cyclotomic integers and to
other number systems, such as the num
bers a + b -v=3 , that arise in proving
the various cases of Fermat's last theo
rem. The theory of ideal factorization
was unquestionably one of the major
achievements of 19th-century mathe
matics. Through a rather bizarre evolu
tion of terminology Kummer's ideal
prime numbers and certain classes of
numbers that are related to them are
now called ideals. Today ideal theo
ry is a separate branch of mathemat
ics, which is further testimony to the
far-reaching importance of Kummer's
ideas. His work illustrates a strange fact
about mathematical research, namely
that it is impossible to predict in ad
vance which lines of investigation will
lead to useful discoveries. His study of
extremely pure. or theoretical, ques
tions in the field of number theory led
him to formulate concepts of unforesee
able value and versatility for mathemat
ics in general.
Kummer's theory led in particular to
the greatest advance that has ever been
made in the study of Fermat's last theo
rem. Only a few years earlier the proofs
for the cases n equals 5 and n equals 7
had been considered major accomplish
ments, but in 184 7 Kummer was able to
prove .the theorem true for all prime ex
ponents smaller than 3 7 , thereby, of
course, establishing the theorem for all
exponents smaller than 3 7 . Moreover,
he came close to proving the theorem
for all prime exponents smaller than
100. Only the exponents 3 7, 59 and 6 7
eluded his method.
Although many historians of mathe
matics have stated that Kummer's theo
ry grew out of his work on Fermat's last
theorem, a careful study of Kummer's
work and correspondence indicates that
Fermat's last theorem was rather inci
dental. Kummer's main aim was to find
the solution to another problem in the
area of higher arithmetic, or number
theory: the problem of the higher reci
procity laws posed by Gauss. (The prob
lem of the higher reciprocity laws con
sists in generalizing to higher powers the
famous quadratic reciprocity law that
Gauss proved for second powers. Brief
ly, the quadratic reciprocity law states
that if p and q are odd prime integers,
then there is a simple relation between
the answers to the questions "Does p dif
fer from the square of an integer by a
multiple of q ?" and "Does q differ from
the square of an integer by a multiple
of p ?") In 184 7 Kummer's work on the
higher reciprocity laws was still in its ear
ly stages, but by 1859 he had achieved

great success. proving a general theorem

that was the culmination of his work in
number theory. The traditional view
that Kummer was motivated by interest
in Fermat's last theorem is not entirely
false. however. because the theorem is
closely related to the problem of the
higher reciprocity laws. Gauss himself.
although he always denied that he was
interested in Fermat's last theorem per
se. expressed the hope that from his re
sults concerning the higher reciprocity
laws he would one day be able to deduce
Fermat's theorem easily.


Regular Primes
Kummer's theory of 1 847 was partic
ularly valuable because it provided a
sufficient condition for an odd prime p
to be an exponent for which Fermat's
last theorem is true. In other words. if an
odd prime p satisfies Kummer's condi
tion. then xP + yP = zp has no solution.
In modern terminology a prime that sat
isfies the condition is called regular.
(Specifically. a prime p is regular if and
only if it does not evenly divide the nu
merator of any of the first p - 3 num
bers in the series of fractions called the
Bernoulli numbers. ) Although regulari
ty is a sufficient condition for Fermat's
last theorem. it is not a necessary condi
tion. There are primes p that are not
regular for which xP + yP = zp has been
proved impossible. Of the primes small
er than 1 00 all but 3 7 . 59 and 67 are
Kummer jumped to the conclusion
that the set of regular primes was infi
nite. but he soon realized he could not
prove it to be true. Indeed. no subse
quent effort has yielded a proof. al
though on the basis of intuition and nu
merical evidence the assertion that there
are infinitely many regular primes seems
to be as certain as an unproved state
ment can be. (Oddly enough. it has been
proved that there are infinitely many ir
regular primes. About 60 percent of the
primes within the present range of com
putation are regular. and there are good
reasons to believe the majority of all
primes are regular. Hence all the prime
numbers can be divided into two sub
sets. the regular primes and the irregular
primes. where the set that is expected to
be larger cannot be proved infinite but
its complement can be proved infinite. )
In later years Kummer established
other sufficient conditions for Fermat's
last theorem that covered even more
prime numbers. including the irregular
primes 3 7 . 59 and 6 7 . Since Kummer's
time even more inclusive sufficient con
ditions have been found. some of the
most inclusive of them by H. S. Van
diver of the University of Texas. Even
the most inclusive conditions. however.
have still not been shown to apply to an
infinite number of prime exponents.
Thus it is still conceivable. no matter

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how unlikely , that Fermat's last theo

rem is true only for a finite number of
prime exponents and consequently that
there is some very large number M such
that the theorem is false for all prime
exponents greater than M.
On the other hand, the sufficient con
ditions for the theorem are now so in
clusive as to cover all the prime num
bers that have ever been tested. In other
words, all the primes within the range of
comp utation have been proved to be ex
ponents for which Fermat's last theorem
is true. The algorithms for checking
whether primes satisfy the conditions
are quite simple, and over the past 2 5
years intensive checking has been done
on large modern computers, notably by
D. H. Lehmer of the University of Cali
fornia at Berkeley, R . G. Selfridge of the
University of Florida, Wells Johnson of
Bowdoin College and Samuel S. Wag
staff of the University of Illinois. In the
early 1970's Johnson's computations es
tablished that Fermat's last theorem was
true for all prime exponents smaller
than 3 0,000. M ore recently Wagstaff,
working with sophisticated techniques
and a large computer at the University
of Illinois, has pushed the limit past
1 2 5 , 000.

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The Present Situation

These computations also show that a

counterexample to Fermat's last theo
rem, a set of integers x, y, z and p for
which xP + yP
zP, would consist of
numbers so immense that they would be
far beyond the range of hand computa
tion or even of computation on the larg
est existing computers, indeed far be
yond the range of any computer that is
remotely conceivable . If p is a prime be
yond Wagstaff's limit, say p is about
3 00,000, then it can be shown that
xP + yP
zp is impossible unless x, y or
z is divisible by p. Therefore zp must be
greater than 3 00,000300,000, a number of
at least a million digits. Other results
show that a counterexample would
involve even more outlandishly large
Thus in a certain sense Fermat's last
theorem is empirically true. If there is a
solution for xn + yn
zn, then the num
bers in it are so inconceivably large that
human beings will never be able to deal
with them . From a philosophical and
mathematical point of view, however ,
the size of the numbers has no bearing
on the validity of Fermat's last theorem.
When a mathematician says a statement
is true for all numbers, he does not mean
that it is true only for all the numbers
anyone has ever encountered or ever
will encounter . On the contrary, since
Fermat's last theorem has not even been
proved true for an infinite number of
prime exponents, one might say that all
the work on the problem has done noth
ing more than verify the theorem in a

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few thousand special cases, and that

there is not even any reliable indication
of whether the theorem is true.
What, then, is the present status of the
work on Fermat's last theorem? It is cer
tainly one of the most famous unsolved
problems in mathematics today, but it is
not the object of a great deal of mathe
matical research because no one knows
how to attack it. It has withstood Kum
mer's powerful methods and their many
subsequent refinements, and it has still
not even been proved for infinitely many
prime exponents. If someone were to
put forward a concept that promised to
bring about some real advances on the
problem. it would surely elicit a great
deal of interest and activity. At the
present time. however. the subject of
Fermat's last theorem is more or less
Interest i n the Theorem
In the past 100 years Fermat's last the
orem has become a favorite topic of
mathematical amateurs. taking a place
beside the problems of squaring the cir
cle and trisecting the angle with ruler
and compass. Fermat's last theorem dif
fers from circle squaring and angle tri
secting in that those tasks are known to
be impossible, and so any purported so
lutions can be rejected out of hand. No
such assurance exists for Fermat's last
theorem. On the contrary, there is just
an outside chance that there is a reason
ably elementary proof of the theorem
that Fermat did discover. Amateurs
rarely succeed, however. in proving
even the case n equals 3 of Fermat's last
theorem. which was proved by Euler
two centuries ago. much less the harder
but also well-known cases n equals 5. 7
and 11.
Perhaps the persistence of amateurs
the belief that mathematical results are
just "out there" waiting to be noticed
has to do with a general lack of under
standing about what it is that mathema
ticians do. In this article I have attempt
ed to show that mathematics is certainly
not cut and dried. that. on the contrary,
mathematicians are usually adrift in a
sea of unanswered questions. In fact. re
search in mathematics routinely takes
such unexpected turns that the investi
gator often ends up finding answers to
questions quite different from those he
set out to answer. Moreover. as the his
tory of Fermat's last theorem shows.
even great mathematicians make mis
takes. Fermat's last theorem remains
in itself an intriguing question. and al
though its importance is largely a histor
ical accident. it will surely continue to
attract the attention of mathematicians
until the proof is found. There is every
reason to believe that in the future. as in
the past. attempts to solve Fermat's last
theorem will bring about important ad
vances in mathematics.

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