(Solutions to this equation are called P
YTHAGOREAN
TRIPLES
.) Around 1637 P
IERRE DE
F
ERMAT
conjectured
that no positive integer solutions exist, however, for
the equations
xn+ yn= zn
with ngreater than two. In his copy of the translated
works of D
IOPHANTUS OF
A
LEXANDRIA
, next to a prob-
lem about Pythagorean triples, Fermat wrote his now
famous note:
On the other hand, it is impossible to separate
a cube into two cubes, or a fourth power into
two fourth powers, or generally any power
except a square into two powers with the
same exponent. I have discovered a truly
admirable proof of this, but the margin is too
narrow to contain it.
For over 350 years mathematicians tried to reproduce
Fermat’s alleged proof. The claim itself became known
as Fermat’s last theorem, and it was one of the greatest
unsolved problems of all time. Although the problem
lends itself to no obvious practical applications,
attempts to solve it helped motivate the development of
a great deal of important mathematics.
It is generally believed that Fermat did not have a
proof of the theorem. In his correspondences with col-
leagues he mentions only the cases nequals 3 and 4
and provides no details of proof even for those special
cases. Fermat, again as a marginal note in his copy of
Diophantus’s work, does provide a detailed proof of
another challenge posed by Diophantus, one about tri-
angles of rational side length. Although not explicitly
mentioned, the proof of the nequals 4 case follows
readily from mathematical argument he provides. It is
thought that Fermat was aware of this.
With the case n= 4 taken care of, it is not difficult
to see that one need only study the cases where nis an
odd prime. For example, if it is known that x7+ y7= z7
has no positive integer solutions, then x42 + y42 = z42
can have no positive integer solutions either. (Rewrite
the latter equation as (x6)7+ (y6)7= (z6)7.)
In the mid-1700s, L
EONHARD
E
ULER
proved that
the equation with n= 3 has no positive integer solu-
tions. The extensive work of M
ARIE
-S
OPHIE
G
ERMAIN
(1776–1831) during the turn of the century allowed
mathematicians to later show that the theorem holds
for all values of nless than 100. During the 19th and
20th centuries mathematicians developed the fields of
algebraic geometry and arithmetic on curves. In 1983,
Gerd Faltings proved the so-called Mordell conjecture,
an important result with the following immediate con-
sequence: any equation of the form xn+ yn= znwith n
> 3 has, at most, a finite number of positive integer
solutions. This led mathematicians a significant step
closer to proving Fermat’s last theorem for all values of
n: is it possible to show that that finite number is zero
in every case? Finally, in 1995, almost 360 years since
Fermat’s claim, the English mathematician A
NDREW
W
ILES
, with the assistance of Richard Taylor, presented
a completed proof of Fermat’s last theorem. It is, not
surprisingly, very long and highly advanced, relying
heavily on new mathematics of the century. Needless to
say, the proof is certainly beyond Fermat’s abilities.
Although Wiles’s proof is deservedly regarded as a high
point of 20th-century mathematics, mathematicians
still search for a simplified argument.
Ferrari, Ludovico (1522–1565) Italian Algebra Born
on February 2, 1522, in Bologna, Italian scholar
Ludovico Ferrari is remembered as the first person to
solve the
QUARTIC EQUATION
. He worked as an assis-
tant to G
IROLAMO
C
ARDANO
(1501–76), who pub-
lished Ferrari’s solution in his famous 1545 work Ars
magna (The great art).
Assigned to be a servant at the Cardano household
at age 14, Ferrari soon impressed his master with his
agile mind and with his ability to read and write. Car-
dano decided to train Ferrari in the art of mathematics.
In exchange, Ferrari helped Cardano prepare his
manuscripts. Four years after his arrival, and with the
blessing of Cardano, Ferrari accepted a post at the Piatti
Foundation in Milan as public lecturer in geometry. Fer-
rari, however, continued to work closely with Cardano.
Ferrari discovered his solution to the quartic equa-
tion in 1540, but it relied on the methods of solving the
CUBIC EQUATION
that had been developed by N
ICCOLÒ
T
ARTAGLIA
(ca. 1499–1557) and revealed to Cardano in
secrecy. (Mathematicians at the time were supported by
patrons and protected their methods as trade secrets:
they were often required to prove their worth by solving
challenges no other scholar could solve.) Unable to pub-
lish the result without breaking a promise, Ferrari and
Cardano felt stymied. However, a few years later, Fer-
190 Ferrari, Ludovico