FERMAT'S
LAST THEOREM

In the [SCIENCE
FOR EVERYONE] column last week, I had three bonus problems posted and no
one had solved any of them. All I heard was all kinds of discussion about
the first bonus, the Fermat's last theorem. It asserts that "For any
integer n greater than 2, the equation (a^n + b^n = c^n) has no solutions
for which a, b, and c are integers greater than zero."

The discussion
was interesting. Actually, I had heard that someone had found a solution
to the theorem sometime last year. A few months ago, I heard that the proof
had some holes in it; some are small like pin-holes and some are as big
as black holes. All the pin-holes, potholes, manholes were filled, but
the biggest hole, the black hole, was not filled. I guess that there is
no way to fill a black hole. It just swallows everything you throw at it
and gets bigger. I didn't pay much attention until last week when I saw
that many people were discussing it. I decided to do some research into
it, and here is what I found. This story is rather long, so I'm going to
present it in an unusual way by summarizing the results first. This is
so people who do not have time to follow the whole story, grasp at least
grasp some idea.

SUMMARY: --
Sometime in the 1630s, Fermat wrote his famous "Last Theorem".

-- In 1955,
Taniyama proposed a conjecture about elliptic curves. It was made more
precise by Shimura later, and it is now known as the Taniyama-Shimura conjecture.
The conjecture contends that "every elliptic curve with rational coefficients
is modular".

-- In 1985,
Frey stated that elliptic curves arising from counterexamples to Fermat's
Last Theorem could not be modular.

-- In 1986,
Ribet proved the Frey's curves.

-- In 1993,
Wiles attempted to prove the Taniyama-Shimura conjecture. A proof of the
conjecture would imply that there exists no counterexample to Fermat's
Last Theorem. Unfortunately (?), Wiles' proof is incomplete.

If you don't
quite grasp the summary and if you have time then read on. Here is the
somewhat longer version of the story.

LONG STORY:
Pierre de Fermat was a successful lawyer and judge in France in the 17th
century, but he is known today almost entirely for his amateur activities:
mathematics. He did not publish his mathematical work, but he carried on
extensive correspondence with other scholars and made major contributions
to number theory and the study of probability. Indeed, the main source
of knowledge about his mathematical work is his correspondence and his
annotations in the margins of books.

Sometime in
the 1630s Fermat was reading the "Arithmetic" of Diophantus of
Alexandria which discusses various problems to be solved in whole numbers
or in rational numbers. Fermat made numerous notes in the book. On one
page where Diophantus asked, "Given a number that is a square, write
it as a sum of two other squares." Fermat's note, translated from
the Latin, reads: "It is impossible to separate a cube into two cubes,
or a fourth power into two fourth powers, or in general, any power higher
than the second into two like powers. I have discovered a truly marvelous
proof of this, which this margin is too narrow to contain." Did Fermat
really have the proof that he could have written out if only the margin
had been a little wider? A likely answer is that Fermat probably thought
he had a proof but later discovered a flaw in it. In subsequent letters
to colleagues, he referred to proofs of the specific cases where n=3 and
n=4, but the general proof was never mentioned again. Why is it called
the last theorem? It surely was not Fermat's last; he lived on until 1665
and made many further contributions to mathematics. The word "last"
probably arose in the 18th or 19th century and was apparently meant to
identify the theorem as the last of Fermat's questions to be answered.

There are
infinite integer solutions to the equation (a^n+b^n=c^n) when n is 1 or
2. So why is it so hard to find a solution when n>2 ? For more than
350 years, many great mathematicians have tried to find the proof for the
theorem and Andrew Wiles of Princeton University came closest to solving
it. Fermat himself mentioned the specific cases where n=3 and n=4, but
only the proof for n=4 is known (found in another marginal note). The idea
underlying the proof is a technique Fermat invented, called "the method
of infinite descent". Begin by assuming there are indeed integer solutions
of the equation (a^4+b^4=c^4=C^2 where C=c^2), Fermat found a sequence
of operations that, given any such solution, generates a smaller one. From
the new solution the same sequence of operations yields a still smaller
solution. The process can be continued without limit, creating an infinite
series of ever smaller solutions. But such a series of continually diminishing
numbers can not exist in the positive integers, which have a lower limit
of 1. That means the solution does not exist.

In the early
18th century, Leonhard Euler, a Swiss mathematician, undertook the case
of n=3. His proof also relies on the method of infinite descent. In the
1820s, the French mathematician Adrien-Marie Legendre and the German P.G.
Lejeune Dirichlet produced proofs for n=5. Dirichlet went on to complete
the proof for n=14. The proof for n=7 was devised by Gabriel Lame of France.
In 1847, the German Ernst E. Kummer proved Fermat last theorem is true
for all n<100. In recent years, 1993, J. Buhler, R. Crandall, E. Ernvall,
and T. Metsankyla, with the help of computers have pushed the lower limit
to n = 4 million. Isn't the range from three to 4 million suffice to say
that the problem is solved? No! No mathematician would consider the question
settled. 4 million is not infinite!

The modern
approach to Fermat's last theorem is an indirect one. Instead of attacking
the equation (a^n+b^n=c^n) directly, one analyzes a new equation of different
form which involves the numbers a^n and b^n. Wiles approached the problem
by setting out to prove another proposition called the Taniyama-Shimura
conjecture, which in turn establishes the truth of Fermat's Last Theorem.
The Taniyama-Shimura conjecture, introduced by the Japanese Yutaka Taniyama
in 1955 and later improved by Goro Shimura of Princeton University, stated
that "all elliptic curves with rational coefficients are modular".
The Taniyama-Shimura conjecture belongs to a realm of mathematics called
"arithmetic algebraic geometry", or "modern arithmetic".
In his approach, Wiles followed up several key discoveries, particularly
the modularity of elliptic curves, made by other mathematicians during
the 1980s.

Before proceeding
further, the terms "elliptic curves" and "modularity"
need some explanation. First, an elliptic curve is not an ellipse. The
name reflects a connection with elliptic functions. Elliptic curves are
nonsingular plane curve defined by a certain class of cubic equations.
(Note that not every cubic equation generates an elliptic curve.) Each
elliptic curve gives rise to a mathematical object called an L-series.
Since a complete L-series is an infinite product, it is hard to use in
further calculations. Modular form is an analytical function and it offer
a short cut to the analysis of L-series. Unlike elliptic curves which are
algebraic objects, modular forms come from a rather different realm of
mathematics: complex analysis. The modular forms are essentially invariant
under certain transformations known as "fractional-linear transformations".
For example, each modular form is invariant under integer translations,
i.e., f(z+1)=f(z) for z is a complex number. The connection between the
elliptic curves and modular forms is that modular forms also give rise
to L-series analogous to those coming from elliptic curves. Studying the
L-series attached to an elliptic curve becomes much easier if a modular
form with the same L-series is obtained. And the Taniyama-Shimura conjecture
contends that "for each elliptic curve, there is a modular form whose
L-series is the same as that of the elliptic curve".

Now, let's
restate the Fermat's Last Theorem as follows, "If n>2, then a^n+b^n=c^n
has no solutions in nonzero integers a, b, and c."

Suppose the
theorem were false, then there would exist nonzero integers a, b, and c,
and n>2 such that a^n+b^n=c^n. Write down the cubic curve [y^2 = x(x+a^n)(x-b^n)]
and it represents elliptic curves.

In 1985, the
German Gerhard Frey drew attention to this equation by stating that the
elliptic curve, which is now often called the Frey curve, could not be
modular. However, he was not able to give a rigorous proof. A year later,
Kenneth A. Ribet of UC Berkeley supplied the proof. Ribet proved that "If
n is a prime greater than three, a, b, and c are nonzero integers, a^+b^n=c^n,
then the elliptic curve represented by y^2=x(x+a^n)(x-b^n) is not modular."

On June 23,
1993, Andrew Wiles put before an audience at the Newton Institute in Cambridge,
England a theorem that says: "If A and B are distinct, nonzero, relative
prime integers, and AB(A-B) is divisible by 16, then the elliptic curve
y^2=x(x+A)(x+B) is modular." (Relative primes are numbers with no
factors in common, e.g., 6 and 5 are relative primes while 6 and 10 are
not.)

Taking A=a^n
and B=-b^n with a,b,c, and n coming from the hypothetical solution to Fermat's
Last Theorem, one can see that the conditions for the theorem are satisfied
if n>4. What does this mean? It means that if Wiles' theorem is true,
together with Ribet's result on Frey's elliptic curve, would imply the
solution to Fermat's Last Theorem does not exist for n>4. Since the
cases for n=3 and n=4 have been proved, the new result would complete the
proof for Fermat's Last Theorem.

Was Andrew
Wiles able to supply the proof for this theorem? Almost. The main point
of Wiles' proof is the connections between the elliptic curves and the
modular forms. It is possible to link up various collections of elliptic
curves and modular forms by means of a technique called "deformation
theory" introduced by Barry Mazur of Harvard University a few years
earlier. One of the steps in his calculations is to show that there are
no more deformations than there are modular forms. This entails calculating
an upper bound on the size of an object called the Selmer group. And this
is where the weaknesses of the proof appear.

In June 1993,
Wiles announced his result at the new Isaac Newton Institute in Cambridge,
England. He then submitted a 200-page manuscript of the proof to the journal
"Inventiones Mathematicae". Copies of the manuscript were in
turn sent to about half a dozen referees for checking. The referees pointed
out the problems in his theory and he was able to correct them all, except
one. In December 1993, he posted a note in the Usenet news group acknowledging
a gap in his proof. The problem involved calculating a precise upper limit
on the size of the Selmer group.

The setback
is disappointing. However, his work has inspired the mathematical community
on many aspects. As mathematicians begin to understand the details of what
Wiles has accomplished, it is possible that someone else (or Wiles himself)
may find a way to bridge the gap, or rework the argument using some of
Wiles' ideas in a different way, or set out to search for a counterexample
to Fermat's Last Theorem.

Are you still
with me after all these mathematical forms and ideas? Then here are the
bonuses.

1/ We now
know that Wiles' proof is incomplete. That means a counterexample to Fermat's
Last Theorem may exist. We may try to fill the gap left by Wiles. This
may be a very difficult job since most of us are not great mathematicians.
The alternate method is much easier, that is searching for a counterexample
to Fermat's Last Theorem. Since most of us can program a computer, the
job shouldn't be too difficult. If you decide to do it then here are some
suggestions:

a/ n must
be a prime, because any solution to the theorem in which n is composite
would imply the existence of a smaller solution with a prime exponent.
That is if a^n+b^n=c^ for n=l*m then A^m+B^m=C^m where A=a^l, B=b^l, and
C=c^l.

b/ a, b, and
c are relative prime, i.e., they have no factor in common. Because if they
do then you can always divide both sides of the equation by that factor
to get a smaller solution.

c/ Exactly
one of a, b, and c must be even. This can be seen easily from a^n+b^n=c^n
or a^n+b^n-c^n=0.

d/ Start from
n greater than 4 million since Buhler et. al. already got to the 4 million
mark.

e/ Assume
b>a, then c < b(1+ln(2)/n) = b(1+0.693/n) for large n.

f/ Need to
find an efficient way to do arithmetic on huge integers.

g/ You are
intelligent and will find more ways to reduce the computational time. Even
if you don't find a counterexample but could push the limit for n to 4
billion or 4 trillion then you can still become famous.

2/ a^n+b^n+c^n=c^n.
It is trivial for n=1. For n=2 then 1^2+2^2+2^2=3^2. For n=3 then 3^3 +
4^3 + 5^3 = 6^3. Now try to find n>3 that would satisfy the equation.

3/ There are
a^2+b^2=c^2 and a^3+b^3+c^3=d^3. Is there a^4+b^4+c^4+d^4=e^4?

Happy number
crunching, everyone!!! -----------------

References:
"A report on Wiles' Cambridge lectures", K. Rubin and A. Silverberg,
Bulletin of the American Mathematical Society, Vol. 31, No. 1, pp 15-38
(July 1994) and references therein. "Fermat's Last Theorem and Modern
Arithmetic", K.A. Ribet and B. Hayes, American Scientist, Vol. 82,
pp 144-156 (Mar-Apr 1994) and references therein. "Last word not yet
in on Fermat's conjecture", R. Lipkin, Science News, Vol. 145, (Jun
25, 1994).