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"Science for Everyone"

"Science for Everyone" was a technical column posted regularly on the VACETS forum. The author of the following articles is Dr. Vo Ta Duc. For more publications produced by other VACETS  members, please visit the VACETS Member Publications page or Technical Columns page.

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Mon, 24 Oct 1994


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).

Duc Ta Vo, Ph.D.

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Copyright © 1996 by VACETS and Duc Ta Vo


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