Marilyn's Writings

Marilyn Writes More Than Just Her Column

Fermat's Last Theorem

Sometimes Marilyn may be misunderstood because she doesn't say enough. She doesn't have much space in her weekly column and her replies may be too brief. In some such cases clarification may be obtained by referring to her other writings. Here we refer to her book The World's Most Famous Math Problem, published in 1993 by St. Martin's Press in order to provide further background for her views in the area of number theory and Wiles's proof of Fermat's Last Theorem.

The proof of Fermat's Last Theorem has been one of the most challenging problems in mathematics. It arises in the course of the study of Diophantine equations, which are equations the solutions of which are restricted to integers. Diophantus himself was interested in defining the class of right triangles that have integral sides. In the study of geometry you soon come across the "3, 4, 5" right triangle, but there are many others; in fact, an infinite number of them. Using the Pythagorean theorem these are expressed as the integer solutions of the equation x2 + y2 = z2 and Diophantus was able to derive general expressions for x, y, and z. Fermat studied Diophantus and in the margin of his book he scribbled the remark that the equation has no solutions in integers if the exponent is greater than 2. He added that he had found a remarkable proof of this statement "Which this margin is too small to contain." The promise that such a proof existed started a search that lasted several hundred years. Many proofs were offered. Some were obvious cranks and others eventually were found to contain flaws. In 1995 Andrew Wiles published in the Annuals of Mathematics a paper entitled "Modular Elliptic Curves and Fermat's Last Theorem," another proof of Fermat's Last Theorem, and Marilyn's book is a pre-publication review of it.

How it came about may be inferred from the Acknowledgments

I also want to thank Robert Weil, my editor at the St. Martin's Press, for having the courage to ask that I write this book in three weeks. If it weren't for him, I wouldn't have known I could do it.
Wiles's paper had gained a reputation prior to its publication and Marilyn apparently was asked to write a review. Apparently she worked not from the paper itself but from Karl Rubin's abstract, which appears in the Appendix. The paper itself would not appear in print until two years later. Considering the fact that she is not widely known as an expert in number theory, the material she had to work with, and the time constraint, she did an excellent job. She first gives an account of the early proofs of the theorem for particular values of the exponent: The proof for 4 by Fermat himself, for 3 by Euler, for 5 by Legendre, for 7 by Lamé, and so on. She then discusses "What is a theorem?" and "What is a proof?" She proceeds to the immediate background of the Wiles theorem, discussing Taniyama's conjecture, Frey's postulate, and Ribet's proof of Frey's postulate. She points out that the chronological order of events is not the same as the logical order. Finally, here is Marilyn's attempt to put Wiles's theorem into "plain English"
Equations of the form xn + yn = zn, called Diophantine equations when n is an integer, can be translated to describe a certain set of elliptic curves. These curves represent the surface of a torus, an object shaped like a smooth doughnut.

Taniyama suggested that for that certain set of elliptic curves in Euclidean geometry (where parallel lines never meet, even if infinitely extended) there are corresponding structures in the hyperbolic (non-Euclidian) plane (where parallel lines can both converge and diverge).

Frey suggested a connection between that certain set of elliptic curves and Fermat's last theorem, namely that if there were solutions in violation of the theorem, they would generate a subset of "semistable" elliptic curves, curves that could not be represented in the hyperbolic plane.

Wiles accepted Ribet's proof of Frey and reasoned that if he (Wiles) could prove Taniyama, at least for the "Fermat subset" of semistable elliptic curves (if not for that larger certain set of elliptic curves), no solutions to Fermat's last theorem could exist, thereby implying a proof of FLT.

Wiles then developed an unconventional method of counting both the Euclidian semistable elliptic curves and their hyperbolic (non-Euclidian) counterparts in such a way as to demonstrate a one-to-one correspondence between the two groups. In this way, he claims to have proved Taniyama for the "Fermat subset" of semistable elliptic curves.

Marilyn explained in detail the implications of a mathematical proof and that errors could be obscure but in the book did not express an opinion on the validity of the proof. Marilyn's critics on this particular topic congregate here.

Here are some thoughts on Fermat's Last Theorem not expressed in Marilyn's book:

In an attempt to commission a proof of Fermat's Last Theorem, the Wolfskehl Prize was established at the University of Göttingen, Germany in 1908. Initially it was DM 100,000, comparable in amount to the Nobel Prize, but wartime inflation has reduced it to DM 10,000. Since Fermat claimed to have a proof, one assumes that a proof is possible in the mathematics of Fermat's time, which is the high-school mathematics of today. Consequently, there have been many thousands of submissions and the university mathematics staff has had the burden of evaluating them. Apparently the offering of the prize has done more harm than good.

Among the many faulty proofs of Fermat's Last Theorem is one by an eminent mathematician that was accepted as correct for a number of years. Might history repeat itself? There is a parallel between that proof and the one by Wiles. The proof to which we refer involved a new mathematical discipline, Complex Analysis, and at the time there were few experts in it. Wiles's proof uses the latest developments in non-Euclidean geometry, and there are few experts today. The reports of the Boston University conference on Wiles's proof comment on this. No doubt Wiles's proof will get closer scrutiny when more people are able to hack it. Surely it will serve as an incentive for further study.

Just as it is possible to prove a large integer to be composite without finding its factors, so it is possible to prove a conjecture false without finding a counter-example. An "existence proof" that there must be some number greater than two for which Fermat's equation holds would not initially be unacceptable to the scientific community. Also, one might try for a proof that FLT is unprovable. Fermat's claim that he had a proof (which many historians now doubt) has led to a one-sided development of the topic.

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