Mathematics for the liberal arts (2013)

Part II. TWO PILLARS OF MATHEMATICS

Chapter 5. NUMBER THEORY

5.16 Fermat’s Last Theorem

We saw in Section 5.11 that the Diophantine equation x² + y² = z² has an infinite number of solutions, the Pythagorean triples. Given this, it is natural to ask about a similar Diophantine equation, x³ + y³ = z³, or more generally,

where n is some fixed integer greater than 2. One type of solution is pretty easy: one of x or y is 0, and z equals the other. But this is too easy; let us call these trivial solutions. So the challenge is to find nontrivial solutions.

The great mathematician Pierre de Fermat studied these equations in the 17th century. After his death, his son discovered the following, written in the margin of his copy of Diophantus’ book Arithmetica.

I have discovered a truly marvelous proof that 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. This margin is too narrow to contain it.

In other words, Fermat claimed that these Diophantine equations had no nontrivial solutions, for any n > 2. It was a remarkable assertion, given that there are lots of solutions with n = 2. Fermat did have a proof of the case n = 4, but no one could find anywhere in his papers the proof that couldn’t fit into the margin.

Assertions of theorems without published proofs were not uncommon at this time. In fact, Fermat himself provided a number of them. Fermat’s record of being correct was very good, however. Several of his theorems were first supplied with published proofs only 100 years later by the great Leonhard Euler. But even Euler couldn’t prove this one. It became known as Fermat’s Last Theorem.

Over the years, many mathematicians attempted to prove or disprove Fermat’s Last Theorem. They succeeded in proving it for some values of n, starting with Euler, who gave a proof for n = 3 in 1770. In the early 19th century, one of the greatest women mathematicians, Sophie Germain (1776–1831), made a number of important advances, which inspired further work, although she did not succeed in proving the theorem. Later in the 19th century, Ernst Kummer (1810–1893) managed to prove many cases, in the process inventing a whole area of mathematics, the theory of ideals, that has proven useful in parts of mathematics far beyond its original inspiration. In the 20th century, computers were enlisted. By 1993 computers (with the assistance of mathematical theory) had verified the theorem for all n up to 4,000,000.

The fame of Fermat’s Last Theorem spread beyond mathematics. In 1908 German industrialist Paul Wolfskehl offered 100,000 marks as a prize for a proof. This inspired many amateurs to pursue the proof. In addition to providing hours of instructive labor to the amateurs, this also wasted a fair bit of time for the professionals. Edmund Landau (1877–1938), who was for a time responsible for evaluating entries for the Wolfskehl prize, received so many incorrect proofs that he had cards printed that read:

Dear… ,

Thank you for your manuscript on the proof of Fermat’s Last Theorem.
The first mistake is on: Page … Line …
This invalidates the proof.

Professor E. M. Landau

He assigned the task of filling in the page and line numbers to his students.

The theorem was finally proven in 1995. The final steps were provided by a number of mathematicians. The general approach was suggested by Gerhard Frey in 1984, based on work from earlier in the century. An important advance was made by Ken Ribet in 1986. After Andrew Wiles, a British mathematician working at Princeton University, learned of this work, he began trying to complete Frey’s program. He worked in secrecy, not wanting anyone else to get the credit for the proof of Fermat’s Last Theorem. In 1993 Wiles announced that he had succeeded. He had, however, made a mistake in his complicated proof. The following year, with the assistance of Richard Taylor, Wiles fixed the error and completed the proof, about 357 years after Fermat made his assertion.

The history of the search for this proof is quite revealing about the nature of mathematics. Perhaps most remarkable is the sustained effort over 350 years, in many countries. Also notable is the way that effort spent to solve this one problem, with no applications in sight, produced the theory of ideals, which has proven applicable. Finally, although the proof was completed by Wiles, it built upon the work of many earlier mathematicians. The proof has many authors, not just one. In the end, mathematics is a collective enterprise.

EXERCISES

5.89 In Fermat’s equation, xⁿ + yⁿ = zⁿ, we can assume that n is a prime number. Why?

5.90 Watch the 1996 Nova film The Proof, about the proof of Fermat’s Last Theorem. Who are some of the mathematicians interviewed in the film? Who are some of the mathematicians, throughout history, who took part in proving Fermat’s Last Theorem? What are some of the key ideas involved in the proof?