Fermat's Last Theorem

About 3 pages (886 words)

Fermat's Last Theorem Summary

Fermat's Last Theorem

Andrew Wiles worked secretly and mostly alone to solve Fermat's Last Theorem. Despite his proof offered in 1994, no one has yet found the proof that Pierre de Fermat claimed to possess because no one has found a proof that used the mathematics and mathematical tools available to Fermat in the seventeenth century.

The proof of Fermat's Last Theorem involves two people separated by over 350 years. The first is the French lawyer and mathematician Pierre de Fermat, who, in about 1637, left a note written in the margin of a book. His note said that the equation aⁿ + bⁿ = cⁿ has no solutions when a, b, and c are whole numbers and n is a whole number greater than 2. The note went on to say that he had marvelous proof of this statement, but the book margin was too narrow for him to write out his proof.

In the twentieth century, a 10-year-old British boy named Andrew Wiles read about this problem and was intrigued and challenged by it. No wonder: Fermat's Last Theorem has been called the world's greatest and hardest mathematical problem. Wiles's childhood dream became "to solve it myself … such a challenge … such a beautiful problem."

The Challenge

Before finding out how Wiles proved Fermat' Last Theorem, consider the equation aⁿ + bⁿ = cⁿ. Working with the Pythagorean theorem and right triangles reveals that, in every right triangle, a² + b² = c². There are also certain whole number values for a, b, and c that are called Pythagorean triples, such as 3, 4, and 5; 5, 12, and 13; 27, 36, and 45; or 9, 40, and 41. Many such triples can be found, and there are formulas that can be used to grind them out endlessly.

Notice, however, that the exponent (n) for these Pythagorean triples is2. Fermat's Last Theorem says that such triples cannot be found for any whole number greater than 2. By the 1980s, mathematicians using computers had proven that Fermat's Last Theorem was correct for all the whole number values of n less than 600, but that is not the same as a general proof that the statement must always be true. Fermat's statement—that the equation aⁿ + bⁿ = cⁿ has no solutions when a, b, and c are whole numbers and n is a whole number greater than 2—may be fairly simple to state, but it has not been so simple to prove.

The Solution

After 7 years of working on the problem in complete secrecy, Andrew Wiles used modern mathematics and modern methods to prove Fermat's Last Theorem. The modern mathematics and methods he used did not, in general, entail making long and elaborate calculations with the aid of computers. Instead, Wiles, while a researcher at Princeton, used modern number theory and thousands of hours of writing by hand on a chalkboard as he thought, tried, and failed, and thought some more about how to prove Fermat's Last Theorem.

As Wiles worked on finding a proof, he looked for patterns; he tried to fit in his ideas with previous broad conceptual ideas of mathematics; he modified existing work; he looked for new strategies. He used the work of many mathematicians, reading their papers to see if they contained ideas or methods he could use. It took him 3 years to accomplish the first step. In spring of 1993, he felt that he was nearly there, and in May of 1993, he believed he had solved the problem.

Wiles asked a friend to review his work, and, in September of 1993, a fundamental error was found in his proof. He worked until the end of November 1993 trying to correct this error, but finally he announced that there was a problem with part of the argument in the proof. Wiles worked almost another whole year trying to correct this flaw. After months of failure while working alone, he was close to admitting defeat. He finally asked for help from a former student, and together they worked for 6 months to review all the steps in the proof without finding a way to correct the flaw.

In September of 1994, a year after the error was originally found, Wiles went back one more time to look at what was not working. On Monday morning, September 19, 1994, Andrew Wiles saw how to correct the error and complete his proof of Fermat's Last Theorem. He called the insight that completed this proof "indescribably beautiful—so simple and elegant."

The solution of Fermat's Last Theorem involves the very advanced mathematical concepts of elliptic curves and modular forms. Wiles's subsequent paper on the proof, "Modular Elliptic Curves and Fermat's Last Theorem," was published in 1995 in the Annals of Mathematics.

Why Is It Called the "Last" Theorem?

Fermat's Last Theorem was not called so because it was his last work. (He apparently wrote the marginal note about this theorem in 1637, and he died in 1665.) Rather, this statement came to be called Fermat's Last Theorem because it was the last remaining statement from Fermat's mathematical work that had not yet been proved.