The Proof of Fermat's Last Theorem

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The Proof of Fermat's Last Theorem

Overview

But one cannot split a cube into two cubes, nor a fourth power into two fourth powers, nor in general any power in infinitum beyond the square into two like powers. I have uncovered a marvelous demonstration indeed of this, but the narrowness of the margin will not contain it.

These words, written by Pierre de Fermat (1601-1665) in the margin of his copy of Diophantus's Arithemetica, have challenged and sometimes haunted mathematicians for more than 350 years. When a successful proof of Fermat's Last Theorem was finally found in 1993, it ended centuries of interesting and often controversial attempts to solve this famous problem.

Background

In modern algebraic terms, the theorem states that the equationxⁿ + yⁿ = zⁿhas no whole number solutions for n > 2. If n = 2, we have the famous Pythagorean Theorem:x² + y² = z²For instance, if x = 3, y = 4, and z = 5, we have, 3² + 4² = 5²9 + 6 = 25 This is only one of an infinite number of solutions, called Pythagorean triples. But Fermat claimed that if n were three or four or any other whole number larger than 2, then there were no solutions to the equation. For example, the equationsx³ + y³ = z³andx⁴ + y⁴ = z⁴or any other equation with larger exponents, have no whole number solutions.

Why does this seemingly simple statement rank among the most famous theorems produced by the mind of a mathematician? The list of mathematicians who have tried to prove it reads like a "who's who" of mathematics: Leonhard Euler (1707-1783), Carl Friedrich Gauss (1777-1855), Niels Abel (1802-1829), Sophie Germain (1776-1831), Adrien-Marie Legendre (1752-1833), Lejeune Dirichlet (1805-1859), Henri Lebesgue (1875-1941), Joseph Liouville (1809-1882), Augustin Cauchy (1789-1857), Carl Jacobi (1804-1851), Gabriel Lamé (1795 1870), David Hilbert (1862-1943), Richard Dedekind (1831-1916), Ernst Kummer (1810-1893), and many, many others. This theorem, which has been tagged "Fermat's Last Theorem" because it was the last of Fermat's theorems to be proved, has played a pivotal role in the history of mathematics.

Fermat's Last Theorem is enigmatic in several ways. First of all, Fermat himself was not very interested in proving theorems and rarely found time to do so, claiming, "I am content to have discovered the truth and to know the means of proving it whenever I shall have the leisure to do so." It's ironic that proving one of his theorems should become the Holy Grail of mathematics. Secondly, although Fermat claimed to have a proof of the theorem, most mathematicians doubt this. The techniques used by mathematician Andrew Wiles (1953- ) to finally prove Fermat's Last Theorem were not available to seventeenth-century mathematicians. That does not make Fermat a liar, however. He was probably only a little overconfident in his ability to provide a legitimate proof. Fermat used a technique called the method of infinite descent to prove his theorem for the case n = 4, and he may have believed (incorrectly) that this method would work for the general case where n is any integer. Whether Fermat actually had a proof or not, his famous statement has contributed to the lore surrounding the theorem. It is reported that on a graffiti-covered wall at a New York train station the following words were found:xⁿ + yⁿ = zⁿ

I have found a truly remarkable proof of this, but I can't write it out now because my train is coming.

Impact

Fermat's Last Theorem belongs to a branch of mathematics called number theory, a field in which few seventeenth-century mathematicians showed any interest. The theorem received little attention until almost one hundred years after Fermat's death, when the famous mathematician Leonhard Euler revived interest in number theory. Euler proved the theorem for n = 3, and later Sophie Germain, one of the first great women mathematicians, did important work in establishing the theorem for a certain class of prime numbers.

The nineteenth century saw still more advances in the search for a proof. Legendre proved the theorem for n = 5 and Lamé for n = 7. The most important advance came from the German mathematician Ernst Kummer, who proved the theorem for a class of numbers called regular primes in 1850. Kummer showed that the theorem was true for all prime exponents between 3 and 100 except for 37, 59, and 67. It had taken approximately two centuries to accumulate proofs for n = 3, 4, 5, and 7, and then, remarkably, Kummer proved the theorem for nearly all of the other values up to 100. Ironically, Kummer had presented a complete but erroneous proof of Fermat's Last Theorem a few years earlier; he had now made the biggest advance in the search for a general proof.

The twentieth century saw the eventual proof of Fermat's Last Theorem, a solution aided by the use of modern technology. Using long-established mathematical methods and high-speed computers, the theorem was proved for values of n up to 150,000 by 1987, 1 million by 1991, and 4 million by 1993. This strengthened the belief of many mathematicians who intuitively believed that Fermat's Last Theorem was true. To nonmathematicians, it seemed to settle the question entirely. If this theorem is true for exponents as large as we can realistically calculate, isn't that enough evidence to pronounce the theorem true for all values of n? A famous example refutes this type of thinking.

Euler stated the following theorem over 200 years ago: The equationx⁴ + y⁴ + z⁴ = k⁴has no whole number solutions for x, y, z, or k. Although never proved, this theorem was believed to be true by Euler and most subsequent mathematicians. Then, two centuries after the theorem was stated, a whole number solution was found to exist, nullifying the theorem. Why did it take so long for mathematicians to find a solution? Because the solutions (the problem actually has many solutions) involve extremely large numbers. The smallest such solution is x = 95,800, y = 217,519, z = 414,560, and k = 422,481. Examples like this are part of the reason mathematicians stubbornly refuse to accept the truth of a theorem until it is proved for all cases.

Although love of mathematics has been the main motivation for most mathematicians who searched for a proof of Fermat's Last Theorem, fame and financial awards also played a part. In 1908 the German mathematician Paul Wolfskehl left 100,000 marks in his will as a prize for a proof of Fermat's Last Theorem. At one time in his life, Wolfskehl had been on the verge of suicide. He had made his plans, written his farewell letters, and was awaiting the time he had appointed to take his own life. While waiting, he began to read a recent publication on Fermat's Last Theorem. He became so engrossed in the work that the time appointed for his suicide came and went. Wolfskehl had a change of heart and canceled his suicide plans. Many years later, he expressed his gratitude by leaving the prize money to the eventual conqueror of Fermat's Last Theorem.

Because of the prizes and notoriety attached to the problem, interest in proving Fermat's Last Theorem became intense among amateur mathematicians. Edmund Landau, a mathematician appointed to administer the Wolfskehl prize, had the following form letter made:

Dear Sir or Madam: Your proof of Fermat's Last Theorem has been received. The first mistake is on page _______, line _______.

Landau's students were then given the job of reviewing the many incorrect proofs received and returning the postcards.

The search for a proof of Fermat's Last Theorem finally ended when Andrew Wiles announced his proof in the summer of 1993. Wiles, a Princeton mathematician, had worked in isolation on the problem for many years and his announcement came as a surprise to the mathematics community. Wiles's work combined two fields of mathematics, elliptical functions and modular forms, to solve the elusive problem.

In proving Fermat's Last Theorem, Wiles had actually solved another problem in mathematics, the Taniyama-Shimura Conjecture. Goro Shimura and Yutaka Taniyama were two Japanese mathematicians who, in the 1950s, conjectured that there was a relationship between elliptical equations and modular forms. Later, thanks to the work of mathematicians Gerhard Frey, Ken Ribet, and Barry Mazur, it was shown that if the Taniyama-Shimura Conjecture were true, then so was Fermat's Last Theorem.

In a dramatic series of lectures at a conference in Cambridge, England, in the summer of 1993, Wiles presented the proof to his colleagues. He did not announce the true intention of his lectures, but by the third day, it became apparent to the mathematicians attending the conference that Wiles's work was leading toward a proof of Fermat's Last Theorem. The excitement grew in the packed room where Wiles was presenting his lectures. Finally, he reached the conclusion of his proof, and put to rest the most famous problem in the history of modern mathematics.

Or had he? Not long after Wiles announced his discovery, an error was found in one section of his long and difficult proof. At first it seemed that Wiles would be able to fix the error and save the proof, but as time went on the "correction" became more and more difficult. Finally, with the help of one of his former students, Richard Taylor, Wiles was able to make the necessary corrections. These corrections took over a year to complete, however, illustrating the complexity of the proof that Wiles had constructed.

Although the search for a proof of Fermat's Theorem is over, its impact upon mathematics is not. Many mathematical advances are credited to mathematicians who were attempting to solve this problem. As in most theorems, finding one proof does not end useful work on the problem. Other proofs using other mathematic tools may be found, and of course the question remains: "Did Fermat really have a proof, and if so, what was it?"

TODD TIMMONS