The Enduring and Revolutionary Impact of Pierre de Fermat's Last Theorem
The Enduring and Revolutionary Impact of Pierre de Fermat's Last Theorem
Pierre de Fermat (1601-1665) was a contemporary of the renowned philosopher and mathematician René Descartes (1596-1650). Fermat, like Descartes, was fascinated with numbers and their properties and relationships, and indeed corresponded with Descartes about his insights and conjectures. Unlike Descartes, however, Fermat was neither a professional mathematician nor a professional philosopher. Nevertheless, though he was considered an amateur mathematician, Fermat now is known as the "Prince of Amateurs."
Despite his amateur status, Fermat contributed much to mathematics, including providing the necessary groundwork for the fields of analytic geometry and infinitesimal analysis. What is more, Fermat is credited with founding number theory, the calculus, and, along with Blaise Pascal (1623-1662), inventing probability theory.
Notwithstanding these incredible accomplishments, Fermat perhaps is most famous for his Last Theorem, a theorem whose solution evaded the brightest minds of mathematics for over 350 years, but whose solution—and quest for the same—revolutionized number theory. According to one contemporary mathematician, the proof of Fermat's Last Theorem, which was finally completed in the fall of 1994, is the historical and intellectual equivalent of "splitting the atom or finding the structure of DNA."
Fermat was born into a wealthy family in the town of Beaumont-de-Lomagne in southwest France. He was well educated both at a Franciscan monastery and the University of Toulouse, although there is no indication that he was particularly attracted to mathematics during this period.
Due in large part to familial influence, Fermat eventually became a lawyer and later entered the civil service. Upon entering the civil service, he served the local parliament as a lawyer, then as a councillor, acting mainly as a liaison between the local municipality and the king. Thus, it appears that politics was his profession. Although Fermat succeeded well in his professional life, his true love and devotion increasingly turned toward the study of numbers.
Fermat's fascination with numbers was rooted in the writings of the ancient Greek mathematician and philosopher Diophantus (fl. a.d. 250), author of the Arithmetica. The Arithmetica originally was comprised of 13 books, though only six are known to have survived the Dark Ages. These books comprised a collection of mathematical problems for which only whole number solutions are possible. As Diophantus was the author of these mathematical problems, they now are known, not surprisingly, as Diophantine problems. Pythagoras's famous theorem is an example.
Interestingly, in ancient Greece, arithmetica, meaning "arithmetic," did not mean mere computation, as it presently does. Rather, arithmetic meant "theory of numbers," and as such was more philosophical in its approach toward understanding numbers and their properties than straightforward mathematical theory.
So it was through Fermat's study of Diophantus's Arithmetica that he became interested in exploring further the mysteries of numbers and their properties, especially with regard to Diophantine problems. Indeed, one of Fermat's favorite pastimes was to write to other mathematicians, asking them whether they could prove a particular theorem, some of which he found in other texts, and some of which he created on his own. Fermat then would taunt these mathematicians, who struggled to provide proofs to Fermat's challenging theorems, by stating that he had the proofs although he would refuse to reveal them.
Needless to say, the mathematicians with whom Fermat corresponded grew increasingly frustrated. Indeed, such taunting moved Descartes to label Fermat a braggart, and the English mathematician John Wallis (1616-1703) to refer to Fermat as "that damned Frenchman." Such responses proved only to motivate Fermat even more as his challenges became ever more complicated and frustrating. These challenges, however, paled in comparison to his ultimate taunt, known as Fermat's Last Theorem, referred to as such because it was the last of Fermat's theorems to be proved.
Pythagoras's theorem states that the square of the hypotenuse of a right triangle is equal to the sum of the square of its sides. Symbolically, the theorem may be expressed as x2 + y2 = z2. If x is assigned the value 3; y, the value 4; and z, the value 5, the equation may be solved: 9 + 16 = 25. Of course, the combination of 3, 4, and 5—known as a Pythagorean triple—is not the only solution to this theorem. Indeed, mathematicians have proved that there exists an infinity of Pythagorean triples that provide solutions to the equation.
Around 1637, when he was 36 years old, Fermat wondered whether there might also be an infinity of Pythagorean Triples to a slight variation of Pythagoras's theorem. Specifically, Fermat wondered whether a cubed version of Pythagoras's theorem—x3 + y3 = z3—had any solutions. Surprisingly, Fermat could not find any Fermatian triples, that is, Fermat was unable to find any solutions to his cubed variation of Pythagoras's theorem. Indeed, Fermat could not find any triples for any Pythagorean variation where the exponents were integers greater than two.
As a result, Fermat noted in the margin of his prized copy of Arithmetica that "It is impossible for a cube to be written as a sum of two cubes or a fourth power to be written as the sum of two fourth powers or, in general, for any number which is a power greater than the second to be written as a sum of two like powers." This statement became known as Fermat's Last Theorem. Another way of stating the theorem is that there are no integer solutions to the equation xn + yn = zn, where n is an integer greater than two. (The equation contained within the Last Theorem was henceforth referred to as Fermat's Equation.)
However, a significant problem remained: Could Fermat actually prove this theorem? In a cryptic notation left in his copy of Arithmetica Fermat claimed to have found a proof for the theorem: "I have discovered a truly remarkable proof which this margin is too small to contain." Unfortunately, Fermat died on January 9, 1665, without ever revealing the proof of his Last Theorem, thereby leaving both an enduring mystery and an incredible taunt for the world to ponder.
For years afterward mathematicians tried in vain to prove the theorem (which more appropriately should have been labeled a conjecture, inasmuch as no proof existed). Partial proofs up to n=7 were achieved, but as there are an infinity of numbers, such proofs amounted to very little toward proving the theorem completely. Some were so intrigued by the possibility of a proof that they offered monetary rewards to those who discovered a proof for all n greater than two. In the early 1900s, for example, Dr. Paul Wolfskehl, a German industrialist and amateur mathematician, offered a prize of 100,000 marks (approximately equivalent to $1 million today) to the first person to solve Fermat's Last Theorem. Needless to say, no serious contenders for the prize money came forward. Indeed, many mathematicians grew skeptical of whether a proof even was possible.
Despite overwhelming skepticism and seemingly impossible odds, on June 27, 1997, approximately 360 years after Fermat conjured up his infamous theorem, and nearly 100 years after the establishment of the Wolfskehl Prize, the prize money was finally bestowed on an unassuming, yet profoundly brilliant mathematics professor from Princeton University. After struggling in secrecy and the isolation of his attic for eight years, Andrew Wiles (1953-) published a 100-page proof of Fermat's Last Theorem in the Annals of Mathematics (May 1995).
What was just as amazing as the fact that Fermat's Last Theorem was finally proved, was the process by which Wiles achieved the proof. To prove Fermat's Last Theorem, he had to unite two completely disparate branches of mathematics, thereby developing a completely novel approach to number theory in order to achieve the proof.
The two disparate areas of mathematics Wiles united concerned elliptic curves and modular forms. Elliptic curves are equations used to measure the perimeter of ellipses and, as a result, have been famously used to compute the elliptical trajectory of planetary orbits. Elliptic equations take the form of y2 = x3 + ax2 + bx + c, where a, b, and c are any whole numbers. Hence, elliptic equations, like the Fermat's Equation, are also Diophantine equations; indeed, Diophantus devoted a large portion of the Arithmetica to the study of elliptic equations.
In contrast, modular forms are highly abstract, complex mathematical objects that display an unusual amount of symmetry. Modular forms are symmetric in the sense that they can be moved about in mathematical space in any conceivable way, but remain unchanged. To take a highly simplified example, imagine a perfect square drawn on a sheet of paper. If one were to rotate the sheet of paper exactly one-quarter turn, the square would appear to remain completely unchanged. Indeed, the square would appear to remain unchanged if one were to turn the paper one-half turn, three-quarters, or even, of course, one full turn. This unchanging aspect of the square as it is moved about on the two-dimensional surface of the sheet of paper demonstrates the square's symmetry.
The study of elliptic equations and of modular forms have historically been completely unrelated endeavors: elliptic equations were used to study real world phenomena, whereas modular forms were studied for their interesting properties in imaginary, mathematical space. However, in September 1955 two young mathematicians at the University of Tokyo—Goro Shimura and Yutaka Taniyama—posed the following conjecture: for every elliptic equation there is an equivalent modular form. This conjecture, generally known as the Taniyama-Shimura conjecture, was so shocking that, initially, few paid much attention to it. Over time, however, the mathematical community discovered that if the Taniyama-Shimura conjecture was correct, many useful applications could be developed, including solving mathematical problems that remained resistant to resolution.
The problem with the Taniyama-Shimura conjecture, like the problem with Fermat's Last Theorem, was that nobody knew how to prove the conjecture. Twenty-nine years later, in the fall of 1984, and half a world away, a small group of number theorists met in Oberwolfach, Germany, to discuss current research in elliptic equations. Work was still being done toward proving the Taniyama-Shimura conjecture, but little progress had been made. Gerhard Frey, a mathematician from Saarbrücken, had discovered something interesting. Frey discovered that Fermat's Equation could be translated into the form of an elliptic equation, but this elliptic equation was quite odd in that it suggested a possible solution to Fermat's Equation, and furthermore, the elliptic equation had no modular form equivalent. Consequently, if Frey's elliptic version of Fermat's Equation was valid, then two conclusions followed: (1) it was possible for there to be a solution to Fermat's Equation, which would prove that the Last Theorem was false; and (2), the Taniyama-Shimura conjecture was false because there existed an elliptic equation without a modular form equivalent.
Stated conversely, what Frey essentially had discovered was that if the Taniyama-Shimura conjecture was correct, then the elliptic version of Fermat's Equation was invalid, which meant that, indeed, there were no solutions to Fermat's Last Theorem. So, in a nutshell, Frey had proved that if the Taniyama-Shimura conjecture was correct, then so was Fermat's Last Theorem.
Ultimately, proving the Taniyama-Shimura conjecture was precisely what Andrew Wiles did in order to finally prove Fermat's Last Theorem. As a result, Wiles opened up new avenues of research previously unavailable, indeed, unknown, to mathematicians. Thereafter problems in elliptic equations could be solved in the modular forms world, and vice versa. Furthermore, the impact of recognizing the underlying unity between these two branches of mathematics would allow mathematicians to better understand each branch.
In the history of knowledge, such unification of disparate branches of thought is not unknown. For example, prior to the nineteenth century physicists studied magnetism and electricity completely separately. But then it was discovered that these two areas of physics were inextricably linked, thereby creating the study of electromagnetism. From the study of electromagnetism came the further discovery that light was nothing more than electromagnetic radiation, which in turn allowed physicists to better understand the nature of the world. Likewise, the quest for the proof of Fermat's Last Theorem has greatly impacted the future of mathematics, and our knowledge of the world generally.
Despite the fact that the mystery of Fermat's Last Theorem has finally been solved, the mystery of whether Fermat actually solved it remains. Although it is highly improbable that Fermat did in fact have a proof, there nevertheless remains the possibility. Consequently, the mystery of whether there exists Fermat's proof to his Last Theorem persists. If in fact Fermat did prove his Last Theorem, the discovery of just what that proof is may further illuminate our understanding of mathematics.
In any event, there are many more mathematical mysteries challenging—indeed, taunting—the world's best mathematicians to this day. On May 24, 2000, the Clay Mathematics Institute published the Millenium Prize Problems, offering a $1 million prize for the solution to each of seven unsolved mathematical problems. As some of these problems are over 100 years old, the Institute hopes that the prize money, like that of the Wolfskehl Prize, will motivate and inspire current and future mathematicians to work on these seemingly intractable mysteries. Importantly, like the revolution sparked by the quest for the proof of Fermat's Last Theorem, solutions to these problems, should they be found, likely will provide us with a better understanding of the nature of numbers. As Pythagoras (580?-500? b.c.) understood so long ago, the nature of numbers is the language of nature.
MARK H. ALLENBAUGH
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Boyer, Carl B. A History of Mathematics. 2nd ed. New York: John Wiley & Sons, 1991.
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Singh, Simon. Fermat's Enigma. New York: Anchor Books, 1997.