Drink to Me (Carolan, sequenced by Barry Taylor)


Timeline of Fermat's Last Theorem

when who what
1900 BC Babylonians           A clay tablet, now in the museum of Columbia University, called Plimpton 322, contains 15 triples of numbers. They show that a square can be written as the sum of two smaller squares, e.g., 52 = 32 + 42.
circa 530 Pythagoras           Pythagoras was born in Samos. Later he spent 13 years in Babylon, and probably learned the Babylonian's results, now known as the Pythagorean triples.

          Pythagoras was also the founder of a secret society that studied among others "perfect" numbers. A perfect number is one that is the sum of its multiplicative factors. For instance, 6 is a perfect number (6 = 1 + 2 + 3). Pythagoreans also recognized that 2 is an irrational number.

circa 300 BC Euclid of Alexandria           Euclid is best known for his treatise Elements.
circa 400 BC Eudoxus           Eudoxus was born in Cnidos, and became a colleague of Plato. He contributed to the theory of proportions, and invented the "method of exhaustion." This is the same method employed in integral calculus.
circa 250 AD Diophantus of Alexandria           Diophantus wrote Arithmetica, a collection of 130 problems giving numerical solutions, which included the Diophantine equations, equations which allow only integer solutions (e.g, ax + by = c, x2 - Dy2 = 1, x3 + y3 = z3, etc.). Of 13 volumes only six survived and the rest were destroyed in the fire that burned the library of Alexandria. The copy Fermat had was the one translated by Claude Bachet in 1621. Diophantus' Problem 8 in Volume II asks how to divide a given square number into the sum of two smaller squares. Subsequently, this problem inspired Fermat to write his famous Last Theorem.
830 Al-Khowarizmi           Abu Ja'far Mohammed ibn MŻs‚ al-Khow‚rizmÓ (Father of Ja'far, Mohammed, son of Moses, native of the town of Al-Khow‚rizmÓ), a Persian author, had written a book which included the rules of arithmetic, called Kitab al jabr w'al-muqabala (Rules of restoration and reduction) dating from about 825 AD. From the title of this book we derive our modern word algebra.
1225 Fibonacci (or Leonardo) of Pisa           Leonardo (1180-1250) grew up in North Africa under the Moors and later travelled extensively around the Mediterranean coast. He recognized the advantages of the "Hindu-Arabic" system. He is best known for the sequence, the Fibonacci Numbers, which appear in a book he wrote, Liber Abaci. In this sequence each term after the first is obtained by adding together the two preceding numbers: 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, ...

The ratio of two successive numbers in the sequence:

          1/1, 1/2, 2/3, 3/5, 8/13, 13/21, 21/34, 34/55, 55/89, 89/144, 144/233, ...

tends to (5 - 1)/2, which is called the Golden Section.

early 1500s The Cossists           The Arabs called the unknown quantity shai (thing), which is cosa in Italian. These Italian algebraists came to be known as the Cossists (Aczel). Well known Cossists included Geronimo Cardano (1501-1576) and Niccolo Tartaglia (1500-1557).
early 1600s Claude Bachet           Bachet acquired a copy of the Greek Arithmetica of Diophantus, and translated and published it as Diophanti Alexandrini Arithmeticorum Libri Sex in Paris in 1621.
circa 1637 Pierre de Fermat           The French jurist, Fermat (1601-1665), studied Viéte's unpublished writings and studied them carefully. van Schooten published Viéte's Opera Mathematica in 1646.

          Fermat moved to Paris in 1636 and formed a scientific circle around Father Mersenne and Etienne Pascal. Some of his correspondence with this group has been preserved. Fermat published anonymously a dissertation on the rectification of curves Lalouvé published in 1660 as an appendix to his book on the cycloid. J. E. Hoffmann in 1943 also discovered 8 anonymous pages appended to a copy of Frenicle's rare pamphlet of 1657 on Pell's equation and other topics.

          Fermat pondered publication of his work on a few occasions, but he insisted on anonymity because the amount of supervision required to produce an adequate copy.

          Fermat's Last Theorem (FLT) states that nth power of a positive integer cannot be expressed as the sum of nth powers of two smaller positive integers, where n > 2, i.e., there are no positive integers x, y, and z such that

xn + yn = zn.           Fermat jotted down in the margins of his personal copy of Diophantus's Arithmetica that for great n no such triples can be found. He added that he had a marvelous proof for this, but the margin was too small to contain (Alf van der Poorten).

Cubum autem in duos cubos, aut quadrato-quadratum in duos quadrato-quadratos, et generaliter nullam in infinitum ultra quadratum potestatem in duos ejudem nominis fas est dividere; cujus rei demonstrationem mirabilem sane detexi. Hanc marginis exiguitas non caperet.

          Fermat proved his Last Theorem for n = 4, using the method called "infinite descent" to prove that there are no positive integers, a, b, and c such that a4 + b4 = c4. Moreover, if a solution exists for some n, the same solution also works for any multiple of n. Hence, only prime numbers have to be considered. Fermat also proved the theorem for n = 3.

Leonhard Euler (1707-1783)           Euler proved FLT for n = 3 and 4 independently. Euler invented the imaginary number, i, and created a new field, topology.
1801 Carl Friedrich Gauss (1777-1855)           Gauss was born in Brunswick, Germany, and he found no fellow mathematical collaborators and worked alone for most of his life. Gauss published Disquisitiones Arithmeticae. Gauss studied the behavior of functions on the complex plane. Some of these analytic functions, called modular forms, turned out to be crucial to the new approaches to the FLT.
Sophie Germain (1776-1831)           Sophie Germain assumed a man's name, Mr. Leblanc. Sophie Germain's theorem states that if a solution of Fermat's equation for n = 5 existed, all three numbers must be divisible by 5. The theorem divides FLT into two cases: Case I for numbers that are not divisible by 5, and Case II for numbers that are.
1828 Peter G. L. Dirichlet           Dirichlet proved FLT for n = 5, and 14 (1832). He also proposed a modern definition of functions.
1840 Gabriel Lamé and Henri Lebesque           They proved the FLT for the case n = 7.
Joseph Fourier (1768-1830)           In his research on heat, Fourier developed the theory of periodic functions. Such a series is called a Fourier Series. He discovered that most functions can be estimated to any degree of accuracy by the sum of many sine and cosine functions. Subsequently, Fourier series plays an important role as the tool for transforming mathematical elements from one area to another in the work of Goro Shimura.
1839, 1847 Gabriel Lamé (1795-1870)           Lamé proved FLT for n = 7 in 1839. Subsequently, he suggested a general approach to the problem and factored the left side of Fermat's equation, xn + yn, into linear factors using complex numbers, but the factorization he suggested was not unique, and hence there was no solution.
Ernst Eduard Kummer (1810-1893)           Kummer attempted to restore the uniqueness of factorization by introducing 'ideal' numbers. Kummer proved that FLT was true for an infinite number of exponents, those that are divisible by "regular" primes. As a result, FLT was known to be true for all exponents less than 100.
1825, 1829 Janos Bolyai (1802-1860)
Ivanovitch Lobachevsky (1793-1856)
          Bolyai developed non-Euclidean geometry. He published his strange new world as a 24 page appendix to his father's book. Lobachevsky published a similar work.
1846 Évariste Galois (1811-1831)           Galois is known for his contributions to group theory. He produced a method of determining when a general equation could be solved by radicals. He jotted down his theory the night before his duel and his paper was sent to his friend Auguste Chevalier. Joseph Liouville published his work in 1846.
1824 Niels Henrik Abel (1802-1829)           In 1824 Abel gave the first accepted proof of the insolubility of the quintic. Abelian group is a crucial element in the modern treatment of the Fermat problem. It is a group where the order of mathematical operations can be reversed without affecting the outcome.
Richard Dedekind (1831-1916)           Dedekind introduced the notion of an ideal which is fundamental to ring theory. Subsequently, ideals were destined to inspire Barry Mazur, and Andrew Wiles would utilize Mazur's work.
Henri Poincaré (1854-1912)           It is said that Poincaré was the originator of algebraic topology. He elevated the status of topology with his publication of Analysis Situs. He studied periodic functions in the complex plane. These were functions that remained unchanged when the complex variable z was changed according to f(z) ó> f(az+b/cz+d), and (a,b,c,d) arranged as a matrix formed an algebraic group. These were called automorphic forms.

          Poincaré then extended them to modular forms. These modular forms reside on the upper half of the complex plane with a hyperbolic geometry. In this space, the non-Euclidean geometry of Bolyai and Lobachevsky rules and Euclid's fifth postulate does not hold (which states that given a line and a point not on the line, a unique line can be drawn through the point parallel to the given line.)

1922 Louis J. Mordell           Mordell discovered the connection between the solutions of algebraic equations and topology.

          Two-dimensional surfaces in three-dimensional space can be classified according to their genus, which is the number of holes in the surface. For example, the genus of a doughnut or a (wedding) ring is one. If the surface of solutions had two or more holes, then the equations had only finitely many whole number solutions, and this came to be known as Mordell's conjecture.

1955 Yutaka Taniyama (1927-1958)
Goro Shimura
          Taniyama and Shimura helped organize the Tokyo-Nikko Symposium on Algebraic Number Theory. André accepted the invitation to attend the symposium. Jean Pierre Serre also attended. Taniyama's problem constituted a conjecture about zeta functions. He seemed to connect Poincaré's automorphic functions of the complex plane with the zeta function of an elliptic curve. It was an intuition, a gut feeling that the automorphic functions with symmetries on the complex plane were somehow connected with the equations of Diophantus.

          Taniyama suggested that automorphic functions are associated with the elliptic curves, whereas Weil did not believe that there was such a connection in general.

early 1960s Goro Shimura           Shimura conjectured that every elliptic curve over the rational numbers is uniformized by a modular form. Shimura declared his conjecture that an elliptic curve should always be uniformized by a modular curve, but André Weil did not believe Shimura conjecture.

          The conjecture became misquoted as Weil-Taniyama conjecture instead of Shimura- Taniyama conjecture. Weil showed reluctance to refer to Shimura. Even in 1967, in a paper written in German, Weil did not attribute the theory to its originator, Shimura.

early 1970s André Weil           Since Weil had written about the conjecture, modular elliptic curves became know as "Weil curves." After Taniyama's problems became known in the West, the conjecture came to be called erroneously the "Taniyama-Weil" conjecture, and Shimura's name was left out. According to Aczel, even in 1979, Weil spoke against the 'Mordell conjecture' on Diophantine equations.
1977 Barry Mazur           Mazur's paper on the Eisenstein Ideal suggests that it is possible to switch from one set of elliptic curves to another. This paper implies, for instance, that it is possible to transform a problem with elliptic curves based on the prime number 3 to another using the prime number 5.
1983 Gerd Faltings           Faltings proved the Mordell conjecture. Since the genus of the Fermat equation for n > 3 was 2 or more, it became evident that the integer solutions to the Fermat equation were finite, if they existed at all. Granville and Heath-Brown further showed that the number of solutions of Fermat's equation, if they existed, decreased at the exponent n increased.

          The theorem was proved for n up to a million in 1983. For larger n, the solutions were very few and decreasing with n, if they existed at all (Aczel).

1984 Gerhard Frey           Frey gave a talk at a number theory conference in Oberwolfach. His paper seemed to imply that if the Shimura-Taniyama conjecture were true, FLT would be proved. Frey's reasoning:

          Suppose that Fermat's Last Theorem is not true. Then for some power n > 2, there is an integer solution to Fermat's equation, (x,y,z) = (a,b,c). This particular solution results in a specific elliptic curve, now called Frey curve, was very strange, and definitely not modular. However, if Shimura-Taniyama conjecture were true, an elliptic curve that was not modular could not exist. Thus, Frey's curve, an elliptic curve that was not modular could not exist, and hence the solutions to Fermat's equation could not exist either. This is called the Frey conjecture. (Aczel, 1996, p. 112)

1985 Kenneth Ribet           Ribet proved a theorem that establishes that if the Shimura-Taniyama conjecture was true, FLT necessarily follows as a direct consequence. However, the Shimura-Taniyama conjecture must be proved.
1987 Andrew Wiles           Wiles tried to show that the number of elliptic curves and that of modular elliptic curves are the same. He limited the Shimura-Taniyama conjecture to the case of semistable elliptic curves with rational numbers as coefficients. Wiles then tried to count sets of Galois representations associated with the semi-stable elliptic curves, thereby showing that they and modular forms are the same.

          In 1993, with the help of Nick Katz, Wiles began to teach a course "Calculations with Elliptic Curves."

1993 Andrew Wiles           Wiles had already proved that elliptic curves based on five were modular. Wiles wrote up 200 pages to prove FLT. Wiles also worried about the refereeing process in journals, and presented it at a conference. The paper was sent to a number of leading experts.
1993 Nick Katz           Katz found a flaw in the proof; there were no Euler System.
1993 Richard Taylor           Taylor, a former student of Wiles, joined him to salvage the proof.
1993 Andrew Wiles           Wiles wrote up his proof using the corrected Horizontal Iwasawa Theory approach.
1995 Andrew Wiles           May 1995 issue of the Annals of Mathematics published Wiles' original Cambridge paper and the correction by Taylor and Wiles.

References
Amir D. Aczel, Fermat's Last Theorem: Unlocking the Secret of Ancient Mathematical Problem, Four Walls Eight Windows, New York, October 1996. (a delightful book!)
E. Kwan Choi, "Fermat's Last TheoremóWas It a Right Question?", October 1998.
Alex Lopez-Ortiz, Fermat's Last Theorem, February 20, 1998.
MacTutor History of Mathematics Archive, Fermat's Last Theorem. Alf van der Poorten, Notes on Fermat's Last Theorem, Canadian Mathematical Society Series of Monographs and Advanced Texts, John Wiley & Sons, Inc., New York, 1995.
André Weil, Number Theory, An approach through history, From Hammurapi to Legendre, Birkhäuser, Boston, 1983.

Acknowledgment
Some statements are quoted from Amir D. Aczel, Fermat's Last Theorem, © Four Walls Eight Windows, permission to quote from Four Walls Eight Windows.