Fermat’s Last Theorem

Fermat’s Last Theorem, classically referenced as Fermat’s conjecture, states that the equation

xⁿ + yⁿ = zⁿ, where xyz ≠ 0,

has no integer solutions when n > 2. The theorem was written in the margin of Pierre de Fermat’s copy of Arithmetica, an ancient Greek text on mathematics, in 1637. Fermat’s famous conjecture is known as a Diophantine equation, and there are infinite solutions to the equation where n = 1 and n = 2. The solution set for n = 2 are referred to as Pythagorean triples. However, a general proof for n > 2 remained unsolved for more than 350 years. Sir Andrew Wiles provided a proof of Fermat’s Last Theorem in 1995. Wiles’s proof uses the modularity theorem for semistable elliptical curves, which, together with the epsilon conjecture, or Ribet’s theorem, implied that Fermat’s conjecture is true. Fermat’s Last Theorem stimulated additional areas of mathematical study, most notably algebraic number theory and a proof of the modularity theorem.

Background

Pierre de Fermat wrote the conjecture that came to be famously known as Fermat’s Last Theorem in a copy of Arithmetica in 1637. Diophantus of Alexandria, the author of Arithmetica, published the ancient text of problems in elementary number theory. In general, the problems are now known as Diophantine equations. Fermat noted that his proof of a Diophantine equation was too large to fit in the margins of his copy of Diophantus’s text. In Latin, Fermat’s statement reads

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

When translated, in English, reads

• It is impossible for a cube to be the sum of two cubes, a fourth power to be the sum of two fourth powers, or in general for any number that is a power greater than the second to be the sum of two like powers. I have discovered a truly marvelous demonstration of this proposition that this margin is too narrow to contain.

The relation described by Fermat is known as a Diophantine equation. Diophantine equations are a set of algebraic equations in which only integer solutions are studied. Integers are numbers that can be written without a fractional component. These equations generally contain two or more unknowns and are solved using Diophantine approximation. The sets from which solutions to Diophantine equations are drawn include the number zero, the set of natural numbers, and their additive inverses. Fermat claimed that there were no solutions to his conjecture.

The seemingly simple configuration of Fermat’s conjecture made it a standard and especially famous problem in the field of mathematics. The cases where n = 0 and n = 1 are known as linear Diophantine equations. The general equation for Fermat’s Last Theorem is based on the case of n = 2, also known as the Pythagorean theorem. Pythagoras’s theorem is the Diophantine equivalence relation for all right triangles with integer side-lengths. The equation

a² + b² = c²

has an infinite solution set, known as Pythagorean triples, where c is the length of the hypotenuse and a and b are the lengths of the triangle’s legs. Fermat claimed that no solutions existed when n > 2. The only case for Fermat’s Last Theorem in which Fermat provided a written solution was for n = 4. In his solution to this case, Fermat introduced a method that is widely used in Diophantine analyses. His method is known as proof by infinite descent.The proof depends on the well-ordered nature of the natural numbers. Fermat’s proof by infinite descent demonstrates that the given equation had no solutions and it is a proof by contradiction.

Fermat’s proof for the case of n = 4 made it sufficient to prove those cases in which n is an odd prime. In 1753, Leonard Euler provided a proof for n = 3. Subsequent solutions for n = 5 and n = 7, and a two-case method to understand Fermat’s Last Theorem, as noted by Sophie Germain in 1820, suggested that more advanced theoretical lenses would be required. Gerhard Frey’s research in number theory was central to Wiles’s later proof, as it generated a set of special elliptical curves. The Frey curve was an elliptical curve construction associated with a solution to Fermat’s equation. One primary insight of Frey curves was that they could not be modular. The Taniyama-Shimura conjecture states that these elliptical curves over the set of rational numbers are modular and, as true, shows that Frey curves cannot exist. The proof to Fermat’s conjecture would naturally follow from this modularity theorem, formerly termed the Taniyama-Shimura-Weil conjecture, which was proposed around 1955. Frey, Jean-Pierre Serre, and Ken Ribet made the connections between the modularity theorem and Fermat’s conjecture during the 1980s. Wiles’s proof of the modularity theorem for semistable elliptical curves, the proof that implied Fermat’s conjecture, was published in 1995.

Impact

Fermat’s Last Theorem contributed to many advances in the field of mathematics. Historians have separated the string of mathematical developments and influences related to Fermat’s conjecture into periods. One such distinction divides impacts from Diophantus to Euler, then Euler to Frey, and finally from Frey to Wiles. The primary impact of Fermat’s Last Theorem is situated within the latter two periods, extending from Euler to Wiles. Subsequent analyses leading to the proof were developed during the 350-year period after Fermat’s theorem was originally published. These analyses informed the development of additional research areas across the discipline. The unresolved problem encouraged the development of algebraic number theory and a full proof of the modularity theorem in the twentieth century.

Bibliography

Cai, Tianxin, Deyi Chen, and Yong Zhang. "A New Generalization of Fermat’s Last Theorem." Journal of Number Theory 149 (2015): 33–45. Print.

Cox, David. "Introduction to Fermat’s Last Theorem," American Mathematical Monthly, 101.1 (1994): 3–4. Print.

Devlin, Keith. The Unfinished Game: Pascal, Fermat, and the Seventeenth-Century Letter that Made the World Modern. New York: Basic Books, 2008. Print.

Freitas, Nuno, and Samir Siksek. "The Asymptotic Fermat’s Last Theorem for Five-Sixths of Real Quadratic Fields." Compositio Mathematica 151.8 (2015): 1–21. Print.

Krantz, Steven G., and Harold R. Parks. "Fermat’s Last Theorem." A Mathematical Odyssey: Journey from the Real to the Complex. New York: Springer US, 2014. 309–38. Print

Nova Online. The Proof. Public Broadcasting Services. N.d. Web. 30 Dec. 2015. www.pbs.org/wgbh/nova/proof.

Saito, Takeshi. Fermat’s Last Theorem: The Proof. Providence: American Mathematical Soc., 2014. Print. Translations of Mathematical Monographs 245.

Singh, Simon L. Fermat's Enigma. New York: Anchor Books, 1998. Print.