Fermat’s Last Theorem

Not only that the case ##n=2## not generalize to the case ##n=4,## we even made wide use of solutions for ##n=2## to disprove the existence of solutions for ##n=4.## Assuming a minimal solution, finding an even smaller solution, and thus generating a contradiction, is a standard proof technique in the domain of natural numbers since they are bounded from below. A geometric proof can be found in [2]. It shifts the cumbersome zoo of variables into the statement that the area of a right triangle with integer side lengths is never a square number, and the useful observation that the integer equation system
$$
x^2+y^2=z^2\text{ and }x^2-y^2=w^2
$$
cannot be solved. It is the proof that is assumed to be known to Fermat [4].

The proof for the case ##n=3## is even more complicated but basically along the same lines of assuming a minimal solution, considering congruences of now Eisenstein numbers, i.e. numbers of the form ##\mathbb{Z}+\mathbb{Z}\cdot e^{2\pi i/3}##, and divisibility; for details, see [1]. Leonhard Euler (1707-1783) published proofs for both cases. For the case ##n = 4##, at least ##20## different proofs have been found. At least ##14## different proofs exist for ##n = 3.##

$$
a^{rs}+b^{rs}=c^{rs} \Longleftrightarrow (a^r)^s+(b^r)^s=(c^r)^s
$$
So if there are no solutions for ##n=s## then there are no solutions for ##n=rs.## Hence, it is sufficient to show that there are no solutions for ##s=4## and all odd primes ##s## because every positive integer greater than two has either an odd prime as a divisor or is divisible by four. This leaves us with the case of odd primes ##n=p.##

In 1825, Adrien-Marie Legendre (1752–1833) and, independently of him, the young Peter Gustav Lejeune Dirichlet (1805–1859) succeeded in proving the Fermat conjecture for the exponent ##n = 5.## In 1839, Gabriel Lamé (1795–1870) followed with a proof for the exponent ##n = 7.##

The Three Worlds

We have solved the cases ##n=2,4,## and the case ##n=3## already required the introduction of certain concepts of abstract algebra, Eisenstein numbers.

The hope expressed by Cauchy and Lamé in 1847 for a quick (and general) proof was dashed, however, by Ernst Eduard Kummer, who discovered a logical error in Lamé’s and Cauchy’s considerations: They had tacitly, and falsely assumed that in the entire closure of integers of the extension fields of rational numbers they considered (cyclotomic fields of order ##p,## which arises from the adjunction of the ##p##-th roots of unity), the unique prime factorization still holds.

Kummer developed a theory in which the unique prime factorization could be salvaged by combining certain sets of numbers from the number field (ideals) and investigating the arithmetic of these new “ideal numbers.” He was thus able to prove Fermat’s Last Theorem for regular prime numbers in 1846. A prime number ##p## is called regular if no numerator of the Bernoulli numbers ##B_0, B_2,\ldots B_{p-3}## is divisible by ##p.## In this case, the class number, i.e., the number of non-equivalent ideal classes of the cyclotomic field of order ##p## is not divisible by ##p.## However, it is not known whether there are infinitely many regular prime numbers. With the help of computers and by further developing Kummer’s methods, Harry Vandiver succeeded in proving the theorem for all prime numbers less than 2,000 in the early 1950s. [5]

Thus, until the beginning of the 1980s, it remained essentially a matter of refinements to Kummer’s work, and after computer technology had improved more and more, numerical verifications of the Fermat conjecture; for example, in 1976, Wagstaff proved that Fermat’s conjecture was correct for prime exponents smaller than 125,000. [3]

This demonstrates several phenomena that are typical of Fermat’s Last Theorem. First, even the greatest mathematicians sometimes make mistakes, and even Wiles’s first draft of the proof was flawed. Second, one suddenly and quickly finds oneself in algebraic realms where new concepts are required and specialized terminology is indispensable, even for partial results.

We will briefly outline this complexity along Kramer’s presentation in [3]. He coined the term of the three worlds to describe the progress in different areas, trying to prove Fermat’s Last Theorem. In a way, we already met them: the non-existence character of the statement, elliptic curves like ##y^2=x^3-1## that lead to ##e^{2\pi i /3},## and generalized divisibility considerations, modules.

The Anti-Fermat World …

… outlines the path to a proof. We want to show that there is no solution, and the only way we can do this is to assume the existence of a solution and derive a contradiction; simply because we cannot test all infinitely many possibilities. We have seen that we can make a few assumptions, such as coprime ##a,b,c,## and ##n=p## being an odd prime greater than ##7,## and eventually impose some kind of minimality of a solution due to the lower bound of positive integers. Many partial results for certain exponents and certain bases have been found, e.g., in the case where ##p\nmid a,b,c.## Subclasses of the problem statement simply mean that people used additional properties and therewith additional constraints and equations to approach the problem. However, the union of all these results never exhausted the entire problem within 350 years of attempts prior to Wiles’ proof. The union of all subclasses remained a proper subset of all cases. It should be noted that the exhaustion method is not an alternative to the contradiction method. In fact, it is merely a different perspective since the contradiction method does the same: excluding cases so that more equations can be used to finally arrive at a contradiction. The case ##n=4## was such an example since it widely used the primitivity of Pythagorean triples. The restriction to odd primes is another such condition.

The Elliptic World

Every odd prime number is congruent ##1## or congruent ##3## modulo ##4.## This led already for ##n=3## into the consideration of elliptic curves, i.e., algebraic curves of the form
$$
Y^2=X^3+A X+B
$$
with integer coefficients ##A,B.## It turned out that the projective version of elliptic curves, i.e., the extension by the infinite points of these curves, with three pairwise different zeros of the cubic polynomial, their continuous deformations, and their images modulo ##p## played a central role in the examination of Fermat’s equation. It also automatically involves complex numbers. A big part of Wiles’ proof, and its prior partial results, deals with invariants of such algebraic varieties.

Elliptic curves occur naturally in the ancient congruence problem: Find to a given natural number ##M## all right triangles with rational side lengths and area ##M.## The solution to this problem uses the elliptic curve
$$
Y^2=X^3-C^2X=X(X^2-C^2)=X(X-C)(X+C)
$$
and results in ##a=\dfrac{X^2-C^2}{Y}\, , \,b=\dfrac{2CX}{Y}\, , \,c=\dfrac{X^2+C^2}{Y}.##

The similarity to Pythagorean triples is no coincidence.

The Modular World …

… deals with modular forms and moduli of algebraic curves. A modular form ##f## of weight ##k\in \mathbb{Z}## and level ##N## is a complex, meromorphic function of the upper complex plane ##\mathcal{H}=\left\{z\in \mathbb{C}\,|\,\operatorname{Im}(z)\geq 0\right\},## such that
$$
f\left(\dfrac{\alpha z+\beta}{\gamma z+\delta}\right)=(\gamma z+\delta)^kf(z)
$$
for all elements of the subgroup
$$
\Gamma(N)=\left\{ \left. \begin{pmatrix}\alpha&\beta\\\gamma&\delta\end{pmatrix}\in \operatorname{SL}_2(\mathbb{Z})\, \right| \, \gamma\equiv 0\pmod{N}\right\} \leq \operatorname{SL}_2(\mathbb{Z}),
$$
and that has a representation as
$$
f(z)=\sum_{m=0}^\infty f_me^{2\pi i mz}.
$$
A moduli space of algebraic curves in algebraic geometry is a geometric space (typically a scheme or an algebraic stack) whose points represent isomorphism classes of algebraic curves. Short: It is about topological surfaces and their genus, the number of their holes or handles. [7]

The modular world leads us directly into algebraic geometry with its own terminology, and we could ask what this has to do with Fermat’s Last Theorem. A look into the history of modular forms gives us a first insight.

The origins of the theory of modular forms go back to Carl Friedrich Gauß (1777-1855), who considered transformations of special modular forms under the modular group, the level in the context of his theory of the arithmetic-geometric mean in complex systems (1805). The founders of the classical (purely analytical) theory of modular forms in the 19th century are Richard Dedekind, Felix Klein, Leopold Kronecker, Karl Weierstrass, Carl Gustav Jacobi, Gotthold Eisenstein, and Henri Poincaré. A well-known example of the application of modular forms in number theory was Jacobi’s theorem about the representations of a number by four squares. The modern theory of modular forms emerged in the first half of the 20th century by Erich Hecke and Carl Ludwig Siegel, who pursued applications in number theory. [8]

The Bridges Between The Worlds

The proof of Fermat’s Last Theorem is now the path from one world to the next one, setting up a mathematical environment that provides tools, constraints, and further equations to finally derive a contradiction to an assumed solution. The transition from Fermat’s statement into an algebraic and topological environment turned out to be the key that embedded four numbers ##a,b,c,n## into a mathematical framework. We therefore need bridges between these worlds. The first bridge can be attributed to Gerhard Frey [9], the second bridge is Andrew John Wiles’ [10] great achievement.

We assume a solution to the equation
$$
a^p+b^p=c^p
$$
with coprime positive integers ##a,b,c,## and an odd prime ##p>5.## We next define an elliptic curve ##F##, a Frey curve with these data by
$$
F\, : \,Y^2=X\left(X-a^p\right)\left(X+b^p\right)=X^3+\left(b^p-a^p\right)X^2-(ab)^pX.
$$
Note that ##F## is an elliptic curve since the transformation ##X’=X+(b^p-a^p)/3## eliminates the quadratic term and ##F## is of the required form. Now it can be shown that
$$
N_F=2\prod_{\substack{p \text{ prime}\\ p\,|\,abc\, , \,p\neq 2}} p
$$
is an invariant of the Frey curve. Every Frey curve allows the definition of a modular form ##f_F## of level ##N_F## depending on its prime number ##p,## see definition 2.9 in [2] for details. Frey supposed in 1986, and Ken Ribet [11] has proven it in the same year, that the Frey curve would provide a counter-example to the Taniyama-Shimura conjecture about modular forms from 1958. The conjecture was proven in 2001 by Richard Taylor [12] and others after Wiles had provided the proof of the most important and difficult case of so-called semistable elliptic curves in 1994. This is our second bridge, the Taniyama-Shimura conjecture, now known as Modularity Theorem.

The Modularity Theorem and its proof are considered one of the great mathematical advances of the 20th century. Consequences of the modularity theorem include Fermat’s Last Theorem and the well-defined nature of the Birch-Swinnerton-Dyer conjecture, since the modularity theorem guarantees an analytic continuation of the L-function associated with an elliptic curve. Today, the Modularity Theorem is considered a special case of the much more general and important Serre conjecture on Galois representations. This conjecture was proven in 2006 by Chandrashekhar Khare, Jean-Pierre Wintenberger, and Mark Kisin, building on the work of Andrew Wiles. [13]

Modularity Theorem

If ##F## is a Frey curve and ##f_F## its modular form of weight ##2## and level ##N_F## then

1. $$
   f_F\left(\dfrac{\alpha z+\beta}{\gamma z+\delta}\right)=\left(\gamma z+\delta\right)^2 f_F\left(z\right)\text{ for all }\begin{pmatrix}\alpha&\beta\\\gamma&\delta\end{pmatrix}\in \Gamma(N_F)
   $$
2. $$
   f_F\left(-\,\dfrac{1}{N_F \,z}\right)=\pm N_F \,z^2\, f_F(z)
   $$

holds for all ##z\in \mathcal{H}.##

Summary

There are essentially three things that are remarkable about Fermat’s Last Theorem. It took about 350 years for Wiles and Taylor to prove it finally, and shortly after Wiles’s publication, rumors circulated that at most a handful of other mathematicians worldwide could even understand the proof. Well, that may have changed. Nevertheless, countless mathematicians have striven for a proof over those 350 years. Quite a few of them were of the stature of brilliant scientists, such as Euler, Cauchy, Legendre, Dirichlet, Kummer, and finally Wiles. Until the late 20th century, partial results were achieved and published, not least thanks to the development of increasingly powerful computers. Moreover, the further development of many mathematical concepts and areas was required before a proof could be provided. And this form of mathematics is anything but simple and easily understandable. So if someone today believes they have found a “simple” proof, they would have achieved what mathematical luminaries have failed to achieve in 350 years. He would have also found a “simple” approach to all the profound results which were necessary to even understand the proof we have, or to obtain partial results. These three things – time, geniuses, and new mathematics – all of which were necessary to ultimately provide a proof would all seem unnecessary if someone were to find a “simple” proof. Such a scenario is extremely unlikely.

Sources