Fermat's Last Theorem

In number theory, Fermat's Last Theorem (sometimes called Fermat's conjecture, especially in older texts) states that there are no positive integers \(a, b, c, n\) with \(n > 2\) such that \(a^n + b^n = c^n\). The cases \(n=1\) and \(n=2\) have been known since antiquity to have infinitely many solutions.

The proposition was first stated as a theorem by Pierre de Fermat around 1637 in the margin of a copy of Arithmetica. Fermat added that he had a proof that was too large to fit in the margin. Although other statements claimed by Fermat without proof were subsequently proven by others and credited as theorems of Fermat (for example, Fermat's theorem on sums of two squares), Fermat's Last Theorem resisted proof, leading to doubt that Fermat ever had a correct proof. Consequently, the proposition became known as a conjecture rather than a theorem. After 358 years of effort by mathematicians, the first successful proof was released in 1994 by Andrew Wiles and formally published in 1995. It was described as a "stunning advance" in the citation for Wiles's Abel Prize award in 2016. It also proved much of the Taniyama-Shimura conjecture, subsequently known as the modularity theorem, and opened up entire new approaches to numerous other problems and mathematically powerful modularity lifting techniques.

The unsolved problem stimulated the development of algebraic number theory in the 19th and 20th centuries. For its influence within mathematics and in culture more broadly, it is among the most notable theorems in the history of mathematics.

The Pythagorean equation, \(x^2+y^2=z^2\), has an infinite number of positive integer solutions for \(x\), \(y\), and \(z\); these solutions are known as Pythagorean triples (with the simplest example being 3, 4, 5). Around 1637, Fermat wrote in the margin of a book that the more general equation \(a^n+b^n=c^n\) had no solutions in positive integers if \(n\) is an integer greater than 2. Although he claimed to have a general proof of his conjecture, Fermat left no details of his proof, and none has ever been found. His claim was discovered some 30 years later, after his death. This claim, which came to be known as Fermat's Last Theorem, stood unsolved for the next three and a half centuries and was solved with mathematics Fermat would not have known.

The claim eventually became one of the most notable unsolved problems of mathematics. Attempts to prove it prompted substantial development in number theory, and over time Fermat's Last Theorem gained prominence as an unsolved problem in mathematics.

The special case \(n=4\), proved by Fermat himself, is sufficient to establish that if the theorem is false for some exponent \(n\) that is not a prime number, it must also be false for some smaller \(n\), so only prime values of \(n\) need further investigation. Over the next two centuries (1637-1839), the conjecture was proved for only the primes 3, 5, and 7, although Sophie Germain innovated and proved an approach that was relevant to an entire class of primes. In the mid-19th century, Ernst Kummer extended this and proved the theorem for all regular primes, leaving irregular primes to be analyzed individually. Building on Kummer's work and using sophisticated computer studies, other mathematicians were able to extend the proof to cover all prime exponents up to four million, but a proof for all exponents was considered exceedingly difficult or unachievable with the knowledge of the time.

Around 1955, Japanese mathematicians Goro Shimura and Yutaka Taniyama suspected a link might exist between elliptic curves and modular forms, two completely different areas of mathematics. Known at the time as the Taniyama-Shimura conjecture, it had no apparent connection to Fermat's Last Theorem. It was widely seen as significant and important in its own right, but was (like Fermat's theorem) considered completely inaccessible to proof.

In 1984, Gerhard Frey noticed an apparent link between these two previously unrelated and unsolved problems, and he gave an outline suggesting this could be proved. The full proof that the two problems were closely linked was accomplished in 1986 by Ken Ribet, building on a partial proof by Jean-Pierre Serre, who proved all but one part known as the "epsilon conjecture" (see: Ribet's Theorem and Frey curve). These papers by Frey, Serre and Ribet showed that if the Taniyama-Shimura conjecture could be proven for at least the semi-stable class of elliptic curves, a proof of Fermat's Last Theorem would also follow automatically. The connection is described below: any solution that could contradict Fermat's Last Theorem could also be used to contradict the Taniyama-Shimura conjecture. So if the Taniyama-Shimura conjecture were found to be true, then no solution contradicting Fermat's Last Theorem could exist, meaning that Fermat's Last Theorem must also be true.

Although both problems were daunting and widely considered to be "completely inaccessible" to proof at the time, this was the first suggestion of a route by which Fermat's Last Theorem could be extended and proved for all numbers, not just some numbers. Unlike Fermat's Last Theorem, the Taniyama-Shimura conjecture was a major active research area and viewed as more within reach of contemporary mathematics. However, general opinion was that this simply showed the impracticality of proving the Taniyama-Shimura conjecture. Mathematician John Coates' quoted reaction was a common one:

Condensed: the full section is in Wikipedia.

There are several alternative ways to state Fermat's Last Theorem that are mathematically equivalent to the original statement of the problem.

In order to state them, we use the following notations: let \(\N\) be the set of natural numbers \(1,2,3,\dots,\) let \(\Z\) be the set of integers \(0,\pm 1,\pm 2,\dots,\) and let \(\Q\) be the set of rational numbers \(a/b\), where \(a\) and \(b\) are in \(\Z\) with \(b \neq 0\). In what follows we will call a solution to \(x^n+y^n=z^n\) where one or more of \(x\), \(y\), or \(z\) is zero a trivial solution. A solution where all three are nonzero will be called a non-trivial solution.

For comparison's sake we start with the original formulation.

• Original statement. With \(n,x,y,z\in\N\) (meaning that \(n,x,y,z\) are all positive whole numbers) and \(n>2\), the equation \(x^n+y^n=z^n\) has no solutions.

Most popular treatments of the subject state it this way, though it is sometimes stated over \(\Z\):

• Equivalent statement 1: x + y = z, where \(n\geq 3\), has no non-trivial solutions \(x, y, z \in \Z\).

The equivalence is clear if n is even. If n is odd and all three of x, y, z are negative, then we can replace x, y, z with −x, −y, −z to obtain a solution in N. If two of them are negative, it must be x and z or y and z. If x, z are negative and y is positive, then we can rearrange to get (−z) + y = (−x) resulting in a solution in N; the other case is dealt with analogously. Now if just one is negative, it must be x or y. If x is negative, and y and z are positive, then it can be rearranged to get (−x) + z = y again resulting in a solution in N; if y is negative, the result follows symmetrically. Thus in all cases a nontrivial solution in Z would also mean a solution exists in N, the original formulation of the problem.

• Equivalent statement 2: x + y = z, where integer n ≥ 3, has no non-trivial solutions x, y, z ∈ Q.

This is because the exponents of x, y, and z are equal (to n), so if there is a solution in Q, then it can be multiplied through by an appropriate common denominator to get a solution in Z, and hence in N.

• Equivalent statement 3: x + y = 1, where integer n ≥ 3, has no non-trivial solutions x, y ∈ Q.

A non-trivial solution a, b, c ∈ Z to x + y = z yields the non-trivial solution a/c, b/c ∈ Q for v + w = 1. Conversely, a solution a/b, c/d ∈ Q to v + w = 1 yields the non-trivial solution ad, cb, bd for x + y = z.

• Equivalent statement 4, reduction to prime exponents: If ⁠\(n=4\)⁠ or ⁠\(n\)⁠ is an odd prime number, then x + y = z has no non-trivial solutions \(x, y, z \in \Z.\)

• Equivalent statement 5, connection to elliptic curves: If a, b, c is a non-trivial solution to a + b = c, p odd prime, then y = x(x − a)(x + b) (Frey curve) will be an elliptic curve without a modular form.

Condensed: the full section is in Wikipedia.

The cubic Fermat equation x + y = z was studied by Abu-Mahmud Khujandi (c. 940-1000), in the period of mathematics in the medieval Islamic world. He claimed to have proven it to have no integer or rational number solutions, but his proof is now considered faulty. Another mathematician from later in the medieval Islamic period, Abdallah ibn Muhhammad ibn Abd al-Razzāq Ibn al-Khawwām (1243: after 1324), a student of Nasir al-Din al-Tusi, published a list of open problems among which he included both the degree 3 and degree 4 cases of Fermat's last theorem.

A modern algebraic reformulation of Problem II.8 of the Arithmetica asks how a given squared rational number is split into two other squares of rational numbers; in other words, for a given rational number k, it asks to find rational numbers u and v such that k = u + v. Diophantus shows how to solve this sum-of-squares problem for k = 4, the solutions being u = 16/5 and v = 12/5.

Around 1637, Fermat wrote his Last Theorem in Latin in the margin of his copy of the Arithmetica next to Diophantus's sum-of-squares problem:

After Fermat's death in 1665, his son Clément-Samuel Fermat produced a new edition of the book (1670) augmented with his father's comments. Although not actually a theorem at the time (meaning a mathematical statement for which proof exists), the marginal note became known over time as Fermat's Last Theorem, as it was the last of Fermat's asserted theorems to remain unproved.

It is not known whether Fermat had actually found a valid proof for all exponents n, but it appears unlikely. Only one related proof by him has survived, namely for the case n = 4, as described in the section § Proofs for specific exponents.

While Fermat posed the cases of n = 4 and of n = 3 as challenges to his mathematical correspondents, such as Marin Mersenne, Blaise Pascal, and John Wallis, he never posed the general case. Moreover, in the last thirty years of his life, Fermat never again wrote of his "truly marvelous proof" of the general case, and never published it. Van der Poorten suggests that while the absence of a proof is insignificant, the lack of challenges means Fermat realised he did not have a proof; he quotes Weil as saying Fermat must have briefly deluded himself with an irretrievable idea. The techniques Fermat might have used in such a "marvelous proof" are unknown.

Wiles and Taylor's proof relies on 20th-century techniques. Fermat's proof would have had to be elementary by comparison, given the mathematical knowledge of his time.

While Harvey Friedman's grand conjecture implies that any provable theorem (including Fermat's last theorem) can be proved using only elementary function arithmetic, such a proof need be 'elementary' only in a technical sense and could involve millions of steps, and thus be far too long to have been Fermat's proof.

The strategy that ultimately led to a successful proof of Fermat's Last Theorem arose from the "astounding" Taniyama-Shimura-Weil conjecture, proposed around 1955, which many mathematicians believed would be near to impossible to prove, and was linked in the 1980s by Gerhard Frey, Jean-Pierre Serre and Ken Ribet to Fermat's equation. By accomplishing a partial proof of this conjecture in 1994, Andrew Wiles ultimately succeeded in proving Fermat's Last Theorem, as well as leading the way to a full proof by others of what is now known as the modularity theorem.

Ribet's proof of the epsilon conjecture in 1986 accomplished the first of the two goals proposed by Frey. Upon hearing of Ribet's success, Andrew Wiles, an English mathematician with a childhood fascination with Fermat's Last Theorem, and who had worked on elliptic curves, decided to commit himself to accomplishing the second half: proving a special case of the modularity theorem (then known as the Taniyama-Shimura conjecture) for semistable elliptic curves.

Wiles worked on that task for six years in near-total secrecy, covering up his efforts by releasing prior work in small segments as separate papers and confiding only in his wife. His initial study suggested proof by induction, and he based his initial work and first significant breakthrough on Galois theory before switching to an attempt to extend horizontal Iwasawa theory for the inductive argument around 1990-91 when it seemed that there was no existing approach adequate to the problem. However, by mid-1991, Iwasawa theory also seemed to not be reaching the central issues in the problem. In response, he approached colleagues to seek out any hints of cutting-edge research and new techniques, and discovered an Euler system recently developed by Victor Kolyvagin and Matthias Flach that seemed "tailor made" for the inductive part of his proof. Wiles studied and extended this approach, which worked. Since his work relied extensively on this approach, which was new to mathematics and to Wiles, in January 1993 he asked his Princeton colleague, Nick Katz, to help him check his reasoning for subtle errors. Their conclusion at the time was that the techniques Wiles used seemed to work correctly.

By mid-May 1993, Wiles was ready to tell his wife he thought he had solved the proof of Fermat's Last Theorem, and by June he felt sufficiently confident to present his results in three lectures delivered on 21-23 June 1993 at the Isaac Newton Institute for Mathematical Sciences. Specifically, Wiles presented his proof of the Taniyama-Shimura conjecture for semistable elliptic curves; together with Ribet's proof of the epsilon conjecture, this implied Fermat's Last Theorem. However, it became apparent during peer review that a critical point in the proof was incorrect. It contained an error in a bound on the order of a particular group. The error was caught by several mathematicians refereeing Wiles's manuscript including Katz (in his role as reviewer), who alerted Wiles on 23 August 1993.

, Andrew Wiles, as quoted by Simon Singh.

On 24 October 1994, Wiles submitted two manuscripts, "Modular elliptic curves and Fermat's Last Theorem" and "Ring theoretic properties of certain Hecke algebras", the second of which was co-authored with Taylor and proved that certain conditions were met that were needed to justify the corrected step in the main paper. The two papers were vetted and published as the entirety of the May 1995 issue of the Annals of Mathematics. The proof's method of identification of a deformation ring with a Hecke algebra (now referred to as an R=T theorem) to prove modularity lifting theorems has been an influential development in algebraic number theory.

Condensed: the full section is in Wikipedia.

The full Taniyama-Shimura-Weil conjecture was finally proved by Diamond (1996), Conrad et al. (1999), and Breuil et al. (2001) who, building on Wiles's work, incrementally chipped away at the remaining cases until the full result was proved. The now fully proved conjecture became known as the modularity theorem.

Several other theorems in number theory similar to Fermat's Last Theorem also follow from the same reasoning, using the modularity theorem. For example: no cube can be written as a sum of two coprime nth powers, n ≥ 3. (The case n = 3 was already known by Euler.)

Fermat's Last Theorem considers solutions to the Fermat equation: \[a^n + b^n = c^n\] with positive integers a, b, and c and an integer n greater than 2. There are several generalizations of the Fermat equation to more general equations that allow the exponent n to be a negative integer or rational, or to consider three different exponents.

The generalized Fermat equation generalizes the statement of Fermat's last theorem by considering positive integer solutions a, b, c, m, n, and k satisfying \[a^m + b^n = c^k.\] In particular, the exponents m, n, and k need not be equal, whereas Fermat's last theorem considers the case m = n = k.

The Beal conjecture states that there are no solutions to the generalized Fermat equation in positive integers a, b, c, m, n, k with a, b, and c being pairwise coprime and all of m, n, k being greater than 2.

The Fermat-Catalan conjecture generalizes Fermat's last theorem with the ideas of the Catalan conjecture. The conjecture states that the generalized Fermat equation has only finitely many solutions (a, b, c, m, n, k) with distinct triplets of values (a, b, c), where a, b, c are positive coprime integers and m, n, k are positive integers satisfying \[\frac{1}{m} + \frac{1}{n} + \frac{1}{k} < 1.\] The statement is about the finiteness of the set of solutions because there are 10 known solutions.

When we allow the exponent n to be the reciprocal of an integer; that is, n = 1/m for some integer m, we have the inverse Fermat equation \[a^\frac1m + b^\frac1m = c^\frac1m.\] All solutions of this equation were computed by Hendrik Lenstra in 1992. In the case in which the mth roots are required to be real and positive, all solutions are given by \[\begin{aligned} a &= rs^m \\ b &= rt^m \\ c &= r(s+t)^m \end{aligned}\] for positive integers r, s, t with s and t coprime.

For the Diophantine equation \[a^\frac{n}{m} + b^\frac{n}{m} = c^\frac{n}{m}\] with n not equal to 1, Bennett, Glass, and Székely proved in 2004 for n > 2, that if n and m are coprime, then there are integer solutions if and only if 6 divides m, and a, b, and c are different complex 6th roots of the same real number.

The abc conjecture roughly states that if three positive integers a, b and c (hence the name) are coprime and satisfy a + b = c, then the radical d of abc is usually not much smaller than c. In particular, the abc conjecture in its most standard formulation implies Fermat's last theorem for n that are sufficiently large. The modified Szpiro conjecture is equivalent to the abc conjecture and therefore has the same implication. An effective version of the abc conjecture, or an effective version of the modified Szpiro conjecture, implies Fermat's Last Theorem outright.