Fermat's Last Theorem

Fermat's Last Theorem says that the equation aⁿ + bⁿ = cⁿ has no solutions in positive whole numbers when n is greater than 2. For n = 2 there are infinitely many solutions, the Pythagorean triples such as 3-4-5. Pierre de Fermat claimed the theorem around 1637; Andrew Wiles proved it in 1994, and the proof was published in 1995.

It is simple enough for a 10-year-old to understand (Wiles first read about it at that age), yet it resisted the best mathematicians for three and a half centuries.

From Pythagorean triples to aⁿ + bⁿ = cⁿ

A Pythagorean triple is a solution of a² + b² = c² in whole numbers. There are infinitely many, and Euclid's formula produces all of them: 3² + 4² = 5², 5² + 12² = 13², 8² + 15² = 17², and so on forever.

The natural next step is to try cubes, looking for whole numbers with a³ + b³ = c³. You can search for a long time and find only near misses. The closest small one is

Another is 9³ + 10³ = 1729 = 12³ + 1. Fermat's Last Theorem says the misses never become hits, for cubes or any higher power:

The word "two" matters. With three cubes it can work: 3³ + 4³ + 5³ = 6³, a cousin of the 3-4-5 triangle. Adding a term is also how Pythagorean quadruples such as 1² + 2² + 2² = 3² arise.

Fermat's margin note (c. 1637)

Pierre de Fermat (died 1665) was a lawyer and magistrate in Toulouse, France, who did mathematics in his spare time and became one of the founders of number theory. He studied a Latin edition of the Arithmetica of Diophantus, a Greek mathematician of Alexandria, published by Claude-Gaspard Bachet in 1621, and wrote comments in its margins. Next to Problem 8 of Book II, which asks how to split a given square into two squares, he wrote, probably around 1637:

Cubum autem in duos cubos, aut quadratoquadratum in duos quadratoquadratos, 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.

"It is impossible to separate a cube into two cubes, or a fourth power into two fourth powers, or in general any power higher than the second into two like powers. I have discovered a truly marvelous proof of this, which this margin is too narrow to contain."

Fermat's own copy of the book is lost. We know the note because his son, Clément-Samuel de Fermat, published a new edition of the Arithmetica in 1670 that included his father's comments. The claim became known as Fermat's "last" theorem because it was the last of his stated results to be either proved or disproved.

Fermat almost certainly did not have a correct proof. None was found among his papers, and the proof that finally worked uses 20th-century mathematics. Most historians think he was mistaken, probably about a general argument with a hidden flaw, as so many later attempts had. He did have a complete proof for one exponent.

Fermat's proof for fourth powers

For n = 4, Fermat's method of infinite descent works, and it is built entirely out of Pythagorean triples. The idea: show that any solution would produce a smaller solution. Then that one would produce a smaller one still, and so on forever. But positive whole numbers cannot keep getting smaller forever, so there was never a solution to begin with. (The same style of argument proves that the square root of 2 is irrational.)

It is actually easier to prove something slightly stronger: x⁴ + y⁴ = z² has no solutions in positive whole numbers. That settles n = 4, because a solution of a⁴ + b⁴ = c⁴ would give one of the stronger equation with z = c². Here is the descent, step by step.

1. Take the smallest solution. Suppose solutions exist and pick the one with the smallest z. Then x and y share no common factor, or we could divide it out and get a smaller solution.
2. Spot a triple. Since (x²)² + (y²)² = z², the numbers x², y², z form a primitive Pythagorean triple. Say x is odd and y is even. Euclid's formula gives coprime m > n, one odd and one even, with x² = m² − n², y² = 2mn and z = m² + n².
3. Spot a second triple. Rearranged, x² + n² = m², so x, n, m form another primitive triple, and n is its even leg. Euclid's formula again gives coprime r and s with n = 2rs and m = r² + s².
4. Find squares. Now y² = 2mn = 4rs(r² + s²). The three numbers r, s and r² + s² share no common factors, and their product is the perfect square (y/2)², so each of them must itself be a perfect square: r = u², s = v², r² + s² = w².
5. Descend. Substituting gives u⁴ + v⁴ = w², a brand-new solution. But w ≤ w² = m < m² + n² = z, so it is smaller than the smallest solution. That is impossible, so no solution exists.

Fermat wrote out this kind of argument to show that the area of a right triangle with whole-number sides can never be a perfect square, one of the very few complete proofs he left, and the n = 4 case follows from the same idea. It also means you only ever need to worry about prime exponents. Any n ≥ 3 is either a multiple of 4 or a multiple of an odd prime p. If n = 4k, a solution would give (a^k)⁴ + (b^k)⁴ = (c^k)⁴, which Fermat ruled out; if n = pk, it would give (a^k)^p + (b^k)^p = (c^k)^p. So it is enough to prove the theorem for n = 4 and for every odd prime.

Three and a half centuries of partial progress

Proving the theorem prime by prime was brutally hard. Each new exponent needed new ideas, and those ideas built large parts of modern number theory.

Lamé's failed proof shows why the problem was hard. He factored aⁿ + bⁿ using special complex numbers and assumed, as with ordinary whole numbers, that the factorization was unique. Kummer had already found that in these number systems it often is not, and repairing that failure led him to "ideal numbers", one of the foundations of modern algebra.

How Andrew Wiles proved it

Wiles, an English mathematician working at Princeton, spent about seven years on the problem, mostly in secret. His proof fills over a hundred pages, but its plan fits in four steps. It connects whole-number equations to two other kinds of objects:

• An elliptic curve is the set of solutions of an equation like y² = x³ + Ax + B. Despite the name it is not an ellipse; it is a curve whose points can be "added" to one another in a surprising way.
• A modular form is a function with an enormous amount of symmetry, studied for completely different reasons. An elliptic curve is called modular if it matches up with a modular form.

1. The conjecture. In the 1950s and 1960s, Yutaka Taniyama, Goro Shimura and others suggested that every elliptic curve (with whole-number coefficients) is modular. Nobody knew how to prove it.
2. The strange curve. Frey noticed that a solution a^p + b^p = c^p with p an odd prime would produce the elliptic curve y² = x(x − a^p)(x + b^p), which would have properties so unusual that it seemed it could not be modular.
3. Ribet's theorem. In 1986 Ribet proved it: the Frey curve really cannot be modular. So if the conjecture is true, the Frey curve cannot exist, and neither can the solution that produced it.
4. Wiles's theorem. Wiles proved the conjecture for a large class of elliptic curves, the "semistable" ones, which includes every Frey curve. Therefore no solution exists, for any odd prime p, and Fermat's Last Theorem is true.

The rest of the conjecture, now called the modularity theorem, was completed in 2001 by Christophe Breuil, Brian Conrad, Fred Diamond and Richard Taylor. Wiles received the Wolfskehl Prize in 1997 and the Abel Prize in 2016. A question about whole numbers was settled by proving something about curves and symmetric functions.

Why squares are special

The exponent 2 has infinitely many solutions and every larger exponent has none. Several facts show how differently squares behave.

The circle can be unwrapped by straight lines

Divide a² + b² = c² by c² and you get x² + y² = 1 with x = a/c and y = b/c. So every Pythagorean triple is a point with fraction coordinates on the unit circle: 3-4-5 gives the point (3/5, 4/5) and 5-12-13 gives (5/13, 12/13). Now draw a line through the point (−1, 0) with a fraction slope t. It meets the circle at exactly one more point, with coordinates

and these are fractions too. Slope t = 1/2 gives (3/5, 4/5); slope t = 2/3 gives (5/13, 12/13). Every fraction slope gives a triple, and every triple comes from one: this is Euclid's formula in disguise.

The same equation for cubes, x³ + y³ = 1, describes a different kind of curve, an elliptic curve. A straight line usually meets it in three points, not two, and the "draw a line through a known point" trick no longer generates new fraction points. For n ≥ 4 the curves xⁿ + yⁿ = 1 are more twisted still, which is the geometric heart of Faltings's theorem.

Differences of squares factor neatly

The algebra that finds triples relies on c² − b² = (c − b)(c + b): two simple factors whose product must be the square a². For cubes, c³ − b³ = (c − b)(c² + cb + b²), and that second factor is the kind that tripped up Lamé.

More terms change everything

Fermat's theorem is about exactly two terms on the left. Euler guessed that you would need at least n terms to make an nth power, but that guess is false. In 1966 Leon Lander and Thomas Parkin found four fifth powers that add up to a fifth power, and in the 1980s Noam Elkies found three fourth powers that add up to a fourth power:

Powers are full of surprises, and the case understood completely is n = 2: see the list of Pythagorean triples or the history of the Pythagorean theorem.