Complex Extensions, Fermat's Last Theorem

Fermat's Last Theorem

In the margin of one of his notebooks, Fermat (biography) asserted that xⁿ+yⁿ = zⁿ has no integer solutions for n > 2, and x, y, and z nonzero. He claimed to have a beautiful proof, but the margin was too small to contain it. After his death, the most brilliant mathematical minds attacked the problem for 350 years, culminating in a proof that involves modular forms and eliptic curves, modern concepts that Fermat could have had no knowledge of.

Most people believe Fermat did not have a valid proof, but we will never know for sure. (Limerick)

PBS has put together a marvelous explanation of Fermat's last theorem, and the towering proof that puts this question to rest once and for all. Click here for the nova special.

Small Exponents and Coprime Solutions

When n = 2, we find the pythagorean triples, so named because x² + y² = z² follows the pythagorean theorem, which describes the lengths of the sides of a right triangle. As you recall from an earlier page, the integer solutions have been completely characterized. But what about n > 2?

When n is even, the variables can be negated without changing anything, so we assume they are positive. For n odd, we could negate y, but that just moves it to the other side of the equation.

xⁿ = yⁿ + zⁿ

Make all variables positive, placing at least two of the three terms on one side of the equation. If all three terms are on one side, and positive, there is no solution. Thus there are two terms on one side and one on the other, hence xⁿ + yⁿ = zⁿ. We can assume all variables are positive.

If there is a solution with n = 15, then there is a solution with n = 3. Replace x with x⁵, y with y⁵, and z with z⁵. So, as we investigate this problem, we can assume n is prime, or 4. The latter allows for the fact that there are indeed solutions when n = 2.

Once n is fixed, the smallest solution has coprime variables. If p divides x and y it divides z, and we can divide through by pⁿ. The same thing happens if x and z are divisible by p, and so on, so we can assume the three variables are pairwise coprime. This was an important step in characterizing the pythagorean triples. Once we found 3,4,5 there was no need to describe 6,8,10.

Mordell's Conjecture

Mordell (biography) proposed a related conjecture that states there are finitely many rational solutions for the polynomial expression p(x,y) = 0 when the genus of p is at least 2. As a rough approximation, this means p has degree 3 or higher. Note that Fermat's last theorem actually describes rational solutions to xⁿ + yⁿ = 1. Mordell's conjecture was proved by Faltings in 1983, and is sometimes called Falting's theorem in honor of the accomplishment. Hence there are finitely many fermat triples for any n > 2. Of course we know there are no such triples, since Fermat's last theorem has also been proved.

Note that some cubic polynomials describe curves of genus 1, such as x³-y²+22, and these equations can have infinitely many rational solutions.

Euler's Conjecture

Intreagued by Fermat's last theorem, Euler proposed a variant which states that the sum of n-1 n^th powers cannot equal another n^th power. When n = 3, we have the sum of two cubes equal to another cube. This can't be, because it contradicts Fermat's last theorem. What about larger values of n?

Now that we have computers, we believe the conjecture fails for all n beyond 3. Here are counter examples for 4 and 5.

95800⁴ + 217519⁴ + 414560⁴ = 422481⁴

27⁵ + 84⁵ + 110⁵ + 133⁵ = 144⁵

Different Exponents

A more recent conjecture allows the exponents to vary. The exponents must exceed 2, and the three integers must be coprime.

x^m + yⁿ = z^p

If we drop the coprime requirement there are lots of solutions.

9³ + 18³ = 9⁴

This is 27 times (1³ + 2³ = 3²). In other words, it is a scaled version of a coprime solution where one of the exponents is 2.

Are there any coprime solutions with exponents > 2? Is this conjecture true or false? Nobody knows.

Exploration

The following pages explore these questions for low exponents. We will prove Fermat's last theorem for n = 3 and 4, along with related equations such as x⁴ + y⁴ = z³. If you're an expert in algebraicc number theory, you probably view this as a waste of time. Powerful theorems push these cases aside like a bulldozer. Meantime, I'm trying to accomplish the same thing with hand tools, because that's all we have. Well - if you know how to drive the bulldozer, you're probably not reading these pages anyways, so let's proceed, using little more than unique factorization in the gaussian integers and basic number theory. That's what Fermat did, and who knows, maybe he stumbled upon a truly elegant proof for his last theorem. Maybe you will too.

First, a couple useful lemmas - the coprime lemma and the xyn lemma.