What was Fermat’s “Marvelous" Proof?

You know the tale about Fermat's Last Theorem.

In 1637, Pierre de Fermat claimed to have the proof of his famous conjecture.

But as the story goes, it was too large to write in the margin of his book.

Yet even after Andrew Wiles proof more than 300 years later, we are still left wondering, what proof did Fermat have in mind?

The mystery surrounding Fermat's last theorem may have to do with the way we understand prime numbers.

You all know what prime numbers are.

An integer greater than one is called prime if it has exactly two factors, one and itself.

In other words, p is prime if whenever you write p as a product of two integers, then one of those integers turns out to be 1.

In fact, this definition works for negative integers, too.

We simply incorporate negative 1.

But the prime numbers satisfy another definition that maybe you haven't thought about.

An integer p is prime if whenever p divides a product of two integers, then p divides exactly one of those two integers.

Let's call this Definition B.

And let's think about it.

Does it sound plausible?

Here's an example.

Suppose our prime is 3.

And notice that 3 divides 12, for instance.

Now, look at the different ways 12 can be factored as a product of two numbers.

What do you see?

No matter how we write 12, 3 always divides one of the two factors.

You may think that's a silly observation, but it does not hold for composite or non-prime numbers.

For example, 4 also divides 12.

But 4 does not divide 2, nor does it divide six.

And the idea is that this observation will hold for all multiples of 3.

For example, 3 also divides 30.

And no matter how you write 30 as a product of two numbers, 3 will always divide one of the factors.

Now, 6 also divides 30.

But it does not have this property.

In particular, 6 is not prime.

So this new definition of prime is perfectly valid, even though it's not the one that we're so used to.

So you might wonder, why don't we ever hear about Definition B?

Is it because these two definitions are actually conveying the same concept?

In other words, is every integer that's prime in the sense of A also prime in the sense of B?

And conversely, is every integer that's prime in the sense of B also prime in the sense of A?

It turns out the answer is no, not always.

That is, the answer is yes if you're working with the integers.

In fact, I encourage you to get out pen and paper, pause the video and prove that an integer satisfies Definition A if and only if it satisfies Definition B.

However-- and here's where it gets interesting-- if we replace the integers by a different number system, a system where we can still add and multiply and factor things just like we do with integers but where those things aren't necessarily integers, then it is not always true that these two definitions coincide.

To see why, let's look at an example.

Let's replace the integers by a different number system.

What exactly?

Well, in Gabe's episode "Beyond the Golden Ratio," he explained how phi, the golden ratio, is just one of a family of metallic means.

But the golden ratio also lives in a different family.

Phi is the number 1/2 plus 1/2 times the square root of 5.

But what about other numbers of the form a fraction plus a fraction times the square root of 5?

There are infinitely many numbers of this form, and the golden ratio is just one of them.

The set of these numbers form what's called a quadratic field, which plays an important role in algebraic number theory.

But for the rest of the episode, let's just focus on the case when A and B are integers.

Collectively, we'll denote these numbers by z adjoin square root of 5.

The nice thing is that we can add and multiply these numbers together.

For example, to add 1 plus 2 root 5 and negative 4 plus 3 root 5, just add the integer parts and the square root parts together.

So their sum is negative 3 plus 5 root 5.

And we can also multiply them together.

We'll just use the familiar distributive law, which some folks like to call FOIL.

So their product is 26 minus 5 root 5.

Moreover, Z adjoined root 5 also has prime numbers given by Definitions A and B.

But because we replaced the integers with Z adjoined root 5, we need to modify the definition a little.

The reason is that Z adjoined root 5 may contain numbers that behave like the number 1, even though they aren't the number 1.

I'll explain.

Here's the new Definition A.

A number p and Z adjoined root 5 is prime if whenever you write p as a product of two numbers then one of them is a unit.

A unit is a word that means "has a multiplicative inverse."

That is, a number u is a unit if there exists some other number v so that u times v is 1.

For example, in the usual integers 3 is not a unit.

It does not have a multiplicative inverse.

OK, yes, 3 times 1/3 is equal to 1.

But 1/3 is not an integer, so that doesn't count.

In fact, the only units in Z are 1 and negative 1.

And that's why "unit" is a good generalization of the number 1.

OK, so we have two definitions, A and B.

If we work with the integers, then these two definitions coincide.

But now I claim that because we are working in Z adjoined root 5, they do not coincide.

In particular, the number 2 is prime by Definition A but not prime by Definition B.

First, let's see why 2 is not prime according to Definition B.

Notice that 4 can be written as 2 times 2, but it can also be written as 1 plus root 5 times negative 1 plus root 5.

This means that 2 divides the product.

But 2 does not divide either factor, 1 plus root 5 or negative 1 plus root 5.

In other words-- and you can verify-- there are no integers a and b so that 1 plus root 5 equals 2 times a plus b root 5.

Similarly if you replace 1 by negative 1.

This shows that 2 is not prime according to Definition B.

However, it is prime by Definition A.

Why?

I'll let you work that one out.

That's a little trickier, but not too much.

I recommend using a proof by contradiction along with something called a norm.

I won't go into the computations now, but if you're interested, check out the references below.

All right, let's summarize.

We have two definitions, A and B.

When working with the integers, these definitions imply each other.

But in Z adjoined root 5, they do not.

Why?

The reason is because the integers possess a very special property that z adjoined root 5 does not have.

Before I tell you what that property is, let me just say that this overall discussion is a part of something called ring theory, the study of rings.

But not this kind of ring.

A ring is a mathematical object, a set of elements that behave a lot like integers even though they may not be integers.

And Z adjoined root 5 is one such example.

The neat thing is that once you have a ring, you have enough mathematical structure to talk about primality.

In particular, our two definitions, A and B, have technical names in ring theory.

An element and a ring is called irreducible if it satisfies Definition A, and it's called prime if it satisfies Definition B.

So earlier, we saw that 2 is irreducible in Z adjoined root 5, but it is not prime.

Now, here's the punchline.

Primality and irreducibility will coincide if and only if your ring has a very special property.

And the integers have that property.

What is it?

The fundamental theorem of arithmetic-- namely, that every integer has a unique factorization into a product of primes.

More generally, if you're looking for a buzzword the integers form a unique factorization domain, or UFD.

And according to abstract ring theory, irreducible and prime are equivalent concepts if and only if your ring is a UFD-- specifically, if each element can be uniquely written as a product of irreducible elements.

What's interesting is that not all rings are UFDs.

And this brings us back to Fermat's last theorem.

The absence of unique factorization is precisely why one of the many attempts to prove Fermat's Last Theorem wasn't successful.

In 1847, French mathematician Gabriel Lame thought he had proof Fermat's conjecture by factoring an expression like this, which occurred in the ring Z adjoined alpha, which is not a unique factorization domain.

And so his technique didn't work.

Fortunately, having a faulty proof isn't always a bad thing.

In fact, the lack of unique factorization was spotted a few years earlier in a different setting by German mathematician Eduard Kummer, who introduced what he called ideal numbers, precisely to get around the issue.

In short, the discovery that not all number systems, or rings, have an analog to the fundamental theorem of arithmetic set the stage for more than a century's worth of brand new mathematics, which then led to Andrew Wiles' proof of Fermat's Last Theorem in 1993.

So what proof did Fermat actually have in mind when he wrote in his margin?

Well, I'm not a historian, but it's very possible that he assumed that properties of the integers, like unique factorization and the equivalence between prime and irreducible, will always hold, just like Lame thought.

But as we saw today, things aren't always what they seem.