Fermat’s Last Theorem is one of the most celebrated problems in mathematics. It is a problem in the field of Number Theory that’s very simple to understand, yet extremely difficult to solve. In fact, it’s so difficult that it remained unsolved for 358 years even though it got a lot of attention from some of the best mathematicians in history. Pierre de Fermat first came up with it in 1637. It remained a conjecture because he didn’t provide any proof. Let’s go ahead and explore this labyrinth, shall we?
Why is it important?
For one, it was in the Guinness Book of World Records for being the world’s most difficult problem. After Fermat proposed the conjecture, it became quite intriguing to people around the world. Over the next three centuries, some of the brightest minds in mathematics became enamored with it. They developed amazingly powerful mathematical tools to address this problem. This, in itself, was enormously useful because those tools are used extensively across many fields today like cryptography, engineering design, material science, and so on.
In order to solve this problem, mathematicians made groundbreaking developments in algebraic number theory, algebraic geometry, ring theory, elliptic curves and modular forms. Elliptic curves are very important in the field of modern cryptography. Modular forms are used a lot in theoretical physics, especially string theory. The related technique called modularity lifting was developed and it is used extensively in category theory, which is the basis for functional programming languages like Scala and Haskell.
What exactly is the theorem?
It’s actually pretty straightforward. Let’s consider the concept of squares. The square of a number is a number multiplied by itself. For example, the square of 5 is 25 and the square of 9 is 81. Now if you sum up the squares, it gives you 106 (25 + 81). This is not very interesting, and you’ll know why that’s so soon.
Let’s consider another pair of numbers, say 6 and 8. The square of 6 is 36 and the square of 8 is 64. If you add them up, we get 100 (36 + 64). Now 100 is actually the square of 10. This is interesting because the sum of two squares resulted in another square. You can represent it as:
62 + 82 = 102
Such triplets (6, 8, 10) are called Pythagorean triplets. In fact, any three positive integers a, b, and c are called Pythagorean triplets if they satisfy the following equation:
a2 + b2 = c2
There are infinite number of such triplets and it’s been well established. Now Fermat started thinking about powers greater than 2. Do we have three positive integers a, b, and c that can satisfy the above equation for powers greater than 2? His initial instinct said that it’s unlikely. After thinking about it further, he realized that it’s so unlikely that he proposed that it’s impossible. He said that there are no positive integers a, b, and c that satisfy the following equation:
an + bn = cn
where n is greater than 2
He didn’t provide any mathematical proof of the above statement. People tried really hard to disprove it, but they couldn’t do that either. So it became a conjecture, and a rather famous one at that.
How was the problem solved?
The British mathematician, Andrew Wiles, published the proof in 1995. The first major breakthrough happened in 1955 when Japanese mathematicians, Goro Shimura and Yutaka Taniyama, found a link between elliptic curves and modular forms.
Elliptic curves and modular forms are two completely distinct branches of mathematics. Elliptic curves are not ellipses. They are actually really important in the field of number theory. I have discussed the concept in this blog post. Modular forms are analytic functions are have special properties that prove to be very useful in number theory. We will not be able to go into the details here, but it’s sufficient to know that modular forms lend themselves nicely to analysis.
When they proposed the link between elliptic curves and modular forms, they called it Taniyama–Shimura-Weil conjecture because it wasn’t proved yet. When it was proved later, it became known as the modularity theorem. The theorem says that elliptic curves over the field of rational numbers are related to modular forms. This theorem proved to be very useful starting step towards solving this colossal problem.
We are almost there . . . .
The next big step is the epsilon conjecture. It was proposed by the French mathematician and Fields Medal winner, Jean-Pierre Serre. The conjecture says that Frey’s elliptic curve (an elliptic curve with certain special properties) is not modular. This conjecture was proved by the American mathematician, Ken Ribet. It’s now known as Ribet’s theorem. It basically states that if the Galois representation associated with an elliptic curve has certain properties, then that curve cannot be modular.
Now all that’s left to do is connect the dots. There are two parts that are essential in proving Fermat’s Last Theorem — epsilon conjecture needs to be true AND the modularity theorem needs to hold for semi-stable elliptic curves. Ken Ribet proved the epsilon conjecture and Taniyama–Shimura-Weil conjecture gave rise to the modularity theorem. The last important piece was to prove the modularity theorem for semi-stable elliptic curves. This is what Andrew Wiles did. He proved the last part, which provided a concrete mathematical proof of Fermat’s Last Theorem. This seemingly simple problem required a lengthy and prodigious proof, but it was finally conquered after 358 years.