One of the central goals of number theory is to find integer solutions to polynomial equations -- this is called the study of Diophantine equations.
This might seem a strange goal, so let's take a step back and ask what the point of mathematics is. The aim of mathematics is to look for hidden structures in mathematical objects. I envision this as similar to the role of an author: the writer tries to tell a particular story in a way which reveals a more general emotional idea; the mathematician tries to solve a particular problem in a way which reveals a more general mathematical idea.
Historically, the theory of Diophantine equations has led to the discovery of many hidden structures in the integers. The articles on this website aim to show how a particular class of Diophantine equations led to discovery (by Langlands, and many others) of some of the deepest, and more profound, structures ever observed by humans. But before getting to the Langlands program, let's start with some simpler examples.
Divisibility
The simplest Diophantine equations are those of the form \(Ax = B\). For example, suppose I ask you to find integer solutions to the equation $$5x = 10,$$ or to $$ 2x = 13. $$
The first equation has an integer solution (\(x=2\)), whereas the second equation has no integer solution: indeed, \(2x\) is always even, but 13 is odd.
As another example, the equation \(3x = 14\) also has no integer solutions; this is because \(14\) divided by \(3\) leaves a remainder of 2: no matter how you try, if you have 14 objects and want to put them into groups of 3, you will always have two objects left over.
Studying equations \(Ax = B\) leads one directly to the ideas of divisibility and remainder.
Modular artihmetic
A systematic way of managing divisibility is modular arithmetic. From some point of view, modular arithmetic is just a type of notation; but it is a very useful notation.
In mod 3 arithmetic, for example, we will consider two numbers to be equal if their difference is divisible by 3. For example, \[7 \equiv 4 \pmod{3},\] because \(7 - 4 = 3\) is divisible by 3; and \[14 \equiv 2 \pmod{3},\] because \(14 - 2 = 12\) is divisible by 3.
You should think of \(\equiv\) as being a fancy sort of equals sign; in mod 3 arithmetic, 5 and 2 are treated as equal. And of course, there are also other types of modular arithmetic: one can work modulo any integer! Here's a sampling of some true equations in modular arithmetic: \[6 \equiv 2 \pmod{4},\] \[3 \equiv -3 \pmod{6},\] \[27 \equiv 0 \pmod{9}.\]
Unique Prime Factorization
After studying Diophantine equations of the form \(Ax = B,\) it is natural to wonder about equations like \(Ax + By = C.\) For example, what integer solutions are there to \[4x - 3y = 1,\] or to \[15x - 18y = 2?\]
The equation \(Ax + By = C\) actually dates all the way back to the Greek geometer Euclid, who devised the Euclidean algorithm: a technique for finding all solutions to \(Ax + By = C.\)
This Diophantine equation might seem a little strange, but it is actually one of the most important in history, for a very simple reason: inventing the Euclidean algorithm is essentially equivalent to inventing unique prime factorization. The integers have unique prime factorizations: every whole number can be written in exactly one way as a product of prime numbers (like \(4725 = 3 \times 3 \times 3 \times 5 \times 5 \times 7 = 3^3 \times 5^2 \times 7\)). Unique prime factorization is a great hidden structure the integers possess, and somehow it is equivalent to knowing how to solve certain Diophantine equations! The link between these two topics is not immediate, but it does become inevitable when one starts thinking more carefully about prime factorization.
The Chinese Remainder Theorem
Unique prime factorization has a consequence in modular arithmetic. The equation \[920 \equiv 2 \pmod{54}\] means that \(920 - 2\) is divisible by \(54.\) As \(54 = 2 \cdot 3^3,\) unique prime factorization tells us that being divisible by \(54\) is equivalent to being divisible by 2, and to being divisible by \(3^3.\)
In other words, the single modular arithmetic equation \[920 \equiv 2 \pmod{54}\] is equivalent to the two equations \[920 \equiv 2 \pmod{2},\] \[920 \equiv 2 \pmod{3^3}.\]
This is an important general principle: any modular arithmetic equation can be written as a system of modular arithmetic equations, but where each equation in the system works modulo the power of a prime.
This observation, called the Chinese remainder theorem might seem a little strange at first, but it is actually very helpful in practice (as we will illustrate by using it in later articles!). The key point is that usually prime power modular arithmetic is easier to handle, and so it is better to study an equation `one prime power at a time,' instead of studying it with all the prime powers combined together at once.
The Langlands program
We saw two examples above of Diophantine equations leading to the discovery of some hidden structure in the integers (divisibility and unique prime factorization).
The purpose of this article was, secretly, to tell the reader about another class of Diophantine equations which leads to the Langlands program, which studies from incredibly intricate hidden structure inside of number theory. The Langlands program studies Diophantine equations of the form \[f(x) = Ny,\] where \(f(x)\) is an integer polynomial. For example, the Langlands program studies equations like \[x^3 - 17x^2 + 5x + 12 = 82y,\] or \[x^2 + 1 = 5y.\]
Just like the equations we saw earlier in this article, these Diophantine equations \(f(x) = Ny\) led to the discovery of great hidden structures in the integers. We hope you'll join us in learning more about these hidden structures!