Article URL: https://hidden-phenomena.com/articles/modular Comments URL: https://news.ycombinator.com/item?id=48882097 Points: 24 # Comments: 6

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. 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. 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.