What is a Diophantine equation, and why should we care about them?

Question

As the question title suggests, what is a Diophantine equation, and why should a high schooler learning about elementary number theory care about them?

Answer 1

A Diophantine equation is an equation, often in several unknowns, where the value of the unknowns are supposed to be integers.

One (modern) reason to care about them is that they're one of the simplest examples of a class of problems where we can prove that there is no possible general procedure for solving them. More precisely, Matiyasevich (1970) proved that there is no algorithm that takes a Diophantine equation as input and determines whether or not it even has a solution.

This is interesting in itself (if you're into that kind of thing), and it is also useful for recognizing futile tasks. If you want to find an algorithm for solving such-and-such problem, you may be able to recognize, "hey, if I could solve every instance of problem X, then I could use that to solve arbitrary Diophantine equations" -- and then you can stop wasting time on finding such a solution procedure and instead try to come up with a more restricted version of your original problem, or otherwise lower your expectations.

Answer 2

Linear Diophantine equation takes the form of $ax+by=c$ where $a,b,c$ are given integers. The main problem is how to find solutions $x,y$. Existence and number of solutions are connected with greatest common divisor. You should look up Euclidean algorithm and Extended Euclidean algorithm(everything you'll ever need about those topics is already written). For better understanding you should be familiar with Bézout's identity.

For me very interesting application is in chinese remainder theorem and modular arithmetic-foundation of cryptography. Cryptography is probably the most wanted profession today(everybody wants to have their files safe), so this is one very cool application! Recently I came across Hasse principle and to quote wikipedia:

Hasse principle is the idea that one can find an integer solution to an equation by using the Chinese remainder theorem to piece together solutions modulo powers of each different prime number. This is handled by examining the equation in the completions of the rational numbers: the real numbers and the p-adic numbers. A more formal version of the Hasse principle states that certain types of equations have a rational solution if and only if they have a solution in the real numbers and in the p-adic numbers for each prime p.

The greatest benifit you will get is fun of exploring divisibilty rules and all kinds of funny little details about numbers you'll come across!