Diophantine geometry

Okay kiddo, imagine you have some marbles. Let's say you have 3 marbles and you want to split them equally between two friends. How many marbles will each friend get? The answer is 1 and a remainder of 1.

Diophantine geometry is kind of like that but with numbers instead of marbles. It's a fancy way of studying equations with whole number solutions. These types of equations are called diophantine equations because they were first studied by a man named Diophantus who lived a really long time ago.

Let's take a simple example of a diophantine equation: 2x + 3y = 7. We want to find whole number solutions for x and y that make this equation true. We can use some fancy math tricks to solve it, but let's keep it simple.

First, we can plug in some values for x and y and see what happens. Let's try x = 1 and y = 1. If we plug those values into the equation, we get 2(1) + 3(1) = 5 which isn't equal to 7. So that doesn't work.

Let's try x = 2 and y = 1. If we plug those values into the equation, we get 2(2) + 3(1) = 7 which is equal to 7! That means we found a solution.

But what if we wanted to find ALL the solutions? Well, that's where diophantine geometry comes in. We use something called a graph to help us find all the possible solutions.

Imagine we make a graph where the x-axis is the values of x and the y-axis is the values of y. Now, for every value of x and y, we can plot a point on the graph. The coordinates of that point will be the value of x and y.

For our example equation, 2x + 3y = 7, we can plot all the possible solutions on this graph. But how do we know which points are solutions? Remember, we want whole number solutions. That means we only want to plot points that have whole number coordinates.

So if we plot all the possible points with whole number coordinates, we get a bunch of dots on the graph. But how do we know which dots are solutions and which ones aren't? Well, we use our trick from before and plug in the values for x and y for each dot to see if they make the equation true.

If we do that for all the dots on the graph, we find that only one of them makes the equation true: the dot at x = 2 and y = 1. That dot represents the solution we found earlier.

So that's diophantine geometry. It's a fancy way of using graphs to find whole number solutions to equations. It's kind of like a treasure hunt where we have to find all the possible solutions by plotting points on a graph and checking each one to see if it works.