Don’t Worry, because we’ll break it down for you in the most casual way possible (because who wants to read math articles with all those fancy words anyway?).
To set the stage: what are elliptic curves and rational points? Well, let’s start with elliptic curves. These are basically graphs that look like a squished oval or an egg-shaped thingy. They have some pretty cool properties for example, they can be used to solve equations in math (which is why mathematicians love them so much).
Now, rational points on elliptic curves. These are the points that you get when you divide an integer by another integer and then take the square root of both sides. Yep, it sounds complicated but trust us, it’s not as bad as it seems! In fact, finding rational points is kind of like solving a puzzle or playing a game (which is why mathematicians love them so much).
So how do you find these elusive rational points? Well, there are actually two ways to go about it: the first involves using some fancy math techniques that we won’t get into here. But if you want to try your hand at finding rational points on elliptic curves (and who doesn’t?!), then we recommend checking out this article by Joseph H. Silverman and John Tate from 1992: “Rational Points on Elliptic Curves.” It’s a bit of a read, but trust us it’s worth it!
But if you don’t have time for all that fancy math stuff (and let’s face it, who does?), then we recommend checking out this article by Lawrence Washington from 2003: “Elliptic Curves: Number Theory and Cryptography.” It’s a bit more casual than the Silverman/Tate article, but still packed with all sorts of interesting information about elliptic curves and rational points.
We hope this has been helpful for those of you who are new to math or just looking for a more casual way to learn about these topics. And if you’re feeling adventurous, why not try your hand at finding some rational points on an elliptic curve yourself? Who knows you might just discover something new and exciting!