Why do prime numbers make these spirals? | Dirichlet’s theorem and pi approximations?

Prime numbers can make cool spirals because they follow special patterns that show up when you draw them out.

Imagine you're coloring a big grid like a coloring book, but instead of following lines, you color every number that's prime, those are numbers like 2, 3, 5, and 7 that can't be divided evenly by anything except 1 and themselves.

Now, if you start at the center of the grid and spiral outwards, coloring each prime number as you go, something magical happens, shapes appear! These spirals look like the swirls in a seashell or the way jellybeans spill out of a bag. It's not magic; it's just how numbers behave when they're prime.

It’s because prime numbers have rules that are kind of hidden. One of those rules is from something called Dirichlet’s theorem, which says that certain kinds of number sequences always include an endless supply of primes, like a never-ending bag of jellybeans, all prime!

And even though we're talking about math here, it’s connected to pi too. Pi is the number you get when you divide a circle's circumference by its diameter (like how many slices of pizza fit around the edge of a pie). Sometimes people use pi in clever ways to guess what prime numbers might look like, and that helps make those spirals appear!