Ooh! Primality tests! I have discovered a primaliy test myself (although quite by accident) with the use of polygons.
First, create any n-gon you’d like. Start at a point, and draw a line from that point to the next point closest to it. Then, do that again in the same direction, but instead of going to the next point, go to the next one, 2 points over. Then 3, then 4, and 5. This eventually creates a pattern.
(Sorry if this seemed confusing, I’ll add a picture below and a better explanation)
• (e) • (b)
• (d) • (c)
So you draw a line from A to B, B to D, D to B, B to A, A to A, and it repeats itself
So as I draw lines from point to point, it increases by increments of 1. For any n-gon where “n” is divisible by 2, the line pattern we’ve made is rotationally symmetrical (180 to 180).
For any n-gon where “n” is prime, the line pattern we made is not symmetrical whatsoever (Now you might be saying, "Ah, but what happens when you have an n-gin where ‘n’ is odd? I have a proof for this, too)
For any n-gon where “n” is odd, the line pattern we made is not symmetrical, but some points that the line travels through get reused, whereas prime n-gons do not. Here’s a picture to help understanding:
I think this has to do with mods, but I haven’t learned them yet. So I have no idea. But I think this is extremely cool, but probably not the best for finding vastly huge prime numbers. I’ll try to figure out a way to represent it (I’ll probably have to do it in different bases, I’m not sure yet) without having to draw out each polygon
Anyways, I’m really excited on this subject