One of my favorite lemmas is “Freshman’s Dream,” which states this: Given a prime \(p\) and integers \(a, b\), the identity \((a+b)^p \equiv a^p + b^p \mod{p}\) holds true. I honestly don’t see it show up too often, but the name is funny, so I’ll certainly never forget this one.... Continue reading...
That is, besides squares. I’ll prove it. Regular n-gons, for \(n \geq 5\) and \(n \neq 6\) We can pretty easily construct an infinite descent proof for regular polygons with obtuse angles. We’ll start with this: Lemma: If three vertices of a parallelogram are lattice points, then the fourth is... Continue reading...
I recently thought of creating a (very simple) 3D graphics display in Processing. So I did that, and here it is. Its current functionalities are placing points based on \((x, y, z)\) coordinates, connecting two points, and rotating around axes. The axes pass through a fixed point on the canvas,... Continue reading...