The Lauren Identity
the most kick-ass identity ever
"The Dice Project"
The funnest math of all. This was my senior thesis in math.
I didn't learn until about the sixth week of the quarter that
our work wasn't supposed to be original. This was a lot of fun to
work on. It can be understood almost entirely visually.
The thesis question was inspired by the intricacies of a game that will remain nameless.
Primitive roots
I saw a passing comment in M.R. Schroeder's "Number Theory in Science
and Communication, 2nd ed." that said that any prime, power of any
prime, 2 times any of these, and 4 have primitive roots (f.y.i. prime
≠2). Moreover, that no composite number has primitive roots.
The audacity!
I couldn't find a proof, so here's mine.
This has been done before, by famous dead people. But here I am, doing it better.
Truth be told I have no idea how those other people did it. Maybe their proofs were shorter?
Summation of powers
In high school my math teacher, Joe Love, told us that:
I wondered if there were a different polynomial for each power, and how to find it.
When putting together The Dice Project I learned that there was a recursive method using Bernoulli polynomials.
So this here is my (non-recursive) closed solution for any power.
