I signed up to present at the Number Theory Learning Seminar at Claremont Graduate University and chose the Fifteen-Theorem to dissect and explain. Here I have attached the paper, my notes from going through each paragraph or so, and my final write up that I will use for my presentation.
The Fifteen-Theorem proves that if we have some quadratic form like x²+y², then it can equal all positive integers for given inputs x and y if and only if it can equal the positive integers 1 to 15 for given inputs x and y.
