Three Quadratic Recurrences Modulo 3: What We Can Prove and What We Can Guess

Overview & Synopsis

We study three innocent-looking recurrences modulo $3$, each built from squares of two earlier terms. One of them turns out to be secretly $\textit{linear}$, and we can compute its periods exactly. One is a $\textit{permutation}$ hiding a small, perfectly orderly piece inside a large, random-looking one. The third looks thoroughly random, but it still has a few provable features. Along the way we meet open problems that need nothing beyond a first course in abstract algebra to state. This paper is a starting point, not the final word in any sense.