Three simple sequences reveal unexpected order and chaos modulo 3
Three Quadratic Recurrences Modulo 3: What We Can Prove and What We Can Guess
Discrete Mathematics
Summary
This paper looks at three simple math sequences where each number depends on the squares of earlier numbers, but with all results simplified by dividing by 3 and keeping remainders. The authors find that one sequence acts like a simple repeating pattern, another mixes order and randomness in a surprising way, and the third seems mostly random but still shows some predictable parts. Along the way, they introduce some open math puzzles that anyone with basic algebra knowledge can understand. The work is an early exploration and doesn’t close off any questions.
What this means in practice
- •For cryptographers: Assess and design number sequences that mix order and randomness for cryptographic algorithms using new characterizations of their structure mod 3.
- •For algorithm designers: Use exact period computations of specific recurrences modulo 3 to optimize algorithms requiring predictable cyclic behaviors.
A theory result. No direct application yet.
Authors
Steven Finch
Abstract
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.