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Three Quadratic Recurrences Modulo 3: What We Can Prove and What We Can Guess
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 linear, and we can compute its periods exactly. One is a 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.
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BibTeX
@misc{finch2026three,
title = {{Three Quadratic Recurrences Modulo 3: What We Can Prove and What We Can Guess}},
author = {Steven Finch},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.34300},
eprint = {2610.34300},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.34300},
}
RIS
TY - UNPB
TI - Three Quadratic Recurrences Modulo 3: What We Can Prove and What We Can Guess
AU - Steven Finch
PY - 2026
DA - 2026/10/25
PB - arXiv
DO - 10.48550/arxiv.2610.34300
UR - https://arxiv.org/abs/2610.34300
AB - We study three innocent-looking recurrences modulo $3$, each built from squares of two earlier terms. One of them turns out to be secretly linear, and we can compute its periods exactly. One is a 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.
ER -